# Creation of a summation formula enmeshed with contiguous relation.

[section]1. Introduction

Generalized Gaussian hypergeometric function of one variable is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where the parameters [b.sub.1], [b.sub.2], ... ,[b.sub.B] are neither zero nor negative integers and A, B are nonnegative integers.

Definition 1.1. Contiguous relation W is defined as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Definition 1.2. Recurrence relation of gamma function is defined as follows

[GAMMA](z + 1) - z[GAMMA](z). (3)

Definition 1.3. Legendre duplication formula t3l is defined as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Definition 1.4. Bailey summation theorem M is defined as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[section]2. Main results of summation formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Derivation of result (8):

Putting

b = -a -47, z = -1/2

in established result (2), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now proceeding same parallel method which is applied in , we can prove the main formula.

References

 L. C. Andrews (1992), Special Function of mathematics for Engineers,second Edition, McGraw-Hill Co Inc., New York.

 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(2008), 335-342.

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

 A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, 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.

 E. D. Rainville, The contiguous function relations for [sub.p][F.sub.q] with applications to Bateman's Juv and Rice's [H.sub.n] ([zeta],p,v), Bull. Amer. Math. Soc., 51(1945), 714-723.

 Salahuddin, M. P. Chaudhary, A New Summation Formula Allied With Hypergeometric Function, Global Journal of Science Frontier Research, 11(2010), 21-37.

 Salahuddin, Evaluation of a Summation Formula Involving Recurrence Relation, Gen. Math. Notes., 2(2010), 42-59.

Salah Uddin

P. D. M College of Engineering, Bahadurgarh, Haryana, India

E-mail: sludn@yahoo.com vsludn@gmail.com