Summation formula for generalized discrete q-Hermite II ... · polynomials (QOP) (also called q-orthogonal polynomials) constitute an interesting set of special functions. They appear

Introduction and motivationOutline

Summation formula for generalized discreteq-Hermite II polynomials

AIMS-Volkswagen Workshop, DoualaOctober 5-12, 2018

African Institute for Mathematical Sciences, CameroonBy

Sama ArjikaFaculty of Sciences and Technics

University of Agadez, Niger

12 septembre 2018Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials


Introduction and motivation

The classical orthogonal polynomial (COP) and the quantum orthogonalpolynomials (QOP) (also called q-orthogonal polynomials) constitute aninteresting set of special functions. They appear in

1 several branches of sciences such as : continued fractions, Eulerianseries, theta functions, elliptic functions,· · · [Andrews (1986), Fine(1988)],

2 quantum groups and quantum algebras [Gasper and Rahman (1990),Koornwinder (1990) and (1994), Nikiforov et al (1991), Vilenkin andKlimyk (1992)].

They have been intensively studied in the last years by several people,[Koekoek and Swarttouw (1998), Lesky (2005), Koekoek et al (2010)],· · · .

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials



Each family of COP and QOP occupy different levels within theso-called, Askey-Wilson scheme and are characterized by the properties :

1 they are solutions of a hypergeometric second order differentialequation,

2 they are generated by a recursion relation,

3 they are orthogonal with respect to a weight function,

4 they obey the Rodrigues-type formula.

In this scheme, the Hermite polynomials are the ground level and most ofthere properties can be generalized.




In their paper, Alvarez-Nodarse et al [Int. J. Pure. Appl. Math. 10 (3)331-342 (2014)], have introduced a q-extension of the discrete q-HermiteII polynomials as :

H(µ)2n (x ; q) : = (−1)n(q; q)n L

(µ−1/2)n (x2; q)

(1)

H(µ)2n+1(x ; q) : = (−1)n(q; q)n x L

(µ+1/2)n (x2; q)

where µ > −1/2, L(α)n (x ; q) are the q-Laguerre polynomials given by

L(α)n (x ; q) :=(qα+1; q)n

(q; q)n1φ1

(q−n

qα+1

∣∣∣q;−qn+α+1x

). (2)

For µ = 0 in (1), the polynomials H(0)n (x ; q) correspond to the discrete

q-Hermite II polynomial

H(0)n (x ; q2) = qn(n−1)hn(x ; q).




Alvarez-Nodarse et al showed that the polynomials H(µ)n (x ; q) satisfy the

orthogonality relation∫ ∞−∞H(µ)

n (x ; q)H(µ)m (x ; q)ω(x)dx = π q−n/2(q1/2; q1/2)n(q1/2; q)1/2 δnm

on the whole real line R with respect to the positive weight functionω(x) = 1/(−x2; q)∞. A detailed discussion of the properties of the

polynomials H(µ)n (x ; q) can be found in [Int. J. Pure. Appl. Math. 10 (3)

331-342 (2014)].




Recently, Saley Jazmat et al [Bulletin of Mathematical Ana. App. 6(4),16-43 (2014)], introduced a novel extension of discrete q-Hermite IIpolynomials by using new q-operators. This extension is defined as :

h2n,α(x ; q) = (−1)n q−n(2n−1)(q; q)2n

(q2α+2; q2)nL(α)n

(x2q−2α−1; q2

)(3)

h2n+1,α(x ; q) = (−1)n q−n(2n+1) (q; q)2n+1

(q2α+2; q2)n+1x L(α+1)

n

(x2q−2α−1; q2

).

For α = −1/2 in (3), the polynomials hn,− 12(x ; q) correspond to the

discrete q-Hermite II polynomials, i.e.,

hn,− 12(x ; q) = hn(x ; q).




The generalized discrete q-Hermite II polynomials hn,α(x ; q) satisfy theorthogonality relation∫ +∞

−∞hn,α(x ; q)hm,α(x ; q)ωα(x ; q)|x |2α+1dqx

=2q−n

2

(1− q)(−q,−q, q2; q2)∞(−q−2α−1,−q2α+3, q2α+2; q2)∞

(q; q)2n(q; q)n,α

δn,m (4)

on the real line R with respect to the positive weight functionωα(x) = 1/(−q−2α−1 x2; q2)∞.




Motivated by Saley Jazmat’s work [Bul. Math. Anal. App. 6(4), 16-43(2014)], our interest in this work is

1 to introduce new family of “generalized discrete q-Hermite IIpolynomials (in short gdq-H2P) hn,α(x , y |q)” which is an extension

of the generalized discrete q-Hermite II polynomials hn,α(x ; q),

2 and investigate summation formula.



Outline

1 Notations and definitions2 Generalized discrete q-Hermite II polynomials {hn,α(x , y |q)}∞n=03 Connection formulae for the generalized discrete q-Hermite II

polynomials {hn,α(x , y |q)}∞n=0



Outline





Outline




Notations and definitionsGeneralized discrete q-Hermite II polynomials {hn,α(x, y|q)}∞n=0

Connection formulae for the generalized discrete q-Hermite II polynomials {hn,α(x, y|q)}∞n=0Conclusion

Notations and definitions

Throughout this paper, we assume that 0 < q < 1, α > −1. For acomplex number a,

F the q-shifted factorials are defined by :

(a; q)0 = 1; (a; q)n =n−1∏k=0

(1− aqk), n ≥ 1; (a; q)∞ =∞∏k=0

(1− aqk).

F The q-number is defined by :

[n]q =1− qn

1− q, n!q :=

n∏k=1

[k]q, 0!q := 1, n ∈ N. (5)





Hahn q-addition and q-subtraction

For x , y ∈ R,

F the Hahn q-addition ⊕q is defined by :(x ⊕q y

)n: = (x + y)(x + qy) . . . (x + qn−1y)

= (q; q)n

n∑k=0

q(k2)xn−kyk

(q; q)k(q; q)n−k, n ≥ 1, (6)

and(x ⊕q y

)0:= 1.

F The q-subtraction q is given by(x q y

)n:=(x ⊕q (−y)

)n(7)

and(x q y

)0:= 1.





1 The generalized backward and forward q-derivative operators Dq,α

and D+q,α, Saley Jazmat et al are defined :

Dq,αf (x) =f (x)− q2α+1f (qx)

(1− q)x, D+

q,αf (x) =f (q−1x)− q2α+1f (x)

(1− q)x.

2 Remark that, for α = − 12 , we have Dq,α = Dq, D+

q,α = D+q where

Dq and D+q are the Jackson’s q-derivative with

Dqf (x) =f (x)− f (qx)

(1− q)x, D+

q f (x) =f (q−1x)− f (x)

(1− q)x. (8)

3 For f (x) = xn, we have

Dq,αxn = [n]q,αx

n−1, D+q,αx

n = q−n[n]q,αxn−1

where [n]q,α := [n + 2α + 1]q, [n]q,−1/2 = [n]q.Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials



Generalized q-shifted factorials

The generalized q-shifted factorials are defined as :

(n + 1)!q,α = [n + 1 + θn(2α + 1)]q n!q,α (9)

(q; q)n+1,α = (1− q)[n + 1 + θn(2α + 1)]q(q; q)n,α, (10)

where

θn =

{1 if n even0 if n odd.

F Remark that, for α = −1/2, we have

(q; q)n,−1/2 = (q; q)n, n!q,−1/2 =(q; q)n

(1− q)n. (11)

F We denote(q; q)2n,α = (q2; q2)n(q2α+2; q2)n, (12)

(q; q)2n+1,α = (q2; q2)n(q2α+2; q2)n+1. (13)




Generalized q-exponential functions

The two Euler’s q-analogs of the exponential functions are given by

eq(x) :=∞∑n=0

xn

(q; q)n=

1

(x ; q)∞(14)

and

Eq (x) :=∞∑n=0

q(n2)

(q; q)nxn = (−x ; q)∞. (15)

For m ≥ 1, we define two generalized q-exponential functions as follows

Eqm,α(x) :=∞∑k=0

qmk(k−1)/2 xk

(qm; qm)k,α, (16)

and

eqm,α(x) :=∞∑k=0

xk

(qm; qm)k,α, |x | < 1. (17)




Generalized q-exponential functions

The two Euler’s q-analogs of the exponential functions are given by

eq(x) :=∞∑n=0

xn

(q; q)n=

1

(x ; q)∞(14)

and

Eq (x) :=∞∑n=0

q(n2)

(q; q)nxn = (−x ; q)∞. (15)

For m ≥ 1, we define two generalized q-exponential functions as follows

Eqm,α(x) :=∞∑k=0

qmk(k−1)/2 xk

(qm; qm)k,α, (16)

and

eqm,α(x) :=∞∑k=0

xk

(qm; qm)k,α, |x | < 1. (17)




Particular case

Remark that, for m = 1 and α = − 12 , we have :

Eq,− 12(x) = Eq(x), eq,− 1

2(x) = eq(x). (18)

Elementary result

For m = 2, the following elementary result is useful in the sequel toestablish summation formula for gdq-H2P :

eq2,− 12(x)Eq2,− 1

2(y) = eq2,− 1

2(x ⊕q2 y), (19)

eq,− 12(x)Eq2,− 1

2(−y) = eq(x q,q2 y), eq2,− 1

2(x)Eq2,− 1

2(−x) = 1, (20)

where

(aq,q2 b)n := n!q

n∑k=0

(−1)kqk(k−1)

(n − k)!q k!q2

an−kbk , (aq,q2 b)0 := 1.




Particular case

Remark that, for m = 1 and α = − 12 , we have :

Eq,− 12(x) = Eq(x), eq,− 1

2(x) = eq(x). (18)

Elementary result

For m = 2, the following elementary result is useful in the sequel toestablish summation formula for gdq-H2P :

eq2,− 12(x)Eq2,− 1

2(y) = eq2,− 1

2(x ⊕q2 y), (19)

eq,− 12(x)Eq2,− 1

2(−y) = eq(x q,q2 y), eq2,− 1

2(x)Eq2,− 1

2(−x) = 1, (20)

where

(aq,q2 b)n := n!q

n∑k=0

(−1)kqk(k−1)

(n − k)!q k!q2

an−kbk , (aq,q2 b)0 := 1.




Generalized discrete q-Hermite II polynomials

Discrete q-Hermite II polynomials

hn(x |q) := (q; q)n

b n/2 c∑k=0

(−1)kq−2nk+k(2k+1) xn−2k

(q; q)n−2k (q2; q2)k. (21)

For α > −1, we define a sequence of generalized discrete q-Hermite IIpolynomials {hn,α(x , y |q)}∞n=0 as follows :

Definition

For x , y ∈ R, a gdq-H2P{hn,α(x , y |q)}∞n=0 are defined by :

hn,α(x , y |q) := (q; q)n

b n/2 c∑k=0

(−1)kq−2nk+k(2k+1) xn−2k yk

(q; q)n−2k,α (q2; q2)k(22)

andhn,α(x , 0|q) := [(q; q)n/(q; q)n,α]xn. (23)






hn(x |q) := (q; q)n

b n/2 c∑k=0

(−1)kq−2nk+k(2k+1) xn−2k

(q; q)n−2k (q2; q2)k. (21)


Definition


hn,α(x , y |q) := (q; q)n

b n/2 c∑k=0


(q; q)n−2k,α (q2; q2)k(22)







hn(x |q) := (q; q)n

b n/2 c∑k=0

(−1)kq−2nk+k(2k+1) xn−2k

(q; q)n−2k (q2; q2)k. (21)


Definition


hn,α(x , y |q) := (q; q)n

b n/2 c∑k=0


(q; q)n−2k,α (q2; q2)k(22)





Particular cases of gdq-H2H hn,α(x , y |q)

1 For y = 1, we have

hn,α(x , 1|q) = hn,α(x ; q) (24)

where hn,α(x ; q) is the generalized discrete q-Hermite II polynomial.

2 For α = −1/2 and y = 1, we have

hn,−1/2(x , 1|q) = hn(x ; q). (25)

where hn(x ; q) is the discrete q-Hermite II polynomial.

3 Indeed since limq→1

(qa; q)n(1− q)n

= (a)n, one readily verifies that

limq→1

hn,− 12(√

1− q2x , 1|q)

(1− q2)n/2=

hα+ 1

2n (x)

2n(26)

where hα+ 1

2n (x) is the Rosenblums generalized Hermite polynomial.






hn,α(x , 1|q) = hn,α(x ; q) (24)



hn,−1/2(x , 1|q) = hn(x ; q). (25)



(qa; q)n(1− q)n


limq→1

hn,− 12(√

1− q2x , 1|q)

(1− q2)n/2=

hα+ 1

2n (x)

2n(26)

where hα+ 1







hn,α(x , 1|q) = hn,α(x ; q) (24)



hn,−1/2(x , 1|q) = hn(x ; q). (25)



(qa; q)n(1− q)n


limq→1

hn,− 12(√

1− q2x , 1|q)

(1− q2)n/2=

hα+ 1

2n (x)

2n(26)

where hα+ 1






Recursion relation

The recursion relation for gdq-H2P {hn,α(x , y |q)}∞n=0 holds true.

xhn,α(x , y |q)− y q−2n+1(1− qn)hn−1,α(x , y |q) =

1− qn+1+θn(2α+1)

1− qn+1hn+1,α(x , y |q).





Theorem 1

We have :

limα→+∞

h2n,α(x , y |q) = q−n(2n−1)(q; q)2n (−y)n Sn(x2y−1q−1; q2

)(27)

and

limα→+∞

h2n+1,α(x , y |q) = q−n(2n+1)(q; q)2n+1 x (−y)n Sn(x2y−1q−1; q2

)(28)

where Sn(x ; q) are the Stieltjes-Wigert polynomials.





Lemma

For α > −1, the sequence of gdq-H2P {hn,α(x , y |q)}∞n=0 can be written

in terms of q-Laguerre polynomials L(α)n (x ; q) as

h2n,α(x , y |q) = q−n(2n−1)(q; q)2n

(q2α+2; q2)n(−y)n L(α)n

(x2y−1q−2α−1; q2

)(29)

and

h2n+1,α(x , y |q) = q−n(2n+1) (q; q)2n+1

(q2α+2; q2)n+1x (−y)n L(α+1)

n

(x2y−1q−2α−1; q2

).

(30)





Proposition

For α > −1, the sequence of gdq-H2P {hn,α(x , y |q)}∞n=0 can be writtenin terms of basic hypergeometric functions as

hn,α(x , y |q) =(q; q)n

(q; q)n,αxn 2φ1

(q−n, q−n−2α

0

∣∣∣ q2; −y q2α+3

x2

).




Connection formulae for the generalized discrete q-HermiteII polynomials {hn,α(x , y |q)}∞n=0

Theorem 2

The sequence of gdq-H2P {hn,α(x , y |q)}∞n=0, satisfies the connectionformula

hn,α(x , ω|q) = (q; q)n

b n/2 c∑k=0

q−2nk+k(2k+1) (−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2khn−2k,α(x , y |q).

(31)




Proof. Summation formula

To prove the above Theorem 2, we need the following generatingfunction

eq2,− 12(−yt2)Eq,α(xt) =

∞∑n=0

q(n2) tn

(q; q)nhn,α(x , y |q), |yt| < 1. (32)

Replacing t by u ⊕q t in the last generating function, we have

Eq,α

[(u ⊕q t)x

]eq2,− 1

2

[− y(u ⊕q t)2

]=∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q)

(33)which can be written as

Eq,α

[(u ⊕q t)x

]= Eq2,− 1

2

[y(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q).

(34)






eq2,− 12(−yt2)Eq,α(xt) =

∞∑n=0

q(n2) tn

(q; q)nhn,α(x , y |q), |yt| < 1. (32)


Eq,α

[(u ⊕q t)x

]eq2,− 1

2

[− y(u ⊕q t)2

]=∞∑n=0

q(n2)(u ⊕q t)n



Eq,α

[(u ⊕q t)x

]= Eq2,− 1

2

[y(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n


(34)






eq2,− 12(−yt2)Eq,α(xt) =

∞∑n=0

q(n2) tn

(q; q)nhn,α(x , y |q), |yt| < 1. (32)


Eq,α

[(u ⊕q t)x

]eq2,− 1

2

[− y(u ⊕q t)2

]=∞∑n=0

q(n2)(u ⊕q t)n



Eq,α

[(u ⊕q t)x

]= Eq2,− 1

2

[y(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n


(34)






eq2,− 12(−yt2)Eq,α(xt) =

∞∑n=0

q(n2) tn

(q; q)nhn,α(x , y |q), |yt| < 1. (32)


Eq,α

[(u ⊕q t)x

]eq2,− 1

2

[− y(u ⊕q t)2

]=∞∑n=0

q(n2)(u ⊕q t)n



Eq,α

[(u ⊕q t)x

]= Eq2,− 1

2

[y(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n


(34)





Replacing y by ω and using various identities, we get :

∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , ω|q) =

eq2,− 12

[− ω(u ⊕q t)2

]Eq2,− 1

2

[y(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n


The r.h.s of the last expression can be written as

eq2,− 12

[(−ω ⊕q2 y)(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q) (35)

or

∞∑r=0

(−ω ⊕q2 y)r (u ⊕q t)2r

(q2; q2)r

∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q). (36)






∞∑n=0

q(n2)(u ⊕q t)n


eq2,− 12

[− ω(u ⊕q t)2

]Eq2,− 1

2

[y(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n



eq2,− 12

[(−ω ⊕q2 y)(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q) (35)

or

∞∑r=0

(−ω ⊕q2 y)r (u ⊕q t)2r

(q2; q2)r

∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q). (36)






∞∑n=0

q(n2)(u ⊕q t)n


eq2,− 12

[− ω(u ⊕q t)2

]Eq2,− 1

2

[y(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n



eq2,− 12

[(−ω ⊕q2 y)(u ⊕q t)2

] ∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q) (35)

or

∞∑r=0

(−ω ⊕q2 y)r (u ⊕q t)2r

(q2; q2)r

∞∑n=0

q(n2)(u ⊕q t)n

(q; q)nhn,α(x , y |q). (36)





Let us substitute n + 2r = k =⇒ r ≤ b k/2 c in the last equation, weget :

∞∑n=0

b n/2 c∑k=0

(q(n−2k2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2khn−2k,α(x , y |q)

(u ⊕q t)n. (37)

Summarizing the above calculations, we obtain

∞∑n=0

q(n2)(u ⊕q t)n


∞∑n=0

b n/2 c∑k=0

(q(n−2k2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2khn−2k,α(x , y |q)

(u ⊕q t)n. (38)

By equating the coefficients of like powers of (u ⊕q t)n/(q; q)n in thelast equation, we get the desired identity.





Let us substitute n + 2r = k =⇒ r ≤ b k/2 c in the last equation, weget :

∞∑n=0

b n/2 c∑k=0

(q(n−2k2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2khn−2k,α(x , y |q)

(u ⊕q t)n. (37)

Summarizing the above calculations, we obtain

∞∑n=0

q(n2)(u ⊕q t)n


∞∑n=0

b n/2 c∑k=0

(q(n−2k2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2khn−2k,α(x , y |q)

(u ⊕q t)n. (38)

By equating the coefficients of like powers of (u ⊕q t)n/(q; q)n in thelast equation, we get the desired identity.




Connection formulae for the gdq-H2P {hn,α(x , y |q)}∞n=0

Particular cases

Letting :

(i) y = 0 in the assertion of Theorem 2, we get the definition ofgdq-H2P, i.e.,

hn,α(x , ω|q) = (q; q)n

b n/2 c∑k=0

(−1)kq−2nk+k(2k+1) xn−2k ωk

(q2; q2)k (q; q)n−2k,α; (39)

(ii) ω = 0 in the assertion of Theorem 2, we get the inversion formulafor gdq-H2P

xn = (q; q)n,α

b n/2 c∑k=0

q−2nk+3k2

yk

(q2; q2)k (q; q)n−2khn−2k,α(x , y |q). (40)




Conclusion

In this work,

(i) we have introduced gdq-H2P hn,α(x , y |q) and derived severalproperties.

(ii) Also, we have derived implicit summation formula for gdq-H2Phn,α(x , y |q) by using different analytical means on their generatingfunction.

(iii) For y = 1, the assertion of Theorem 2 can be expressed in terms ofgeneralized discrete q-Hermite II polynomials hn,α(x ; q). Theassertion of Theorem 2 can be written in terms of discreteq-Hermite II polynomials hn(x ; q) by choosing y = 1 and α = −1/2.

(iv) This process can be extend to summation formula for moregeneralized forms of q-Hermite polynomials. This study is under way.




Conclusion

In this work,








Conclusion

In this work,








Conclusion

In this work,








Thank you for attention ! ! !


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Summation formula for generalized discrete q-Hermite II ... · polynomials (QOP) (also called q-orthogonal polynomials) constitute an interesting set of special functions. They appear