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A Summation Formula of Half Argument Influence with Recurrence Relation

Salahuddin1*

1P.D.M. College of Engineering, Bahadurgarh, Haryana, INDIA

Research Article

Abstract: The main aim of the present paper is to evaluate a summation formula of half argument associated with contiguous relation and recurrence relation of gamma function.

Key words: Gaussian Hypergeometric function, Recurrence relation, Bailey summation theorem, Contiguous relation.

2010 MSC NO: 33C05 , 33C20 , 33C45, 33D50 , 33D60.

Introduction:

Generalized Gaussian Hypergeometric function of one variable is defined by

AFB(,a2,…,aA;b1,b2,…bB;z ) =      …(1)

where the parameters  b1 , b2 , ….,bB are neither zero nor negative integers and A , B are non negative integers.

Contiguous Relation is defined by

[Andrews p.363(9.16)]

(a-b) 2F1(a, b ; c; z) = a 2F1(a+1,b; c; z) - b 2F1(a,b+1; c; z) …(2)

Recurrence relation is defined by

Γ(ξ+1)  = ξ Γ(ξ) …(3)

Bailey summation theorem[Prud, p.491(7.3.7.3)

2F1(a, 1-a ; ; ) =  =         …(4)

Main Summation Formula:

2F1(a, -a-40 ;; ) = [

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Derivations of main result:

Putting b= -a-40 , z= in known result (2) , we get

(2a+40) 2F1(a, -a-40 ; c; ) = a 2F1(a+1,-a-40; c; ) + (a+40) 2F1(a,-a-39; c; )

Now involving the same parallel method of Ref [5], the main result is derived.

References:

[1]   Andrews, L.C., Special Function of mathematics for Engineers,Second Edition, McGraw-Hill Co Inc., New York, 1992.

[2]    Arora, Asish, Singh, Rahul , Salahuddin. ; Development of a family of summation formulae of half argument using Gauss and Bailey theorems  ,Journal of Rajasthan Academy of Physical Sciences., 7, 335-342, 2008.

[3]   Bells, Richard, Wong, Roderick; Special Functions, A Graduate Text. Cambridge Studies in Advanced Mathematics, 2010.

[4]   Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O.I.; Integrals and Series Vol. 3: More Special Functions. Nauka, 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]   Salahuddin, Chaudhary, M.P ; A New Summation Formula Allied With Hypergeometric Function, Global Journal of Science Frontier Research,11,21- 37, 2010.