International Journal of Pure and Applied Sciences and Technology

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1 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), pp International Journal of Pure and Applied Sciences and Technology ISSN Available online at Research Paper Development of a Summation Formula Using Gauss Theorem Salahuddin P.D.M College of Engineering, Bahadurgarh, Haryana, India * Corresponding author, (sludn@yahoo.com) (Received: 17212; Accepted: 14312) Abstract: The main object of present paper is to obtain one summation formula involving Contiguous Relation, Recurrence relation, Gauss second summation theorem, and Legendre duplication formula. Keywords: Contiguous relation, Recurrence relation, Gauss second summation theorem and Legendre duplication formula. 1. Introduction The Pochhammer s Symbol is defined by (α, k ) =(α) k = = Generalized Gaussian Hypergeometric Function of One Variable is defined by (1) AF B (a 1,a 2,,a A ;b 1,b 2, b B ;z ) = (2)

2 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), where the parameters b 1, b 2,.,b B are neither zero nor negative integers and A, B are non negative integers. The series converges for all finite z if A B, converges for z <1 if A=B1, diverges for all z, z 0 if A > B1. Contiguous Relation is defined by [Andrews s p.367 (8), E. D. p.52 (19), H.T. F. I p.103 (38)] (ab) (1z) 2 F 1 (a, b ; c; z) = (cb) 2 F 1 (a,b1; c; z) (ac) 2 F 1 (a1,b; c; z) (3) Recurrence Relation : Γ(ζ1) = ζ Γ(ζ) (4) Legendre s Duplication Formula 2 2z1, z 0,1, 2,. (5) 2 b1 (6) Following Prudnikov, Brychkov & Marichev [ **p.491.( )], we have 2F 1 (a, b ; ; ) = [ ] (7) Now involving Legendre s duplication formula and Recurrence relation of Gamma function, The above formula can be written in the form 2F 1 (a, b ; ; ) = 2 b1 2 ab1 ] (8) It is noted that the above formula Prudnikov, Brychkov & Marichev [ **p.491.( )], i.e. equation (7) or (8) is not correct. The correct form of equation(7) or (8) is obtained Ashish,Rahul & Salahuddin.[** p.337 (10)]

3 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), F 1 (a, b ; ; ) = 2 b1 { } 2 ] (9) 2. Main Result of Summation Formula For a < 1 and a > 16 and b 12 2F 1 (a, b ; ; ) =2 b1 [ { }

4 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), { }] (10) 3. Derivation of the Summation Formula (10) Substituting c = and z = in Eq. (3), we get (ab) 2 F 1 (a, b; ; ) =(ab16) 2 F 1 (a, b1 ; ; )(ab16) 2 F 1 (a1, b ; ; Now involving the derived result from (10), we get 2F 1 (a, b ; ; ) =2 b1 [ {

5 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), } {

6 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), }] ) 2 b1 [ { } {

7 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), }] =2 b1 [ { } {

8 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), }] Thus, we prove the result (10) 4. Conclusion In this paper we have derived a summation formula with the help of contiguous relation. However, the formula presented herein may be further developed to extend this result.thus we can only hope that the development presented in this work will stimulate further interest and research in this important area of classical special functions. Just as the mathematical properties of the Gauss hypergeometric function are already of immense and significant utility in mathematical sciences and numerous other areas of pure and applied mathematics, the elucidation and discovery of the formula of hypergeometric functions considered herein should certainly eventually prove useful to further developments in the broad areas alluded to above. References [1] A. Arora, R. Singh and 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), [2] J. Choi, H. Harsh and A.K. Rathie, Some summation formulae for the Apple s function F1, East Asian Math. Journal, 17(2001),

9 Int. J. Pure Appl. Sci. Technol., 9(1) (2012), [3] A. Erd elyi, W. Magnus, F. Okerhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol.1 (Bateman Manuscript Project), McGrawHill book P. Inc. New York, Toronto and London, [4] K.D. Krupnikov and K.S. K olbig, Some special cases of the generalized hypergeometric function q1fq, Journal of Computational and Applied Math., 78(1997), [5] J.L. Lavoie, Notes on a paper by J. B. Miller, J. Austral. Math. Soc. Ser. B, 29(1987), [6] J.L. Lavoie, Some summation formulae for the series 3F2, Math. Comput., 49(1987), [7] J.L. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Watson s theorem on the sum of a 3F2, Indian J. Math., 34(1992), [8] J.K. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Whipple s theorem on the sum of a 3F2, J. Comput. Appl. Math., 72(1996), [9] J.L. Lavoie, F. Grondin and A.K. Rathie and K. Arora, Generalizations of Dixon s theorem on the sum of a 3F2, Math. Comput., 62(1994), [10] S. C. Mitra.; On the Expansion of the Weierstrassian and Jacobian Elliptic Functions in Powers of the Argument, Bull. Calcutta, Math. Soc., 17(1926), [11] A.P. Prudnikov, A. Yu. 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, [12] E.D. Rainville, The contiguous function relations for pfq with applications to Bateman s Ju,v n and Rice s H n (ζ, ρ, υ), Bull. Amer. Math. Soc., 51(1945), [13] S.S. Sharma and A.K. Rathie, Some summation formulae for the Apple s Function F1, Proc. of the fourth Int. Conf. SSFA, 4(2003), 8184.