Research ArticleExplicit Determinants of the RFP119903L119903R Circulant and RLP119903F119903LCirculant Matrices Involving Some Famous Numbers
Tingting Xu12 Zhaolin Jiang1 and Ziwu Jiang1
1 Department of Mathematics Linyi University Linyi Shandong 276005 China2 School of Mathematical Sciences Shandong Normal University Jinan 250014 China
Correspondence should be addressed to Zhaolin Jiang jzh1208sinacom
Received 28 April 2014 Accepted 5 June 2014 Published 19 June 2014
Academic Editor Tongxing Li
Copyright copy 2014 Tingting Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Circulant matrices may play a crucial role in solving various differential equations In this paper the techniques used herein arebased on the inverse factorization of polynomialWe give the explicit determinants of the RFP119903L119903R circulantmatrices and RLP119903F119903Lcirculant matrices involving Fibonacci Lucas Pell and Pell-Lucas number respectively
1 Introduction
It has been found out that circulant matrices play an impor-tant role in solving differential equations in various fields suchas Lin and Yang discretized the partial integrodifferentialequation (PIDE) in pricing options with the preconditionedconjugate gradient (PCG) method where constructed thecirculant preconditioners By using the FFT the cost foreach linear system is 119874(119899 log 119899) where 119899 is the size of thesystem in [1] Lei and Sun [2] proposed the preconditionedCGNR (PCGNR) method with a circulant preconditionerto solve such Toeplitz-like systems Kloeden et al adoptedthe simplest approximation schemes for (1) in [3] with theEuler method which reads (5) in [3] They exploited thatthe covariance matrix of the increments can be embeddedin a circulant matrix The total loops can be done by fastFourier transformation which leads to a total computationalcost of 119874(119898 log119898) = 119874(119899 log 119899) By using a Strang-typeblock-circulant preconditioner Zhang et al [4] speeded upthe convergent rate of boundary-value methods In [5] theresulting dense linear system exhibits so much structure thatit can be solved very efficiently by a circulant preconditionedconjugate gradient method Ahmed et al used coupled maplattices (CML) as an alternative approach to include spatialeffects in FOS Consider the 1-system CML (10) in [6] Theyclaimed that the system is stable if all the eigenvalues ofthe circulant matrix satisfy (2) in [6] Wu and Zou in [7]
discussed the existence and approximation of solutions ofasymptotic or periodic boundary-value problems of mixedfunctional differential equationsThey focused on (513) in [7]with a circulant matrix whose principal diagonal entries arezeroes
Circulant matrix family have important applications invarious disciplines including image processing communica-tions signal processing encoding and preconditioner Theyhave been put on firm basis with the work of Davis [8] andJiang and Zhou [9] The circulant matrices long a fruitfulsubject of research have in recent years been extended inmany directions [10ndash13] The 119891(119909)-circulant matrices areanother natural extension of this well-studied class and canbe found in [14ndash20] The 119891(119909)-circulant matrix has a wideapplication especially on the generalized cyclic codes in [14]The properties and structures of the 119909119899 minus 119903119909 minus 119903-circulantmatrices which are called RFP119903L119903R circulant matrices arebetter than those of the general 119891(119909)-circulant matrices sothere are good algorithms for determinants
There are many interests in properties and generalizationof some special matrices with famous numbers Jaiswalevaluated some determinants of circulant whose elementsare the generalized Fibonacci numbers [21] Dazheng gavethe determinant of the Fibonacci-Lucas quasicyclic matrices[22] Lind presented the determinants of circulant and skewcirculant involving Fibonacci numbers in [23] Shen et al[24] discussed the determinant of circulant matrix involving
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 647030 9 pageshttpdxdoiorg1011552014647030
2 Abstract and Applied Analysis
Fibonacci and Lucas numbers Akbulak and Bozkurt [25]gave the norms of Toeplitz involving Fibonacci and Lucasnumbers The authors [26 27] discussed some propertiesof Fibonacci and Lucas matrices Stanimirovic et al gavegeneralized Fibonacci and Lucas matrix in [28] Z Zhangand Y Zhang [29] investigated the Lucas matrix and somecombinatorial identities
Firstly we introduce the definitions of theRFP119903L119903Rcircu-lant matrices and RLP119903F119903L circulant matrices and propertiesof the related famous numbers Then we present the mainresults and the detailed process
2 Definition and Lemma
Definition 1 A row first-plus-119903last 119903-right (RFP119903L119903R) circu-lant matrix with the first row (119886
0 1198861 119886
119899minus1) denoted by
RFP119903LRcirc119903fr(1198860 1198861 119886
119899minus1) means a square matrix of the
form
119860 =(
1198860
1198861
sdot sdot sdot 119886119899minus1
119903119886119899minus1
1198860+ 119903119886119899minus1
sdot sdot sdot 119886119899minus2
119903119886119899minus2
119903119886119899minus1
+ 119903119886119899minus2
sdot sdot sdot 119886119899minus3
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199031198861
1199031198862+ 1199031198861
sdot sdot sdot 1198860+ 119903119886119899minus1
) (1)
Note that the RFP119903L119903R circulant matrix is a 119909119899 minus 119903119909 minus 119903circulant matrix which is neither an extension nor specialcase of the circulant matrix [8] They are two completelydifferent kinds of special matrices
We define Θ(119903119903)
as the basic RFP119903L119903R circulant matrixthat is
Θ(119903119903)
=(
0 1 0 sdot sdot sdot 0 0
0 0 1 sdot sdot sdot 0 0
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
0 0 0 sdot sdot sdot 0 1
119903 119903 0 sdot sdot sdot 0 0
)
119899times119899
= RFP119903LRcirc119903fr (0 1 0 0)
(2)
Both the minimal polynomial and the characteristic poly-nomial of Θ
(119903119903)are 119892(119909) = 119909
119899
minus 119903119909 minus 119903 which has onlysimple roots denoted by 120576
119896(119896 = 1 2 119899) In addition
Θ(119903119903)
satisfiesΘ119895(119903119903)
= RFP119903LRcirc119903fr(0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119895
1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119899minus119895minus1
) and
Θ119899
(119903119903)= 119903119868119899+ 119903Θ(119903119903)
Then a matrix 119860 can be written in theform
119860 = 119891 (Θ(119903119903)
) =
119899minus1
sum
119894=0
119886119894Θ119894
(119903119903) (3)
if and only if 119860 is a RFP119903L119903R circulant matrix where thepolynomial 119891(119909) = sum
119899minus1
119894=0119886119894119909119894 is called the representer of the
RFP119903L119903R circulant matrix 119860Since Θ
(119903119903)is nonderogatory then 119860 is a RFM119903L119903R
circulant matrix if and only if119860 commutes withΘ(119903119903)
that is119860Θ(119903119903)
= Θ(119903119903)
119860 Because of the representation RFM119903L119903Rcirculant matrices have very nice structure and the algebraicproperties also can be easily attained Moreover the productof two RFM119903L119903R circulant matrices and the inverse 119860minus1 areagain RFM119903L119903R circulant matrices
Definition 2 A row last-plus-119903first 119903-left (RLP119903F119903L) circu-lant matrix with the first row (119886
0 1198861 119886
119899minus1) denoted by
RLP119903FLcirc119903fr(1198860 1198861 119886
119899minus1) means a square matrix of the
form
119861 =(
1198860
sdot sdot sdot 119886119899minus2
119886119899minus1
1198861
sdot sdot sdot 119886119899minus1
+ 1199031198860
1199031198860
1198862
sdot sdot sdot 1199031198860+ 1199031198861
1199031198861
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
119886119899minus1
+ 1199031198860sdot sdot sdot 119903119886
119899minus3+ 119903119886119899minus2
119903119886119899minus2
) (4)
Let 119860 = RLP119903FLcirc119903fr(1198860 1198861 119886
119899minus1) and 119861 =
RFP119903LRcirc119903fr(119886119899minus1
119886119899minus2
1198860) By explicit computation
we find
119860 = 119861119868119899 (5)
where 119868119899is the backward identity matrix of the form
119868119899=(
1
1
c1
1
) (6)
The Fibonacci Lucas Pell and the Pell-Lucas sequences[30ndash36] are defined by the following recurrence relationsrespectively
119865119899+1
= 119865119899+ 119865119899minus1
where 1198650= 0 119865
1= 1
119871119899+1
= 119871119899+ 119871119899minus1
where 1198710= 2 119871
1= 1
119875119899+1
= 2119875119899+ 119875119899minus1
where 1198750= 0 119875
1= 1
119876119899+1
= 2119876119899+ 119876119899minus1
where 1198760= 2 119876
1= 2
(7)
The first few values of these sequences are given by thefollowing table (119899 ge 0)
119899 0 1 2 3 4 5 6 7
1198651198990 1 1 2 3 5 8 13
1198711198992 1 3 4 7 11 18 29
1198751198990 1 2 5 12 29 70 169
1198761198992 2 6 14 34 82 198 478
(8)
The sequences 119865119899 119871119899 119875119899 and 119876
119899 are given by the
Binet formulae
119865119899=
120572119899
minus 120573119899
120572 minus 120573
119871119899= 120572119899
+ 120573119899
119875119899=
120572119899
1minus 120573119899
1
1205721minus 1205731
119876119899= 120572119899
1+ 120573119899
1
(9)
where 120572 120573 are the roots of the characteristic equation 1199092minus119909minus1 = 0 and 120572
1 1205731are the roots of the characteristic equation
1199092
minus 2119909 minus 1 = 0By Proposition 51 in [14] we deduce the following
lemma
Abstract and Applied Analysis 3
Lemma 3 Let 119860 = RFP119903LRcirc119903fr(1198860 119886
119899minus1) then the
eigenvalues of 119860 are
119891 (120576119896) =
119899minus1
sum
119894=0
(119886119894120576119894
119896) (10)
and in addition
det119860 =
119899
prod
119896=1
119899minus1
sum
119894=0
(119886119894120576119894
119896) (11)
where 120576119896(119896 = 1 2 119899) are the roots of the equation
119909119899
minus 119903119909 minus 119903 = 0 (12)
Lemma 4 Consider
119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886)
= 119888119899
minus 119903119888 [(119886119904)119899minus1
+ (119886119905)119899minus1
]
minus 119903 [(119886119904)119899
+ (119886119905)119899
] + 1199032
119886119899minus1
(119888 minus 119887 + 119886)
(13)
where
119904 =
minus119887 + radic1198872minus 4119886119888
2119886
119905 =
minus119887 minus radic1198872minus 4119886119888
2119886
(14)
and 120576119896(119896 = 1 2 119899) satisfy (12) 119886 119887 119888 isin R 119886 = 0
Proof Consider
119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886) = 119886
119899
119899
prod
119896=1
(1205762
119896+
119887
119886
120576119896+
119888
119886
)
= 119886119899
119899
prod
119896=1
(120576119896minus 119904) (120576
119896minus 119905)
= 119886119899
119899
prod
119896=1
(119904 minus 120576119896) (119905 minus 120576
119896)
(15)
while
119904 + 119905 = minus
119887
119886
119904119905 =
119888
119886
119904 =
minus119887 + radic1198872minus 4119886119888
2119886
119905 =
minus119887 minus radic1198872minus 4119886119888
2119886
(16)
Since 120576119896(119896 = 1 2 119899) satisfy (12) we must have
119909119899
minus 119903119909 minus 119903 =
119899
prod
119896=1
(119909 minus 120576119896) (17)
So119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886)
= 119886119899
(119904119899
minus 119904119903 minus 119903) (119905119899
minus 119905119903 minus 119903)
= 119886119899
[(119904119905)119899
minus 119903119904119905 (119904119899minus1
+ 119905119899minus1
) minus 119903 (119904119899
+ 119905119899
)]
+ 119886119899
[1199032
(119904 + 119905 + 119904119905 + 1)]
= 119886119899
[(
119888
119886
)
119899
minus 119903
119888
119886
(119904119899minus1
+ 119905119899minus1
) minus 119903 (119904119899
+ 119905119899
)]
+ 119886119899
[1199032
(
119888
119886
minus
119887
119886
+ 1)]
= 119888119899
minus 119903119888 [(119886119904)119899minus1
+ (119886119905)119899minus1
] minus 119903 [(119886119904)119899
+ (119886119905)119899
]
+ 1199032
119886119899minus1
(119888 minus 119887 + 119886)
(18)
3 Determinant of the RFP119903L119903R and RLP119903F119903LCirculant Matrices with the FibonacciNumbers
Theorem 5 Let A = RFP119903LRcirc119903fr(1198650 1198651 119865
119899minus1) Then
detA =
(minus119903119865119899)119899
minus (minus119903)119899+1
119865119899minus1
119899minus1
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899minus1
119899minus1119865119899(119892119899minus1
1+ ℎ119899minus1
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899
119899minus1(119892119899
1+ ℎ119899
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(19)
where
1198921=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
+
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
ℎ1=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
minus
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
(20)
Proof Thematrix A can be written as
A =(
1198650
1198651
sdot sdot sdot 119865119899minus1
119903119865119899minus1
1198650+ 119903119865119899minus1
sdot sdot sdot 119865119899minus2
d
1199031198652
1199031198653+ 1199031198652
sdot sdot sdot 1198651
1199031198651
1199031198652+ 1199031198651
sdot sdot sdot 1198650+ 119903119865119899minus1
)
119899times119899
(21)
4 Abstract and Applied Analysis
Using Lemma 3 the determinant of A is
detA =
119899
prod
119896=1
(1198650+ 1198651120576119896+ sdot sdot sdot + 119865
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
120572 minus 120573
120572 minus 120573
120576119896+ sdot sdot sdot +
120572119899minus1
minus 120573119899minus1
120572 minus 120573
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119865119899minus1
1205762
119896+ (1 minus 119903119865
119899minus1minus 119903119865119899) 120576119896minus 119903119865119899
1 minus 120576119896minus 1205762
119896
(22)
Using Lemma 4 we obtain
detA =
(minus119903119865119899)119899
minus (minus119903)119899+1
119865119899minus1
119899minus1
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899minus1
119899minus1119865119899(119892119899minus1
1+ ℎ119899minus1
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899
119899minus1(119892119899
1+ ℎ119899
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(23)
where
1198921=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
+
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
ℎ1=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
minus
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
(24)
Using the method in Theorem 5 similarly we also havethe following
Theorem 6 Let A1015840 = RFP119903LRcirc119903fr(119865119899minus1
1198650) Then
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(25)
Theorem 7 Let F = RLP119903FLcirc119903fr(1198650 119865
119899minus1) Then
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(26)
Proof Thematrix F can be written as
F = (
1198650
sdot sdot sdot 119865119899minus2
119865119899minus1
1198651
sdot sdot sdot 119865119899minus1
+ 1199031198650
1199031198650
1198652
sdot sdot sdot 1199031198650+ 1199031198651
1199031198651
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
119865119899minus1
+ 1199031198650sdot sdot sdot 119903119865
119899minus3+ 119903119865119899minus2
119903119865119899minus2
)
= (
119865119899minus1
119865119899minus2
sdot sdot sdot 1198650
1199031198650
119865119899minus1
+ 1199031198650
sdot sdot sdot 1198651
d
119903119865119899minus3
119903119865119899minus4
+ 119903119865119899minus3
sdot sdot sdot 119865119899minus2
119903119865119899minus2
119903119865119899minus3
+ 119903119865119899minus2
sdot sdot sdot 119865119899minus1
+ 1199031198650
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = A1015840
Γ
(27)
Hence we have
det F = detA1015840 det Γ (28)
where A1015840 = RFP119903LRcirc119903fr(119865119899minus1
119865119899minus2
1198650) and its deter-
minant is obtained fromTheorem 6
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(29)
In addition
det Γ = (minus1)119899(119899minus1)2 (30)
so
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(31)
4 Determinant of the RFM119903L119903R and RLM119903F119903LCirculant Matrices with the Lucas Numbers
Theorem 8 Let B = RFP119903LRcirc119903fr(1198710 1198711 119871
119899minus1) Then
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(32)
Abstract and Applied Analysis 5
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(33)
Proof Thematrix B can be written as
B =(
1198710
1198711
sdot sdot sdot 119871119899minus1
119903119871119899minus1
1198711+ 119903119871119899minus1
sdot sdot sdot 119871119899minus2
d
1199031198712
1199031198713+ 1199031198712
sdot sdot sdot 1198711
1199031198711
1199031198712+ 1199031198711
sdot sdot sdot 1198710+ 119903119871119899minus1
) (34)
Using Lemma 3 we have
detB =
119899
prod
119896=1
(1198710+ 1198711120576119896+ sdot sdot sdot + 119871
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
[2 + (120572 + 120573) 120576119896+ sdot sdot sdot + (120572
119899minus1
+ 120573119899minus1
) 120576119899minus1
119896]
=
119899
prod
119896=1
minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896minus 2 + 119903119871
119899
1 minus 120576119896minus 1205762
119896
(35)
According to Lemma 4 we obtain
119899
prod
119896=1
[minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896+ 2 minus 119903119871
119899]
= (2 minus 119903119871119899)119899
+ (minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
minus (minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
(36)
Then we get
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(37)
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(38)
Using the method in Theorem 8 similarly we also havethe following
Theorem 9 Let B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) Then
detB1015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(39)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(40)
Theorem 10 Let L = RLP119903FLcirc119903fr(1198710 1198711 119871
119899minus1) Then
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(41)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(42)
6 Abstract and Applied Analysis
Proof Thematrix L can be written as
L = (
1198710
sdot sdot sdot 119871119899minus2
119871119899minus1
1198711
sdot sdot sdot 119871119899minus1
+ 1199031198710
1199031198710
d
119871119899minus2
sdot sdot sdot 119903119871119899minus4
+ 119903119871119899minus3
119903119871119899minus3
119871119899minus1
minus 1199031198710sdot sdot sdot 119903119871
119899minus3+ 119903119871119899minus2
119903119871119899minus2
)
= (
119871119899minus1
119871119899minus2
sdot sdot sdot 1198710
1199031198710
119871119899minus1
+ 1199031198710
sdot sdot sdot 1198711
d
119903119871119899minus3
119903119871119899minus4
+ 119903119871119899minus3
sdot sdot sdot 119871119899minus2
119903119871119899minus2
119903119871119899minus3
+ 119903119871119899minus2
sdot sdot sdot 119871119899minus1
+ 1199031198710
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = B1015840
Γ
(43)
Thus we have
detL = detB1015840 det Γ (44)
where matrix B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) and its
determinant can be obtained fromTheorem 9
det1198611015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(45)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(46)
In addition
det Γ = (minus1)119899(119899minus1)2 (47)
so the determinant of matrix L is
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(48)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(49)
5 Determinants of the RFP119903L119903R and RLP119903F119903LCirculant Matrix with the Pell Numbers
Theorem 11 If C = RFP119903LRcirc119903fr(1198750 1198751 119875
119899minus1) then
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(50)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(51)
Proof Thematrix C can be written as
C =(
(
1198750
1198751
sdot sdot sdot 119875119899minus1
119903119875119899minus1
1198750+ 119903119875119899minus1
1198751
119875119899minus2
119903119875119899minus1
+ 119903119875119899minus2
d
1199031198752
sdot sdot sdot 1198751
1199031198751
1199031198752+ 1199031198751
sdot sdot sdot 1198750+ 119903119875119899minus1
)
)119899times119899
(52)
Using Lemma 3 the determinant of C is
detC =
119899
prod
119896=1
(1198750+ 1198751120576119896+ sdot sdot sdot + 119875
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
1205721minus 1205731
1205721minus 1205731
120576119896+ sdot sdot sdot +
120572119899minus1
1minus 120573119899minus1
1
1205721minus 1205731
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119875119899minus1
1205762
119896+ (1 minus 119903119875
119899minus1minus 119903119875119899) 120576119896minus 119903119875119899
1 minus 2120576119896minus 1205762
119896
(53)
Abstract and Applied Analysis 7
According to Lemma 4 we can get
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(54)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(55)
Using the method in Theorem 11 similarly we also havethe following
Theorem 12 If C1015840 = RFP119903LRcirc119903fr(119875119899minus1
119875119899minus2
1198750) then
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(56)
Theorem 13 If P = RLP119903FLcirc119903fr(1198750 1198751 119875
119899minus1) then one
has
detP =
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(57)
Proof Thematrix P can be written as
P =(
(
1198750
sdot sdot sdot 119875119899minus2
119875119899minus1
1198751
c 119875119899minus1
+ 1199031198750
1199031198750
c 1199031198750+ 1199031198751
119875119899minus2
c 119903119875
119899minus3
119875119899minus1
+ 1199031198750sdot sdot sdot 119903119875
119899minus3+ 119903119875119899minus2
119903119875119899minus2
)
)
= (
119875119899minus1
119875119899minus2
sdot sdot sdot 1198750
1199031198750
119875119899minus1
+ 1199031198750
sdot sdot sdot 1198751
d
119903119875119899minus2
119903119875119899minus3
+ 119903119875119899minus2
sdot sdot sdot 119875119899minus1
+ 1199031198750
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
)
(58)
Then we can get
detP = detC1015840 det Γ (59)
where C1015840 = RFPLRcircfr(119875119899minus1
119875119899minus2
1198750) and its determi-
nant could be obtained throughTheorem 12 namely
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(60)
det Γ = (minus1)119899(119899minus1)2 (61)
So
detP = detC1015840 det Γ
=
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(62)
6 Determinants of the RFP119903L119903R andRLP119903F119903L Circulant Matrix with thePell-Lucas Numbers
Theorem 14 If D = RFP119903LRcirc119903fr(1198760 1198761 119876
119899minus1) then
one has
detD =
(2 minus 119903119876119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
(2 minus 119903119876119899) (119892119899minus1
5+ ℎ119899minus1
5) minus 119903119876
119899minus1(119892119899
5+ ℎ119899
5) minus 4119903
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899
119876119899minus1
119899minus1
(63)
where
1198925=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
+
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
ℎ5=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
minus
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
(64)
Proof Themethod is similar to Theorem 11
Certainly we can get the following theorem
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Fibonacci and Lucas numbers Akbulak and Bozkurt [25]gave the norms of Toeplitz involving Fibonacci and Lucasnumbers The authors [26 27] discussed some propertiesof Fibonacci and Lucas matrices Stanimirovic et al gavegeneralized Fibonacci and Lucas matrix in [28] Z Zhangand Y Zhang [29] investigated the Lucas matrix and somecombinatorial identities
Firstly we introduce the definitions of theRFP119903L119903Rcircu-lant matrices and RLP119903F119903L circulant matrices and propertiesof the related famous numbers Then we present the mainresults and the detailed process
2 Definition and Lemma
Definition 1 A row first-plus-119903last 119903-right (RFP119903L119903R) circu-lant matrix with the first row (119886
0 1198861 119886
119899minus1) denoted by
RFP119903LRcirc119903fr(1198860 1198861 119886
119899minus1) means a square matrix of the
form
119860 =(
1198860
1198861
sdot sdot sdot 119886119899minus1
119903119886119899minus1
1198860+ 119903119886119899minus1
sdot sdot sdot 119886119899minus2
119903119886119899minus2
119903119886119899minus1
+ 119903119886119899minus2
sdot sdot sdot 119886119899minus3
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
1199031198861
1199031198862+ 1199031198861
sdot sdot sdot 1198860+ 119903119886119899minus1
) (1)
Note that the RFP119903L119903R circulant matrix is a 119909119899 minus 119903119909 minus 119903circulant matrix which is neither an extension nor specialcase of the circulant matrix [8] They are two completelydifferent kinds of special matrices
We define Θ(119903119903)
as the basic RFP119903L119903R circulant matrixthat is
Θ(119903119903)
=(
0 1 0 sdot sdot sdot 0 0
0 0 1 sdot sdot sdot 0 0
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
0 0 0 sdot sdot sdot 0 1
119903 119903 0 sdot sdot sdot 0 0
)
119899times119899
= RFP119903LRcirc119903fr (0 1 0 0)
(2)
Both the minimal polynomial and the characteristic poly-nomial of Θ
(119903119903)are 119892(119909) = 119909
119899
minus 119903119909 minus 119903 which has onlysimple roots denoted by 120576
119896(119896 = 1 2 119899) In addition
Θ(119903119903)
satisfiesΘ119895(119903119903)
= RFP119903LRcirc119903fr(0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119895
1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119899minus119895minus1
) and
Θ119899
(119903119903)= 119903119868119899+ 119903Θ(119903119903)
Then a matrix 119860 can be written in theform
119860 = 119891 (Θ(119903119903)
) =
119899minus1
sum
119894=0
119886119894Θ119894
(119903119903) (3)
if and only if 119860 is a RFP119903L119903R circulant matrix where thepolynomial 119891(119909) = sum
119899minus1
119894=0119886119894119909119894 is called the representer of the
RFP119903L119903R circulant matrix 119860Since Θ
(119903119903)is nonderogatory then 119860 is a RFM119903L119903R
circulant matrix if and only if119860 commutes withΘ(119903119903)
that is119860Θ(119903119903)
= Θ(119903119903)
119860 Because of the representation RFM119903L119903Rcirculant matrices have very nice structure and the algebraicproperties also can be easily attained Moreover the productof two RFM119903L119903R circulant matrices and the inverse 119860minus1 areagain RFM119903L119903R circulant matrices
Definition 2 A row last-plus-119903first 119903-left (RLP119903F119903L) circu-lant matrix with the first row (119886
0 1198861 119886
119899minus1) denoted by
RLP119903FLcirc119903fr(1198860 1198861 119886
119899minus1) means a square matrix of the
form
119861 =(
1198860
sdot sdot sdot 119886119899minus2
119886119899minus1
1198861
sdot sdot sdot 119886119899minus1
+ 1199031198860
1199031198860
1198862
sdot sdot sdot 1199031198860+ 1199031198861
1199031198861
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
119886119899minus1
+ 1199031198860sdot sdot sdot 119903119886
119899minus3+ 119903119886119899minus2
119903119886119899minus2
) (4)
Let 119860 = RLP119903FLcirc119903fr(1198860 1198861 119886
119899minus1) and 119861 =
RFP119903LRcirc119903fr(119886119899minus1
119886119899minus2
1198860) By explicit computation
we find
119860 = 119861119868119899 (5)
where 119868119899is the backward identity matrix of the form
119868119899=(
1
1
c1
1
) (6)
The Fibonacci Lucas Pell and the Pell-Lucas sequences[30ndash36] are defined by the following recurrence relationsrespectively
119865119899+1
= 119865119899+ 119865119899minus1
where 1198650= 0 119865
1= 1
119871119899+1
= 119871119899+ 119871119899minus1
where 1198710= 2 119871
1= 1
119875119899+1
= 2119875119899+ 119875119899minus1
where 1198750= 0 119875
1= 1
119876119899+1
= 2119876119899+ 119876119899minus1
where 1198760= 2 119876
1= 2
(7)
The first few values of these sequences are given by thefollowing table (119899 ge 0)
119899 0 1 2 3 4 5 6 7
1198651198990 1 1 2 3 5 8 13
1198711198992 1 3 4 7 11 18 29
1198751198990 1 2 5 12 29 70 169
1198761198992 2 6 14 34 82 198 478
(8)
The sequences 119865119899 119871119899 119875119899 and 119876
119899 are given by the
Binet formulae
119865119899=
120572119899
minus 120573119899
120572 minus 120573
119871119899= 120572119899
+ 120573119899
119875119899=
120572119899
1minus 120573119899
1
1205721minus 1205731
119876119899= 120572119899
1+ 120573119899
1
(9)
where 120572 120573 are the roots of the characteristic equation 1199092minus119909minus1 = 0 and 120572
1 1205731are the roots of the characteristic equation
1199092
minus 2119909 minus 1 = 0By Proposition 51 in [14] we deduce the following
lemma
Abstract and Applied Analysis 3
Lemma 3 Let 119860 = RFP119903LRcirc119903fr(1198860 119886
119899minus1) then the
eigenvalues of 119860 are
119891 (120576119896) =
119899minus1
sum
119894=0
(119886119894120576119894
119896) (10)
and in addition
det119860 =
119899
prod
119896=1
119899minus1
sum
119894=0
(119886119894120576119894
119896) (11)
where 120576119896(119896 = 1 2 119899) are the roots of the equation
119909119899
minus 119903119909 minus 119903 = 0 (12)
Lemma 4 Consider
119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886)
= 119888119899
minus 119903119888 [(119886119904)119899minus1
+ (119886119905)119899minus1
]
minus 119903 [(119886119904)119899
+ (119886119905)119899
] + 1199032
119886119899minus1
(119888 minus 119887 + 119886)
(13)
where
119904 =
minus119887 + radic1198872minus 4119886119888
2119886
119905 =
minus119887 minus radic1198872minus 4119886119888
2119886
(14)
and 120576119896(119896 = 1 2 119899) satisfy (12) 119886 119887 119888 isin R 119886 = 0
Proof Consider
119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886) = 119886
119899
119899
prod
119896=1
(1205762
119896+
119887
119886
120576119896+
119888
119886
)
= 119886119899
119899
prod
119896=1
(120576119896minus 119904) (120576
119896minus 119905)
= 119886119899
119899
prod
119896=1
(119904 minus 120576119896) (119905 minus 120576
119896)
(15)
while
119904 + 119905 = minus
119887
119886
119904119905 =
119888
119886
119904 =
minus119887 + radic1198872minus 4119886119888
2119886
119905 =
minus119887 minus radic1198872minus 4119886119888
2119886
(16)
Since 120576119896(119896 = 1 2 119899) satisfy (12) we must have
119909119899
minus 119903119909 minus 119903 =
119899
prod
119896=1
(119909 minus 120576119896) (17)
So119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886)
= 119886119899
(119904119899
minus 119904119903 minus 119903) (119905119899
minus 119905119903 minus 119903)
= 119886119899
[(119904119905)119899
minus 119903119904119905 (119904119899minus1
+ 119905119899minus1
) minus 119903 (119904119899
+ 119905119899
)]
+ 119886119899
[1199032
(119904 + 119905 + 119904119905 + 1)]
= 119886119899
[(
119888
119886
)
119899
minus 119903
119888
119886
(119904119899minus1
+ 119905119899minus1
) minus 119903 (119904119899
+ 119905119899
)]
+ 119886119899
[1199032
(
119888
119886
minus
119887
119886
+ 1)]
= 119888119899
minus 119903119888 [(119886119904)119899minus1
+ (119886119905)119899minus1
] minus 119903 [(119886119904)119899
+ (119886119905)119899
]
+ 1199032
119886119899minus1
(119888 minus 119887 + 119886)
(18)
3 Determinant of the RFP119903L119903R and RLP119903F119903LCirculant Matrices with the FibonacciNumbers
Theorem 5 Let A = RFP119903LRcirc119903fr(1198650 1198651 119865
119899minus1) Then
detA =
(minus119903119865119899)119899
minus (minus119903)119899+1
119865119899minus1
119899minus1
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899minus1
119899minus1119865119899(119892119899minus1
1+ ℎ119899minus1
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899
119899minus1(119892119899
1+ ℎ119899
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(19)
where
1198921=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
+
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
ℎ1=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
minus
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
(20)
Proof Thematrix A can be written as
A =(
1198650
1198651
sdot sdot sdot 119865119899minus1
119903119865119899minus1
1198650+ 119903119865119899minus1
sdot sdot sdot 119865119899minus2
d
1199031198652
1199031198653+ 1199031198652
sdot sdot sdot 1198651
1199031198651
1199031198652+ 1199031198651
sdot sdot sdot 1198650+ 119903119865119899minus1
)
119899times119899
(21)
4 Abstract and Applied Analysis
Using Lemma 3 the determinant of A is
detA =
119899
prod
119896=1
(1198650+ 1198651120576119896+ sdot sdot sdot + 119865
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
120572 minus 120573
120572 minus 120573
120576119896+ sdot sdot sdot +
120572119899minus1
minus 120573119899minus1
120572 minus 120573
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119865119899minus1
1205762
119896+ (1 minus 119903119865
119899minus1minus 119903119865119899) 120576119896minus 119903119865119899
1 minus 120576119896minus 1205762
119896
(22)
Using Lemma 4 we obtain
detA =
(minus119903119865119899)119899
minus (minus119903)119899+1
119865119899minus1
119899minus1
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899minus1
119899minus1119865119899(119892119899minus1
1+ ℎ119899minus1
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899
119899minus1(119892119899
1+ ℎ119899
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(23)
where
1198921=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
+
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
ℎ1=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
minus
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
(24)
Using the method in Theorem 5 similarly we also havethe following
Theorem 6 Let A1015840 = RFP119903LRcirc119903fr(119865119899minus1
1198650) Then
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(25)
Theorem 7 Let F = RLP119903FLcirc119903fr(1198650 119865
119899minus1) Then
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(26)
Proof Thematrix F can be written as
F = (
1198650
sdot sdot sdot 119865119899minus2
119865119899minus1
1198651
sdot sdot sdot 119865119899minus1
+ 1199031198650
1199031198650
1198652
sdot sdot sdot 1199031198650+ 1199031198651
1199031198651
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
119865119899minus1
+ 1199031198650sdot sdot sdot 119903119865
119899minus3+ 119903119865119899minus2
119903119865119899minus2
)
= (
119865119899minus1
119865119899minus2
sdot sdot sdot 1198650
1199031198650
119865119899minus1
+ 1199031198650
sdot sdot sdot 1198651
d
119903119865119899minus3
119903119865119899minus4
+ 119903119865119899minus3
sdot sdot sdot 119865119899minus2
119903119865119899minus2
119903119865119899minus3
+ 119903119865119899minus2
sdot sdot sdot 119865119899minus1
+ 1199031198650
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = A1015840
Γ
(27)
Hence we have
det F = detA1015840 det Γ (28)
where A1015840 = RFP119903LRcirc119903fr(119865119899minus1
119865119899minus2
1198650) and its deter-
minant is obtained fromTheorem 6
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(29)
In addition
det Γ = (minus1)119899(119899minus1)2 (30)
so
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(31)
4 Determinant of the RFM119903L119903R and RLM119903F119903LCirculant Matrices with the Lucas Numbers
Theorem 8 Let B = RFP119903LRcirc119903fr(1198710 1198711 119871
119899minus1) Then
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(32)
Abstract and Applied Analysis 5
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(33)
Proof Thematrix B can be written as
B =(
1198710
1198711
sdot sdot sdot 119871119899minus1
119903119871119899minus1
1198711+ 119903119871119899minus1
sdot sdot sdot 119871119899minus2
d
1199031198712
1199031198713+ 1199031198712
sdot sdot sdot 1198711
1199031198711
1199031198712+ 1199031198711
sdot sdot sdot 1198710+ 119903119871119899minus1
) (34)
Using Lemma 3 we have
detB =
119899
prod
119896=1
(1198710+ 1198711120576119896+ sdot sdot sdot + 119871
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
[2 + (120572 + 120573) 120576119896+ sdot sdot sdot + (120572
119899minus1
+ 120573119899minus1
) 120576119899minus1
119896]
=
119899
prod
119896=1
minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896minus 2 + 119903119871
119899
1 minus 120576119896minus 1205762
119896
(35)
According to Lemma 4 we obtain
119899
prod
119896=1
[minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896+ 2 minus 119903119871
119899]
= (2 minus 119903119871119899)119899
+ (minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
minus (minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
(36)
Then we get
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(37)
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(38)
Using the method in Theorem 8 similarly we also havethe following
Theorem 9 Let B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) Then
detB1015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(39)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(40)
Theorem 10 Let L = RLP119903FLcirc119903fr(1198710 1198711 119871
119899minus1) Then
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(41)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(42)
6 Abstract and Applied Analysis
Proof Thematrix L can be written as
L = (
1198710
sdot sdot sdot 119871119899minus2
119871119899minus1
1198711
sdot sdot sdot 119871119899minus1
+ 1199031198710
1199031198710
d
119871119899minus2
sdot sdot sdot 119903119871119899minus4
+ 119903119871119899minus3
119903119871119899minus3
119871119899minus1
minus 1199031198710sdot sdot sdot 119903119871
119899minus3+ 119903119871119899minus2
119903119871119899minus2
)
= (
119871119899minus1
119871119899minus2
sdot sdot sdot 1198710
1199031198710
119871119899minus1
+ 1199031198710
sdot sdot sdot 1198711
d
119903119871119899minus3
119903119871119899minus4
+ 119903119871119899minus3
sdot sdot sdot 119871119899minus2
119903119871119899minus2
119903119871119899minus3
+ 119903119871119899minus2
sdot sdot sdot 119871119899minus1
+ 1199031198710
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = B1015840
Γ
(43)
Thus we have
detL = detB1015840 det Γ (44)
where matrix B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) and its
determinant can be obtained fromTheorem 9
det1198611015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(45)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(46)
In addition
det Γ = (minus1)119899(119899minus1)2 (47)
so the determinant of matrix L is
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(48)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(49)
5 Determinants of the RFP119903L119903R and RLP119903F119903LCirculant Matrix with the Pell Numbers
Theorem 11 If C = RFP119903LRcirc119903fr(1198750 1198751 119875
119899minus1) then
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(50)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(51)
Proof Thematrix C can be written as
C =(
(
1198750
1198751
sdot sdot sdot 119875119899minus1
119903119875119899minus1
1198750+ 119903119875119899minus1
1198751
119875119899minus2
119903119875119899minus1
+ 119903119875119899minus2
d
1199031198752
sdot sdot sdot 1198751
1199031198751
1199031198752+ 1199031198751
sdot sdot sdot 1198750+ 119903119875119899minus1
)
)119899times119899
(52)
Using Lemma 3 the determinant of C is
detC =
119899
prod
119896=1
(1198750+ 1198751120576119896+ sdot sdot sdot + 119875
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
1205721minus 1205731
1205721minus 1205731
120576119896+ sdot sdot sdot +
120572119899minus1
1minus 120573119899minus1
1
1205721minus 1205731
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119875119899minus1
1205762
119896+ (1 minus 119903119875
119899minus1minus 119903119875119899) 120576119896minus 119903119875119899
1 minus 2120576119896minus 1205762
119896
(53)
Abstract and Applied Analysis 7
According to Lemma 4 we can get
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(54)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(55)
Using the method in Theorem 11 similarly we also havethe following
Theorem 12 If C1015840 = RFP119903LRcirc119903fr(119875119899minus1
119875119899minus2
1198750) then
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(56)
Theorem 13 If P = RLP119903FLcirc119903fr(1198750 1198751 119875
119899minus1) then one
has
detP =
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(57)
Proof Thematrix P can be written as
P =(
(
1198750
sdot sdot sdot 119875119899minus2
119875119899minus1
1198751
c 119875119899minus1
+ 1199031198750
1199031198750
c 1199031198750+ 1199031198751
119875119899minus2
c 119903119875
119899minus3
119875119899minus1
+ 1199031198750sdot sdot sdot 119903119875
119899minus3+ 119903119875119899minus2
119903119875119899minus2
)
)
= (
119875119899minus1
119875119899minus2
sdot sdot sdot 1198750
1199031198750
119875119899minus1
+ 1199031198750
sdot sdot sdot 1198751
d
119903119875119899minus2
119903119875119899minus3
+ 119903119875119899minus2
sdot sdot sdot 119875119899minus1
+ 1199031198750
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
)
(58)
Then we can get
detP = detC1015840 det Γ (59)
where C1015840 = RFPLRcircfr(119875119899minus1
119875119899minus2
1198750) and its determi-
nant could be obtained throughTheorem 12 namely
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(60)
det Γ = (minus1)119899(119899minus1)2 (61)
So
detP = detC1015840 det Γ
=
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(62)
6 Determinants of the RFP119903L119903R andRLP119903F119903L Circulant Matrix with thePell-Lucas Numbers
Theorem 14 If D = RFP119903LRcirc119903fr(1198760 1198761 119876
119899minus1) then
one has
detD =
(2 minus 119903119876119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
(2 minus 119903119876119899) (119892119899minus1
5+ ℎ119899minus1
5) minus 119903119876
119899minus1(119892119899
5+ ℎ119899
5) minus 4119903
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899
119876119899minus1
119899minus1
(63)
where
1198925=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
+
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
ℎ5=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
minus
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
(64)
Proof Themethod is similar to Theorem 11
Certainly we can get the following theorem
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Lemma 3 Let 119860 = RFP119903LRcirc119903fr(1198860 119886
119899minus1) then the
eigenvalues of 119860 are
119891 (120576119896) =
119899minus1
sum
119894=0
(119886119894120576119894
119896) (10)
and in addition
det119860 =
119899
prod
119896=1
119899minus1
sum
119894=0
(119886119894120576119894
119896) (11)
where 120576119896(119896 = 1 2 119899) are the roots of the equation
119909119899
minus 119903119909 minus 119903 = 0 (12)
Lemma 4 Consider
119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886)
= 119888119899
minus 119903119888 [(119886119904)119899minus1
+ (119886119905)119899minus1
]
minus 119903 [(119886119904)119899
+ (119886119905)119899
] + 1199032
119886119899minus1
(119888 minus 119887 + 119886)
(13)
where
119904 =
minus119887 + radic1198872minus 4119886119888
2119886
119905 =
minus119887 minus radic1198872minus 4119886119888
2119886
(14)
and 120576119896(119896 = 1 2 119899) satisfy (12) 119886 119887 119888 isin R 119886 = 0
Proof Consider
119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886) = 119886
119899
119899
prod
119896=1
(1205762
119896+
119887
119886
120576119896+
119888
119886
)
= 119886119899
119899
prod
119896=1
(120576119896minus 119904) (120576
119896minus 119905)
= 119886119899
119899
prod
119896=1
(119904 minus 120576119896) (119905 minus 120576
119896)
(15)
while
119904 + 119905 = minus
119887
119886
119904119905 =
119888
119886
119904 =
minus119887 + radic1198872minus 4119886119888
2119886
119905 =
minus119887 minus radic1198872minus 4119886119888
2119886
(16)
Since 120576119896(119896 = 1 2 119899) satisfy (12) we must have
119909119899
minus 119903119909 minus 119903 =
119899
prod
119896=1
(119909 minus 120576119896) (17)
So119899
prod
119896=1
(119888 + 120576119896119887 + 1205762
119896119886)
= 119886119899
(119904119899
minus 119904119903 minus 119903) (119905119899
minus 119905119903 minus 119903)
= 119886119899
[(119904119905)119899
minus 119903119904119905 (119904119899minus1
+ 119905119899minus1
) minus 119903 (119904119899
+ 119905119899
)]
+ 119886119899
[1199032
(119904 + 119905 + 119904119905 + 1)]
= 119886119899
[(
119888
119886
)
119899
minus 119903
119888
119886
(119904119899minus1
+ 119905119899minus1
) minus 119903 (119904119899
+ 119905119899
)]
+ 119886119899
[1199032
(
119888
119886
minus
119887
119886
+ 1)]
= 119888119899
minus 119903119888 [(119886119904)119899minus1
+ (119886119905)119899minus1
] minus 119903 [(119886119904)119899
+ (119886119905)119899
]
+ 1199032
119886119899minus1
(119888 minus 119887 + 119886)
(18)
3 Determinant of the RFP119903L119903R and RLP119903F119903LCirculant Matrices with the FibonacciNumbers
Theorem 5 Let A = RFP119903LRcirc119903fr(1198650 1198651 119865
119899minus1) Then
detA =
(minus119903119865119899)119899
minus (minus119903)119899+1
119865119899minus1
119899minus1
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899minus1
119899minus1119865119899(119892119899minus1
1+ ℎ119899minus1
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899
119899minus1(119892119899
1+ ℎ119899
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(19)
where
1198921=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
+
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
ℎ1=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
minus
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
(20)
Proof Thematrix A can be written as
A =(
1198650
1198651
sdot sdot sdot 119865119899minus1
119903119865119899minus1
1198650+ 119903119865119899minus1
sdot sdot sdot 119865119899minus2
d
1199031198652
1199031198653+ 1199031198652
sdot sdot sdot 1198651
1199031198651
1199031198652+ 1199031198651
sdot sdot sdot 1198650+ 119903119865119899minus1
)
119899times119899
(21)
4 Abstract and Applied Analysis
Using Lemma 3 the determinant of A is
detA =
119899
prod
119896=1
(1198650+ 1198651120576119896+ sdot sdot sdot + 119865
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
120572 minus 120573
120572 minus 120573
120576119896+ sdot sdot sdot +
120572119899minus1
minus 120573119899minus1
120572 minus 120573
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119865119899minus1
1205762
119896+ (1 minus 119903119865
119899minus1minus 119903119865119899) 120576119896minus 119903119865119899
1 minus 120576119896minus 1205762
119896
(22)
Using Lemma 4 we obtain
detA =
(minus119903119865119899)119899
minus (minus119903)119899+1
119865119899minus1
119899minus1
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899minus1
119899minus1119865119899(119892119899minus1
1+ ℎ119899minus1
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899
119899minus1(119892119899
1+ ℎ119899
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(23)
where
1198921=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
+
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
ℎ1=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
minus
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
(24)
Using the method in Theorem 5 similarly we also havethe following
Theorem 6 Let A1015840 = RFP119903LRcirc119903fr(119865119899minus1
1198650) Then
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(25)
Theorem 7 Let F = RLP119903FLcirc119903fr(1198650 119865
119899minus1) Then
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(26)
Proof Thematrix F can be written as
F = (
1198650
sdot sdot sdot 119865119899minus2
119865119899minus1
1198651
sdot sdot sdot 119865119899minus1
+ 1199031198650
1199031198650
1198652
sdot sdot sdot 1199031198650+ 1199031198651
1199031198651
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
119865119899minus1
+ 1199031198650sdot sdot sdot 119903119865
119899minus3+ 119903119865119899minus2
119903119865119899minus2
)
= (
119865119899minus1
119865119899minus2
sdot sdot sdot 1198650
1199031198650
119865119899minus1
+ 1199031198650
sdot sdot sdot 1198651
d
119903119865119899minus3
119903119865119899minus4
+ 119903119865119899minus3
sdot sdot sdot 119865119899minus2
119903119865119899minus2
119903119865119899minus3
+ 119903119865119899minus2
sdot sdot sdot 119865119899minus1
+ 1199031198650
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = A1015840
Γ
(27)
Hence we have
det F = detA1015840 det Γ (28)
where A1015840 = RFP119903LRcirc119903fr(119865119899minus1
119865119899minus2
1198650) and its deter-
minant is obtained fromTheorem 6
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(29)
In addition
det Γ = (minus1)119899(119899minus1)2 (30)
so
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(31)
4 Determinant of the RFM119903L119903R and RLM119903F119903LCirculant Matrices with the Lucas Numbers
Theorem 8 Let B = RFP119903LRcirc119903fr(1198710 1198711 119871
119899minus1) Then
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(32)
Abstract and Applied Analysis 5
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(33)
Proof Thematrix B can be written as
B =(
1198710
1198711
sdot sdot sdot 119871119899minus1
119903119871119899minus1
1198711+ 119903119871119899minus1
sdot sdot sdot 119871119899minus2
d
1199031198712
1199031198713+ 1199031198712
sdot sdot sdot 1198711
1199031198711
1199031198712+ 1199031198711
sdot sdot sdot 1198710+ 119903119871119899minus1
) (34)
Using Lemma 3 we have
detB =
119899
prod
119896=1
(1198710+ 1198711120576119896+ sdot sdot sdot + 119871
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
[2 + (120572 + 120573) 120576119896+ sdot sdot sdot + (120572
119899minus1
+ 120573119899minus1
) 120576119899minus1
119896]
=
119899
prod
119896=1
minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896minus 2 + 119903119871
119899
1 minus 120576119896minus 1205762
119896
(35)
According to Lemma 4 we obtain
119899
prod
119896=1
[minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896+ 2 minus 119903119871
119899]
= (2 minus 119903119871119899)119899
+ (minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
minus (minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
(36)
Then we get
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(37)
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(38)
Using the method in Theorem 8 similarly we also havethe following
Theorem 9 Let B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) Then
detB1015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(39)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(40)
Theorem 10 Let L = RLP119903FLcirc119903fr(1198710 1198711 119871
119899minus1) Then
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(41)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(42)
6 Abstract and Applied Analysis
Proof Thematrix L can be written as
L = (
1198710
sdot sdot sdot 119871119899minus2
119871119899minus1
1198711
sdot sdot sdot 119871119899minus1
+ 1199031198710
1199031198710
d
119871119899minus2
sdot sdot sdot 119903119871119899minus4
+ 119903119871119899minus3
119903119871119899minus3
119871119899minus1
minus 1199031198710sdot sdot sdot 119903119871
119899minus3+ 119903119871119899minus2
119903119871119899minus2
)
= (
119871119899minus1
119871119899minus2
sdot sdot sdot 1198710
1199031198710
119871119899minus1
+ 1199031198710
sdot sdot sdot 1198711
d
119903119871119899minus3
119903119871119899minus4
+ 119903119871119899minus3
sdot sdot sdot 119871119899minus2
119903119871119899minus2
119903119871119899minus3
+ 119903119871119899minus2
sdot sdot sdot 119871119899minus1
+ 1199031198710
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = B1015840
Γ
(43)
Thus we have
detL = detB1015840 det Γ (44)
where matrix B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) and its
determinant can be obtained fromTheorem 9
det1198611015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(45)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(46)
In addition
det Γ = (minus1)119899(119899minus1)2 (47)
so the determinant of matrix L is
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(48)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(49)
5 Determinants of the RFP119903L119903R and RLP119903F119903LCirculant Matrix with the Pell Numbers
Theorem 11 If C = RFP119903LRcirc119903fr(1198750 1198751 119875
119899minus1) then
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(50)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(51)
Proof Thematrix C can be written as
C =(
(
1198750
1198751
sdot sdot sdot 119875119899minus1
119903119875119899minus1
1198750+ 119903119875119899minus1
1198751
119875119899minus2
119903119875119899minus1
+ 119903119875119899minus2
d
1199031198752
sdot sdot sdot 1198751
1199031198751
1199031198752+ 1199031198751
sdot sdot sdot 1198750+ 119903119875119899minus1
)
)119899times119899
(52)
Using Lemma 3 the determinant of C is
detC =
119899
prod
119896=1
(1198750+ 1198751120576119896+ sdot sdot sdot + 119875
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
1205721minus 1205731
1205721minus 1205731
120576119896+ sdot sdot sdot +
120572119899minus1
1minus 120573119899minus1
1
1205721minus 1205731
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119875119899minus1
1205762
119896+ (1 minus 119903119875
119899minus1minus 119903119875119899) 120576119896minus 119903119875119899
1 minus 2120576119896minus 1205762
119896
(53)
Abstract and Applied Analysis 7
According to Lemma 4 we can get
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(54)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(55)
Using the method in Theorem 11 similarly we also havethe following
Theorem 12 If C1015840 = RFP119903LRcirc119903fr(119875119899minus1
119875119899minus2
1198750) then
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(56)
Theorem 13 If P = RLP119903FLcirc119903fr(1198750 1198751 119875
119899minus1) then one
has
detP =
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(57)
Proof Thematrix P can be written as
P =(
(
1198750
sdot sdot sdot 119875119899minus2
119875119899minus1
1198751
c 119875119899minus1
+ 1199031198750
1199031198750
c 1199031198750+ 1199031198751
119875119899minus2
c 119903119875
119899minus3
119875119899minus1
+ 1199031198750sdot sdot sdot 119903119875
119899minus3+ 119903119875119899minus2
119903119875119899minus2
)
)
= (
119875119899minus1
119875119899minus2
sdot sdot sdot 1198750
1199031198750
119875119899minus1
+ 1199031198750
sdot sdot sdot 1198751
d
119903119875119899minus2
119903119875119899minus3
+ 119903119875119899minus2
sdot sdot sdot 119875119899minus1
+ 1199031198750
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
)
(58)
Then we can get
detP = detC1015840 det Γ (59)
where C1015840 = RFPLRcircfr(119875119899minus1
119875119899minus2
1198750) and its determi-
nant could be obtained throughTheorem 12 namely
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(60)
det Γ = (minus1)119899(119899minus1)2 (61)
So
detP = detC1015840 det Γ
=
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(62)
6 Determinants of the RFP119903L119903R andRLP119903F119903L Circulant Matrix with thePell-Lucas Numbers
Theorem 14 If D = RFP119903LRcirc119903fr(1198760 1198761 119876
119899minus1) then
one has
detD =
(2 minus 119903119876119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
(2 minus 119903119876119899) (119892119899minus1
5+ ℎ119899minus1
5) minus 119903119876
119899minus1(119892119899
5+ ℎ119899
5) minus 4119903
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899
119876119899minus1
119899minus1
(63)
where
1198925=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
+
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
ℎ5=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
minus
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
(64)
Proof Themethod is similar to Theorem 11
Certainly we can get the following theorem
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Using Lemma 3 the determinant of A is
detA =
119899
prod
119896=1
(1198650+ 1198651120576119896+ sdot sdot sdot + 119865
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
120572 minus 120573
120572 minus 120573
120576119896+ sdot sdot sdot +
120572119899minus1
minus 120573119899minus1
120572 minus 120573
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119865119899minus1
1205762
119896+ (1 minus 119903119865
119899minus1minus 119903119865119899) 120576119896minus 119903119865119899
1 minus 120576119896minus 1205762
119896
(22)
Using Lemma 4 we obtain
detA =
(minus119903119865119899)119899
minus (minus119903)119899+1
119865119899minus1
119899minus1
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899minus1
119899minus1119865119899(119892119899minus1
1+ ℎ119899minus1
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899+1
119865119899
119899minus1(119892119899
1+ ℎ119899
1)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(23)
where
1198921=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
+
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
ℎ1=
(119903119865119899+ 119903119865119899minus1
minus 1)
minus2119903119865119899minus1
minus
radic1199032(119865119899minus 119865119899minus1
)2
minus 2119903 (119865119899+ 119865119899+1
)
minus2119903119865119899minus1
(24)
Using the method in Theorem 5 similarly we also havethe following
Theorem 6 Let A1015840 = RFP119903LRcirc119903fr(119865119899minus1
1198650) Then
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(25)
Theorem 7 Let F = RLP119903FLcirc119903fr(1198650 119865
119899minus1) Then
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(26)
Proof Thematrix F can be written as
F = (
1198650
sdot sdot sdot 119865119899minus2
119865119899minus1
1198651
sdot sdot sdot 119865119899minus1
+ 1199031198650
1199031198650
1198652
sdot sdot sdot 1199031198650+ 1199031198651
1199031198651
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
119865119899minus1
+ 1199031198650sdot sdot sdot 119903119865
119899minus3+ 119903119865119899minus2
119903119865119899minus2
)
= (
119865119899minus1
119865119899minus2
sdot sdot sdot 1198650
1199031198650
119865119899minus1
+ 1199031198650
sdot sdot sdot 1198651
d
119903119865119899minus3
119903119865119899minus4
+ 119903119865119899minus3
sdot sdot sdot 119865119899minus2
119903119865119899minus2
119903119865119899minus3
+ 119903119865119899minus2
sdot sdot sdot 119865119899minus1
+ 1199031198650
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = A1015840
Γ
(27)
Hence we have
det F = detA1015840 det Γ (28)
where A1015840 = RFP119903LRcirc119903fr(119865119899minus1
119865119899minus2
1198650) and its deter-
minant is obtained fromTheorem 6
detA1015840 =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(29)
In addition
det Γ = (minus1)119899(119899minus1)2 (30)
so
det F =(119903 minus 119865
119899minus1)119899
minus 119903 (119903 minus 119865119899minus1
) (119865119899minus 119903)119899minus1
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
minus
119903(119865119899minus 119903)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
(31)
4 Determinant of the RFM119903L119903R and RLM119903F119903LCirculant Matrices with the Lucas Numbers
Theorem 8 Let B = RFP119903LRcirc119903fr(1198710 1198711 119871
119899minus1) Then
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(32)
Abstract and Applied Analysis 5
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(33)
Proof Thematrix B can be written as
B =(
1198710
1198711
sdot sdot sdot 119871119899minus1
119903119871119899minus1
1198711+ 119903119871119899minus1
sdot sdot sdot 119871119899minus2
d
1199031198712
1199031198713+ 1199031198712
sdot sdot sdot 1198711
1199031198711
1199031198712+ 1199031198711
sdot sdot sdot 1198710+ 119903119871119899minus1
) (34)
Using Lemma 3 we have
detB =
119899
prod
119896=1
(1198710+ 1198711120576119896+ sdot sdot sdot + 119871
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
[2 + (120572 + 120573) 120576119896+ sdot sdot sdot + (120572
119899minus1
+ 120573119899minus1
) 120576119899minus1
119896]
=
119899
prod
119896=1
minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896minus 2 + 119903119871
119899
1 minus 120576119896minus 1205762
119896
(35)
According to Lemma 4 we obtain
119899
prod
119896=1
[minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896+ 2 minus 119903119871
119899]
= (2 minus 119903119871119899)119899
+ (minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
minus (minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
(36)
Then we get
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(37)
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(38)
Using the method in Theorem 8 similarly we also havethe following
Theorem 9 Let B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) Then
detB1015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(39)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(40)
Theorem 10 Let L = RLP119903FLcirc119903fr(1198710 1198711 119871
119899minus1) Then
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(41)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(42)
6 Abstract and Applied Analysis
Proof Thematrix L can be written as
L = (
1198710
sdot sdot sdot 119871119899minus2
119871119899minus1
1198711
sdot sdot sdot 119871119899minus1
+ 1199031198710
1199031198710
d
119871119899minus2
sdot sdot sdot 119903119871119899minus4
+ 119903119871119899minus3
119903119871119899minus3
119871119899minus1
minus 1199031198710sdot sdot sdot 119903119871
119899minus3+ 119903119871119899minus2
119903119871119899minus2
)
= (
119871119899minus1
119871119899minus2
sdot sdot sdot 1198710
1199031198710
119871119899minus1
+ 1199031198710
sdot sdot sdot 1198711
d
119903119871119899minus3
119903119871119899minus4
+ 119903119871119899minus3
sdot sdot sdot 119871119899minus2
119903119871119899minus2
119903119871119899minus3
+ 119903119871119899minus2
sdot sdot sdot 119871119899minus1
+ 1199031198710
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = B1015840
Γ
(43)
Thus we have
detL = detB1015840 det Γ (44)
where matrix B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) and its
determinant can be obtained fromTheorem 9
det1198611015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(45)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(46)
In addition
det Γ = (minus1)119899(119899minus1)2 (47)
so the determinant of matrix L is
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(48)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(49)
5 Determinants of the RFP119903L119903R and RLP119903F119903LCirculant Matrix with the Pell Numbers
Theorem 11 If C = RFP119903LRcirc119903fr(1198750 1198751 119875
119899minus1) then
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(50)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(51)
Proof Thematrix C can be written as
C =(
(
1198750
1198751
sdot sdot sdot 119875119899minus1
119903119875119899minus1
1198750+ 119903119875119899minus1
1198751
119875119899minus2
119903119875119899minus1
+ 119903119875119899minus2
d
1199031198752
sdot sdot sdot 1198751
1199031198751
1199031198752+ 1199031198751
sdot sdot sdot 1198750+ 119903119875119899minus1
)
)119899times119899
(52)
Using Lemma 3 the determinant of C is
detC =
119899
prod
119896=1
(1198750+ 1198751120576119896+ sdot sdot sdot + 119875
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
1205721minus 1205731
1205721minus 1205731
120576119896+ sdot sdot sdot +
120572119899minus1
1minus 120573119899minus1
1
1205721minus 1205731
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119875119899minus1
1205762
119896+ (1 minus 119903119875
119899minus1minus 119903119875119899) 120576119896minus 119903119875119899
1 minus 2120576119896minus 1205762
119896
(53)
Abstract and Applied Analysis 7
According to Lemma 4 we can get
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(54)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(55)
Using the method in Theorem 11 similarly we also havethe following
Theorem 12 If C1015840 = RFP119903LRcirc119903fr(119875119899minus1
119875119899minus2
1198750) then
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(56)
Theorem 13 If P = RLP119903FLcirc119903fr(1198750 1198751 119875
119899minus1) then one
has
detP =
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(57)
Proof Thematrix P can be written as
P =(
(
1198750
sdot sdot sdot 119875119899minus2
119875119899minus1
1198751
c 119875119899minus1
+ 1199031198750
1199031198750
c 1199031198750+ 1199031198751
119875119899minus2
c 119903119875
119899minus3
119875119899minus1
+ 1199031198750sdot sdot sdot 119903119875
119899minus3+ 119903119875119899minus2
119903119875119899minus2
)
)
= (
119875119899minus1
119875119899minus2
sdot sdot sdot 1198750
1199031198750
119875119899minus1
+ 1199031198750
sdot sdot sdot 1198751
d
119903119875119899minus2
119903119875119899minus3
+ 119903119875119899minus2
sdot sdot sdot 119875119899minus1
+ 1199031198750
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
)
(58)
Then we can get
detP = detC1015840 det Γ (59)
where C1015840 = RFPLRcircfr(119875119899minus1
119875119899minus2
1198750) and its determi-
nant could be obtained throughTheorem 12 namely
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(60)
det Γ = (minus1)119899(119899minus1)2 (61)
So
detP = detC1015840 det Γ
=
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(62)
6 Determinants of the RFP119903L119903R andRLP119903F119903L Circulant Matrix with thePell-Lucas Numbers
Theorem 14 If D = RFP119903LRcirc119903fr(1198760 1198761 119876
119899minus1) then
one has
detD =
(2 minus 119903119876119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
(2 minus 119903119876119899) (119892119899minus1
5+ ℎ119899minus1
5) minus 119903119876
119899minus1(119892119899
5+ ℎ119899
5) minus 4119903
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899
119876119899minus1
119899minus1
(63)
where
1198925=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
+
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
ℎ5=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
minus
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
(64)
Proof Themethod is similar to Theorem 11
Certainly we can get the following theorem
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(33)
Proof Thematrix B can be written as
B =(
1198710
1198711
sdot sdot sdot 119871119899minus1
119903119871119899minus1
1198711+ 119903119871119899minus1
sdot sdot sdot 119871119899minus2
d
1199031198712
1199031198713+ 1199031198712
sdot sdot sdot 1198711
1199031198711
1199031198712+ 1199031198711
sdot sdot sdot 1198710+ 119903119871119899minus1
) (34)
Using Lemma 3 we have
detB =
119899
prod
119896=1
(1198710+ 1198711120576119896+ sdot sdot sdot + 119871
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
[2 + (120572 + 120573) 120576119896+ sdot sdot sdot + (120572
119899minus1
+ 120573119899minus1
) 120576119899minus1
119896]
=
119899
prod
119896=1
minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896minus 2 + 119903119871
119899
1 minus 120576119896minus 1205762
119896
(35)
According to Lemma 4 we obtain
119899
prod
119896=1
[minus119903119871119899minus1
1205762
119896minus (1 + 119903119871
119899+ 119903119871119899minus1
) 120576119896+ 2 minus 119903119871
119899]
= (2 minus 119903119871119899)119899
+ (minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
minus (minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
(36)
Then we get
detB =
(2 minus 119903119871119899)119899
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
+
(minus119903)119899
119871119899minus1
119899minus1(2 minus 119903119871
119899) (119892119899minus1
2+ ℎ119899minus1
2)
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
minus
(minus119903)119899
119871119899minus1
119899minus1[119903119871119899minus1
(119892119899
2+ ℎ119899
2) minus 3119903]
1 minus 119903119871119899minus1
minus 119903119871119899+ (minus1)
119899minus1
1199032
(37)
where
1198922=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
+
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
ℎ2=
1 + 119903119871119899minus1
+ 119903119871119899
minus2119903119871119899minus1
minus
radic1199032(119871119899minus 119871119899minus1
)2
+ 10119903119871119899minus1
+ 2119903119871119899+ 1
minus2119903119871119899minus1
(38)
Using the method in Theorem 8 similarly we also havethe following
Theorem 9 Let B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) Then
detB1015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(39)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(40)
Theorem 10 Let L = RLP119903FLcirc119903fr(1198710 1198711 119871
119899minus1) Then
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(41)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(42)
6 Abstract and Applied Analysis
Proof Thematrix L can be written as
L = (
1198710
sdot sdot sdot 119871119899minus2
119871119899minus1
1198711
sdot sdot sdot 119871119899minus1
+ 1199031198710
1199031198710
d
119871119899minus2
sdot sdot sdot 119903119871119899minus4
+ 119903119871119899minus3
119903119871119899minus3
119871119899minus1
minus 1199031198710sdot sdot sdot 119903119871
119899minus3+ 119903119871119899minus2
119903119871119899minus2
)
= (
119871119899minus1
119871119899minus2
sdot sdot sdot 1198710
1199031198710
119871119899minus1
+ 1199031198710
sdot sdot sdot 1198711
d
119903119871119899minus3
119903119871119899minus4
+ 119903119871119899minus3
sdot sdot sdot 119871119899minus2
119903119871119899minus2
119903119871119899minus3
+ 119903119871119899minus2
sdot sdot sdot 119871119899minus1
+ 1199031198710
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = B1015840
Γ
(43)
Thus we have
detL = detB1015840 det Γ (44)
where matrix B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) and its
determinant can be obtained fromTheorem 9
det1198611015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(45)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(46)
In addition
det Γ = (minus1)119899(119899minus1)2 (47)
so the determinant of matrix L is
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(48)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(49)
5 Determinants of the RFP119903L119903R and RLP119903F119903LCirculant Matrix with the Pell Numbers
Theorem 11 If C = RFP119903LRcirc119903fr(1198750 1198751 119875
119899minus1) then
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(50)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(51)
Proof Thematrix C can be written as
C =(
(
1198750
1198751
sdot sdot sdot 119875119899minus1
119903119875119899minus1
1198750+ 119903119875119899minus1
1198751
119875119899minus2
119903119875119899minus1
+ 119903119875119899minus2
d
1199031198752
sdot sdot sdot 1198751
1199031198751
1199031198752+ 1199031198751
sdot sdot sdot 1198750+ 119903119875119899minus1
)
)119899times119899
(52)
Using Lemma 3 the determinant of C is
detC =
119899
prod
119896=1
(1198750+ 1198751120576119896+ sdot sdot sdot + 119875
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
1205721minus 1205731
1205721minus 1205731
120576119896+ sdot sdot sdot +
120572119899minus1
1minus 120573119899minus1
1
1205721minus 1205731
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119875119899minus1
1205762
119896+ (1 minus 119903119875
119899minus1minus 119903119875119899) 120576119896minus 119903119875119899
1 minus 2120576119896minus 1205762
119896
(53)
Abstract and Applied Analysis 7
According to Lemma 4 we can get
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(54)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(55)
Using the method in Theorem 11 similarly we also havethe following
Theorem 12 If C1015840 = RFP119903LRcirc119903fr(119875119899minus1
119875119899minus2
1198750) then
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(56)
Theorem 13 If P = RLP119903FLcirc119903fr(1198750 1198751 119875
119899minus1) then one
has
detP =
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(57)
Proof Thematrix P can be written as
P =(
(
1198750
sdot sdot sdot 119875119899minus2
119875119899minus1
1198751
c 119875119899minus1
+ 1199031198750
1199031198750
c 1199031198750+ 1199031198751
119875119899minus2
c 119903119875
119899minus3
119875119899minus1
+ 1199031198750sdot sdot sdot 119903119875
119899minus3+ 119903119875119899minus2
119903119875119899minus2
)
)
= (
119875119899minus1
119875119899minus2
sdot sdot sdot 1198750
1199031198750
119875119899minus1
+ 1199031198750
sdot sdot sdot 1198751
d
119903119875119899minus2
119903119875119899minus3
+ 119903119875119899minus2
sdot sdot sdot 119875119899minus1
+ 1199031198750
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
)
(58)
Then we can get
detP = detC1015840 det Γ (59)
where C1015840 = RFPLRcircfr(119875119899minus1
119875119899minus2
1198750) and its determi-
nant could be obtained throughTheorem 12 namely
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(60)
det Γ = (minus1)119899(119899minus1)2 (61)
So
detP = detC1015840 det Γ
=
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(62)
6 Determinants of the RFP119903L119903R andRLP119903F119903L Circulant Matrix with thePell-Lucas Numbers
Theorem 14 If D = RFP119903LRcirc119903fr(1198760 1198761 119876
119899minus1) then
one has
detD =
(2 minus 119903119876119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
(2 minus 119903119876119899) (119892119899minus1
5+ ℎ119899minus1
5) minus 119903119876
119899minus1(119892119899
5+ ℎ119899
5) minus 4119903
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899
119876119899minus1
119899minus1
(63)
where
1198925=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
+
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
ℎ5=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
minus
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
(64)
Proof Themethod is similar to Theorem 11
Certainly we can get the following theorem
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Proof Thematrix L can be written as
L = (
1198710
sdot sdot sdot 119871119899minus2
119871119899minus1
1198711
sdot sdot sdot 119871119899minus1
+ 1199031198710
1199031198710
d
119871119899minus2
sdot sdot sdot 119903119871119899minus4
+ 119903119871119899minus3
119903119871119899minus3
119871119899minus1
minus 1199031198710sdot sdot sdot 119903119871
119899minus3+ 119903119871119899minus2
119903119871119899minus2
)
= (
119871119899minus1
119871119899minus2
sdot sdot sdot 1198710
1199031198710
119871119899minus1
+ 1199031198710
sdot sdot sdot 1198711
d
119903119871119899minus3
119903119871119899minus4
+ 119903119871119899minus3
sdot sdot sdot 119871119899minus2
119903119871119899minus2
119903119871119899minus3
+ 119903119871119899minus2
sdot sdot sdot 119871119899minus1
+ 1199031198710
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
) = B1015840
Γ
(43)
Thus we have
detL = detB1015840 det Γ (44)
where matrix B1015840 = RFP119903LRcirc119903fr(119871119899minus1
1198710) and its
determinant can be obtained fromTheorem 9
det1198611015840 =(minus119903 minus 119871
119899minus1)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(45)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(46)
In addition
det Γ = (minus1)119899(119899minus1)2 (47)
so the determinant of matrix L is
detL =
(minus119903 minus 119871119899minus1
)119899
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
(119903 + 119871119899minus1
) (119892119899minus1
3+ ℎ119899minus1
3)
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
(minus1)119899(119899minus1)2
+
2119899minus1
119903119899
[minus2119903 (119892119899
3+ ℎ119899
3) + 119903 (119871
119899minus 119871119899minus1
)]
(minus1)119899
+ 119903119871119899minus1
minus 119903119871119899+ 1199032
times (minus1)119899(119899minus1)2
(48)
where
1198923=
119871119899minus 119903 + radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
ℎ3=
119871119899minus 119903 minus radic(119903 minus 119871
119899)2
+ 8119903 (119903 + 119871119899minus1
)
4119903
(49)
5 Determinants of the RFP119903L119903R and RLP119903F119903LCirculant Matrix with the Pell Numbers
Theorem 11 If C = RFP119903LRcirc119903fr(1198750 1198751 119875
119899minus1) then
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(50)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(51)
Proof Thematrix C can be written as
C =(
(
1198750
1198751
sdot sdot sdot 119875119899minus1
119903119875119899minus1
1198750+ 119903119875119899minus1
1198751
119875119899minus2
119903119875119899minus1
+ 119903119875119899minus2
d
1199031198752
sdot sdot sdot 1198751
1199031198751
1199031198752+ 1199031198751
sdot sdot sdot 1198750+ 119903119875119899minus1
)
)119899times119899
(52)
Using Lemma 3 the determinant of C is
detC =
119899
prod
119896=1
(1198750+ 1198751120576119896+ sdot sdot sdot + 119875
119899minus1120576119899minus1
119896)
=
119899
prod
119896=1
(
1205721minus 1205731
1205721minus 1205731
120576119896+ sdot sdot sdot +
120572119899minus1
1minus 120573119899minus1
1
1205721minus 1205731
120576119899minus1
119896)
=
119899
prod
119896=1
minus119903119875119899minus1
1205762
119896+ (1 minus 119903119875
119899minus1minus 119903119875119899) 120576119896minus 119903119875119899
1 minus 2120576119896minus 1205762
119896
(53)
Abstract and Applied Analysis 7
According to Lemma 4 we can get
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(54)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(55)
Using the method in Theorem 11 similarly we also havethe following
Theorem 12 If C1015840 = RFP119903LRcirc119903fr(119875119899minus1
119875119899minus2
1198750) then
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(56)
Theorem 13 If P = RLP119903FLcirc119903fr(1198750 1198751 119875
119899minus1) then one
has
detP =
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(57)
Proof Thematrix P can be written as
P =(
(
1198750
sdot sdot sdot 119875119899minus2
119875119899minus1
1198751
c 119875119899minus1
+ 1199031198750
1199031198750
c 1199031198750+ 1199031198751
119875119899minus2
c 119903119875
119899minus3
119875119899minus1
+ 1199031198750sdot sdot sdot 119903119875
119899minus3+ 119903119875119899minus2
119903119875119899minus2
)
)
= (
119875119899minus1
119875119899minus2
sdot sdot sdot 1198750
1199031198750
119875119899minus1
+ 1199031198750
sdot sdot sdot 1198751
d
119903119875119899minus2
119903119875119899minus3
+ 119903119875119899minus2
sdot sdot sdot 119875119899minus1
+ 1199031198750
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
)
(58)
Then we can get
detP = detC1015840 det Γ (59)
where C1015840 = RFPLRcircfr(119875119899minus1
119875119899minus2
1198750) and its determi-
nant could be obtained throughTheorem 12 namely
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(60)
det Γ = (minus1)119899(119899minus1)2 (61)
So
detP = detC1015840 det Γ
=
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(62)
6 Determinants of the RFP119903L119903R andRLP119903F119903L Circulant Matrix with thePell-Lucas Numbers
Theorem 14 If D = RFP119903LRcirc119903fr(1198760 1198761 119876
119899minus1) then
one has
detD =
(2 minus 119903119876119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
(2 minus 119903119876119899) (119892119899minus1
5+ ℎ119899minus1
5) minus 119903119876
119899minus1(119892119899
5+ ℎ119899
5) minus 4119903
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899
119876119899minus1
119899minus1
(63)
where
1198925=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
+
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
ℎ5=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
minus
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
(64)
Proof Themethod is similar to Theorem 11
Certainly we can get the following theorem
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
According to Lemma 4 we can get
detC =
(minus119903119875119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
[119875119899(119892119899minus1
4+ ℎ119899minus1
4) + 119875119899minus1
(119892119899
4+ ℎ119899
4) minus 1]
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899+1
119875119899minus1
119899minus1
(54)
where
1198924=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
+
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
ℎ4=
119903119875119899minus1
+ 119903119875119899minus 1
minus2119903119875119899minus1
minus
radic1199032(119875119899minus 119875119899minus1
)2
minus 2119903 (119875119899+ 119875119899minus1
) + 1
minus2119903119875119899minus1
(55)
Using the method in Theorem 11 similarly we also havethe following
Theorem 12 If C1015840 = RFP119903LRcirc119903fr(119875119899minus1
119875119899minus2
1198750) then
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(56)
Theorem 13 If P = RLP119903FLcirc119903fr(1198750 1198751 119875
119899minus1) then one
has
detP =
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(57)
Proof Thematrix P can be written as
P =(
(
1198750
sdot sdot sdot 119875119899minus2
119875119899minus1
1198751
c 119875119899minus1
+ 1199031198750
1199031198750
c 1199031198750+ 1199031198751
119875119899minus2
c 119903119875
119899minus3
119875119899minus1
+ 1199031198750sdot sdot sdot 119903119875
119899minus3+ 119903119875119899minus2
119903119875119899minus2
)
)
= (
119875119899minus1
119875119899minus2
sdot sdot sdot 1198750
1199031198750
119875119899minus1
+ 1199031198750
sdot sdot sdot 1198751
d
119903119875119899minus2
119903119875119899minus3
+ 119903119875119899minus2
sdot sdot sdot 119875119899minus1
+ 1199031198750
)
times(
0 0 sdot sdot sdot 0 1
0 sdot sdot sdot 0 1 0
c c c
0 1 0 sdot sdot sdot 0
1 0 0 sdot sdot sdot 0
)
(58)
Then we can get
detP = detC1015840 det Γ (59)
where C1015840 = RFPLRcircfr(119875119899minus1
119875119899minus2
1198750) and its determi-
nant could be obtained throughTheorem 12 namely
detC1015840 =(119903 minus 119875
119899minus1)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
(60)
det Γ = (minus1)119899(119899minus1)2 (61)
So
detP = detC1015840 det Γ
=
(119903 minus 119875119899minus1
)119899
minus (119875119899minus 119903)119899minus1
(119903119875119899minus 119903119875119899minus1
)
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032
times (minus1)119899(119899minus1)2
(62)
6 Determinants of the RFP119903L119903R andRLP119903F119903L Circulant Matrix with thePell-Lucas Numbers
Theorem 14 If D = RFP119903LRcirc119903fr(1198760 1198761 119876
119899minus1) then
one has
detD =
(2 minus 119903119876119899)119899
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
+
(2 minus 119903119876119899) (119892119899minus1
5+ ℎ119899minus1
5) minus 119903119876
119899minus1(119892119899
5+ ℎ119899
5) minus 4119903
1 minus 119903119876119899minus1
minus 119903119876119899+ 2(minus1)
119899minus1
1199032
times (minus119903)119899
119876119899minus1
119899minus1
(63)
where
1198925=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
+
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
ℎ5=
2 + 119903119876119899minus1
+ 119903119876119899
minus2119903119876119899minus1
minus
radic1199032(119876119899minus 119876119899minus1
)2
+ 12119903119876119899minus1
+ 4119903119876119899+ 4
minus2119903119876119899minus1
(64)
Proof Themethod is similar to Theorem 11
Certainly we can get the following theorem
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
Theorem 15 If D1015840 = RFP119903LRcirc119903fr(119876119899minus1
1198761 1198760) then
one gets
detD1015840 =(minus2119903 minus 119903119876
119899minus1)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032 (65)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(66)
Theorem 16 IfQ = RLP119903LFcirc119903fr(1198760 1198761 119876
119899minus1) then
detQ =
(minus2119903 minus 119903119876119899minus1
)119899
minus 2119899minus1
119903119899
119870
(minus1)119899
+ 119903119876119899minus1
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of