-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
1/9
World Applied Programming, Vol (2), No (2), February 2012.
116-124
ISSN: 2222-2510
2011 WAP journal. www.waprogramming.com
116
Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
Salahuddin
Department of Applied Sciences
P.D.M College of Engineering
Bahadurgarh , Haryana , India
[email protected], [email protected]
Abstract: The main objective of this paper is to evaluate two
summation formulas associated to Contiguous
Relation and involving Recurrence relation.
Keywords: Pochhammers symbol, Contiguous relation, Recurrence
relation.
2010 MSC No: 33C05, 33C45, 33D50, 33D60
I. INTRODUCTION
A. The Pochhammers symbol
(, k) = ()k==
(1)
B. Generalized Gaussian Hypergeometric function of one
variable
AFB( ,a2,,aA;b1,b2,bB;z ) = (2)
or
AFB((aA);(bB);z) AFB((aj)Aj=1;(bj)
Bj=1; z) = (3)
where the parameters b1, b2, .,bBare neither zero nor negative
integers and A , B are non negative integers.
C. Contiguous Relations
[Andrewss p.363 (9.16), E.D. p.51 (10), H.T.F.I. p.103 (32)]
(a-b) 2F1(a, b; c; z) = a 2F1(a+1, b; c; z) - b 2F1(a, b+1; c;
z) (4)
[Abramowitz p.558 (15.2.19)]
(a-b) (1-z) 2F1(a, b; c; z) = (c-b) 2F1(a, b-1; c; z) + (a-c)
2F1(a-1, b; c; z) (5)
D. A New Summation Formula
[Ref.[2] p.337(10)]
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
2/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
117
2F1(a, b ; ; ) = 2b-1 { }+ 2 ]
(6)
E. Recurrence relation
(z+1) = z (z) (7)
II. MAIN SUMMATION FORMULAS
For both the formulas a b
2F1(a, b ; ; ) =2b-1 [ {
+
+
+
+
+ }+
{
+
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
3/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
118
+
+
+
+ }] (8)
2F1(a, b ; ; ) =2b-1 [ {
+
+
+
+
+
+ }+ {
+
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
4/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
119
+
+
+
+
+
+ }] (9)
III. DERIVATIONS OF SUMMATION FORMULAE (8) TO (9):
Derivation of (7): Substituting c = and z = in equation(4) , we
get
(a-b) 2F1(a, b; ; ) =(a-b-15) 2F1(a , b-1 ; ; ) + (a-b+15)
2F1(a-1 , b ; ;
Now with the help of the derived result from equation (6) , we
get
L.H.S = (a-b-15) 2b-2 [
{
+
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
5/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
120
+
+
+ }
+
{
+
+
+
+ }]
+ (a-b+15) 2b-1
[ {
+
+
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
6/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
121
+
+
+ }
+ {
+
+
+
+
+ }]
= 2b-1 [
{
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
7/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
122
+
+
+
+ }
+
{
+
+
+
+
}]
+ 2b-1
[ {
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
8/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
123
+
+
+
+
+ }
+ {
+
+
+
+
+ }]
On simplification, we get the formula (8)
Similarly, we can prove the formula (9).
-
8/10/2019 Two Summation Formulas Related to Contiguous Relation
and Involving Hypergeometric Function
9/9
Salahuddin, World Applied Programming, Vol (2), No (2), February
2012.
124
REFERENCES
[1] Abramowitz, A. Milton, and Irene Stegun, Handbook of
Mathematical Functions with formulas , graphs , and
mathematical Tables, National Bureau of Standards, 1970.
[2] Asish Arora, , Rahul Singh, Salahuddin , Development of a
family of summation formulae of half argument using
Gauss and Bailey theorems, Journal of Rajasthan academy of
Physical Sciences 7(2008), 335-342
[3] J.L. Lavoie, Some summation formulae for the series 3F2Math.
Comput. , 49(1987), 269-274
[4] A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev,
Integrals and Series Vol.3: More Special FunctionsNauka, Moscow,
1986. Translated from the Russian by G.G. Gould, Gordon and Breach
Science Publishers, New
York, Philadelphia,London, Paris, Montreux, Tokyo, Melbourne,
1990.
[5] E. D. Rainville,The contiguous function relations for
pFqwith applications to Batemans Jnu, vand Rices Hn(_,
p, ) Bull. Amer. Math. Soc., 51(1945), 714-723.
![FINDING IDENTITIES WITH THE WZ METHODpeople.brandeis.edu/~gessel/homepage/papers/WZ.pdfindefinite summation algorithm [5], for proving identities for hypergeometric series. (See also](https://img.pdfslide.us/doc/110x75/5f1f56b0e9df115aa0181f8e/finding-identities-with-the-wz-gesselhomepagepaperswzpdf-indeinite-summation.jpg)
