# 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 [6], we can prove the main formula.

References

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

[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(2008), 335-342.

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

[4] 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.

[5] 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.

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

[7] 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