A Summation Formula Tangled with Hypergeometric Function and Recurrence Relation
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- Θεοφύλακτος Δαγκλής
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1 Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 10 Version 1.0 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA Online ISSN: & Print ISSN: A Summation Formula Tangled with Hypergeometric Function and Recurrence Relation By Salahuddin, M. P. Chaudhary & Rahul Singh P.D.M College of Engineering, India Abstract - The main object of the present paper is to establish a summation formula tangled with Hypergeometric function and recurrence relation. Keywords : Contiguous relation, Gauss second summation theorem, Recurrence relation. GJSFR-F Classification : MSC 2000: 33C05, 33C20, 33C45, 33C60, 33C70 A Summation Formula Tangled with Hypergeometric Function and Recurrence Relation Strictly as per the compliance and regulations of :. Salahuddin, M. P. Chaudhary & Rahul Singh. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
2 A Summation Formula Tangled with Hypergeometric Function and Recurrence Relation Salahuddin α, M. P. Chaudhary σ & Rahul Singh ρ 55 Abstract - The main object of the present paper is to establish a summation formula tangled with Hypergeometric function and recurrence relation. Keywords and Phrases : Contiguous relation, Gauss second summation theorem, Recurrence relation. a Generalized Hypergeometric Functions I. Introduction A generalized hypergeometric function p F q (a 1,...a p ; b 1,...b q ; z is a function which can be defined in the form of a hypergeometric series, i.e., a series for which the ratio of successive terms can be written c k+1 c k = P(k Q(k = (k + a 1 (k + a 2...(k + a p (k + b 1 (K + b 2...(k + b q (k + 1 z. (1 Where k + 1 in the denominator is present for historical reasons of notation, and the resulting generalized hypergeometric function is written or pf q pf q a 1, a 2,, a p ; b 1, b 2,, b q ; (a p ; (b q ; z z p F q = k=0 (a j p j=1 ; (b j q j=1 ; (a 1 k (a 2 k (a p k z k (b 1 k (b 2 k (b q k k! z = k=0 ((a p k z k ((b q k k! where the parameters b 1, b 2,, b q are neither zero nor negative integers and p, q are non-negative integers. The p F q series converges for all finite z if p q, converges for z < 1 if p q +1, diverges for all z, z 0 if p > q + 1. (2 (3 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Author α : P.D.M College of Engineering, Bahadurgarh,Haryana, India. sludn@yahoo.com Author σ : International Scientific Research and Welfare Organization, New Delhi, India.. mpchaudhary _ 2000@yahoo.com Author ρ : Greater Noida Institute of Technology, Greater Noida , U.P., India. Global Journals Inc. (US
3 56 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I The p F q series absolutely converges for z = 1 if R(ζ < 0, conditionally converges for p q z = 1, z 0 if 0 R(ζ < 1, diverges for z = 1, if 1 R(ζ, ζ = a i b i. The function 2 F 1 (a, b; c; z corresponding to p = 2, q = 1, is the first hypergeometric function to be studied (and, in general, arises the most frequently in physical problems, and so is frequently known as the hypergeometric equation or, more explicitly, Gauss s hypergeometric function (Gauss 1812, Barnes To confuse matters even more, the term hypergeometric function is less commonly used to mean closed form, and hypergeometric series is sometimes used to mean hypergeometric function. The hypergeometric functions are solutions of Gaussian hypergeometric linear differential equation of second order The solution of this equation is z(1 zy + [c (a + b + 1z]y aby = 0 (4 [ y = A ab a(a + 1b(b + 1 ] z + z 2 + 1! c 2! c(c + 1 This is the so-called regular solution,denoted 2F 1 (a, b; c; z = [ 1 + ab a(a + 1b(b + 1 ] z + z 2 + = 1! c 2! c(c + 1 i=1 i=0 (a k (b k z k k=0 (c k k! which converges if c is not a negative integer for all of z < 1 and on the unit circle z = 1 if R(c a b > 0. It is known as Gauss hypergeometric function in terms of Pochhammer symbol (a k or generalized factorial function. Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are (1 z a = z 2 F 1 (1, 1; 2; z (7 (5 (6 sin 1 z = z 2 F 1 ( 1 2, 1 2 ; 3 2 ; z2 (8 The special case of (1.3.4 when a = c and b = 1, or a = 1 and b = c, yields the elementary geometric series z n = 1 + z + z 2 + z z n + (9 n=0 Hence the term Hypergeometric is given. The term hypergeometric was first used by Wallis in his work Arithmetrica Infinitorum. Hypergeometric series or more precisely Gauss series is due to Carl Friedrich Gauss( who in year 1812 introduced and studied this series in his thesis presented at Gottingen and gave the F-notation for it. Here z is a real or complex variable. If c is zero or negative integer, the series (6 does not exist and hence the function 2 F 1 (a, b; c; z is not defined unless one of the parameters Global Journals Inc. (US
4 a or b is also a negative integer such that c < a. If either of the parameters a or b is a negative integer, say m then in this case (6 reduce to the hypergeometric polynomial defined as m ( m n (b n z n 2F 1 ( m, b; c; z = (10 (c n n! n=0 b Generalized Ordinary Hypergeometric Function of One Variable The generalized Gaussian hypergeometric function of one variable is defined as follows AF B a 1, a 2, a 3,..., a A ; b 1, b 2, b 3,..., b B ; z = n=0 (a 1 n (a 2 n (a 3 n (a A n z n (b 1 n (b 2 n (b 3 n (b B n n! (11 or, AF B (a A ; (b B ; z = n=0 [(a A ] n z n [(b B ] n n! where for the sake of convenience (in the contracted notation, (a A denotes thearray of A number of parameters given by a 1, a 2, a 3,..., a A. The denominatorparameters are neither zero nor negative integers. The numerator parameters may be zero and negative integers. A and B are positive integers or zero. Empty sum is to be interpreted as zero and empty product as unity. b n=a and b n=a [(a A ] n = [(a A ] n = (a 1 n (a 2 n (a 3 n (a A n = are empty if b < a. (12 ( 1nA [1 (a A ] n (13 A (a m n = m=1 A m=1 Γ(a m + n Γ(a m where a 1, a 2, a 3,..., a A ; b 1, b 2, b 3,..., b B and z may be real and complex numbers. 3F 2 { a, b, 1 ; c, 2 ; 2F 1 z = a 1, b 1 ; c 1 ; The convergence conditions of A F B are given below (c 1 (a 1(b 1z z 1 Suppose that numerator parameters are neither zero nor negative integers (otherwise question of convergence will not arise. } (14 (15 57 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Global Journals Inc. (US
5 (i If A B, then series A F B is always convergent for all finite values of z(real or complex i.e., z <. (ii If A = B + 1 and z < 1, then series A F B is convergent. (iii If A = B + 1 and z > 1, then series A F B is divergent. 58 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I (iv If A = B + 1 and z = 1, then series A F B is absolutely convergent, when { B Re m=1 b m A } a n > 0 (v If A = B + 1 and z = 1, then series A F B is convergent, when { B Re m=1 b m n=1 A } a n > 0 (vi If A = B + 1 and z = 1, then series A F B is divergent, when { B Re m=1 b m n=1 A } a n 0 (vii If A = B + 1 and z = 1, then series A F B is convergent, when { B Re m=1 b m n=1 A } a n > 1 (viii If A = B + 1 and z = 1, but z 1, then series A F B is conditionally convergent, when { B 1 < Re m=1 n=1 b m A } a n 0 n=1 (ix If A > B + 1, then series A F B is convergent, when z = 0. (x If A = B + 1 and z 1, then it is defined as an analytic continuation of this series. (xi If A = B + 1 and z = 1, then series A F B is divergent, when { B Re m=1 b m A } a n 1 n=1 Global Journals Inc. (US
6 (xii If A > B + 1, then a meaningful independent attempts were made to define MacRobert s E-function, Meijer s G-function, Fox s H-function and its related functions. (xiii If one or more of the numerator parameters are zero or negative integers, then series A F B terminates for all finite values of z i.e., A F B will be a hypergeometric polynomial and the question of convergence does not enter the discussion. Contiguous Relation is defined by [ E. D. p.51(10] (a b 2 F 1 [ a, b ; c ; ] [ a + 1, b ; z = a 2 F 1 c ; ] [ a, b + 1 ; z b 2 F 1 c ; ] z (16 59 Gauss second summation theorem is defined by [Prud., 491( ] [ a, b ; 2F 1 a+b+1 ; 2 ] 1 = Γ(a+b+1 Γ( Γ( a+1 2 Γ(b+1 (17 2 = 2(b 1 Γ( b 2 Γ(a+b+1 2 Γ(b Γ( a+1 2 (18 In a monograph of Prudnikov et al., a summation theorem is given in the form [Prud., p.491( ] [ ] a, b ; 1 2F 1 a+b 1 = [ Γ( a+b+1 2 π ; 2 2 Γ( a+1 2 Γ(b Γ( ] a+b 1 2 (19 Γ(a Γ(b 2 Now using Legendre s duplication formula and Recurrence relation for Gamma function, the above theorem can be written in the form 2F 1 [ a, b ; a+b 1 2 ; ] 1 = 2(b 1 Γ( a+b Γ(b Recurrence relation is defined by [ Γ( b 2 Γ( a 1 + 2(a b+1 Γ( 2 {Γ(a} 2 a 2 Γ(a Γ(b+2 2 Γ( a+1 2 ] (20 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Γ(z + 1 = z Γ(z (21 Global Journals Inc. (US
7 II. Main Summation Formula 60 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I [ a, b ; 2F 1 a+b+41 2 ; ] 1 = 2b Γ( a+b (a b Γ(b [ Γ( b 2 { a( a Γ( a a( a a 3 [ 19 { } ][ a( a a 5 [ 19 { } ][ a( a a 7 [ 19 { } ][ a( a a a 10 [ 19 { } ][ a( a a a 13 [ 19 { } ][ a( a a a a a 18 + a 19 [ 19 { } ][ a( b ab + [ 19 { } ][ a( a2 b a 3 b [ 19 { } ][ a( a4 b a 5 b [ 19 { } ][ a( a6 b a 7 b [ 19 { } ][ a( a8 b a 9 b a 10 b Global Journals Inc. (US
8 a( a11 b a 12 b a 13 b [ 19 { } ][ a( a14 b a 15 b a 16 b 75582a 1 7b + 741a 18 b [ 19 { } ][ a( b ab 2 [ 19 { } ][ a( a2 b a 3 b 2 [ 19 { } ][ a( a4 b a 5 b 2 [ 19 { } ][ a( a6 b a 7 b 2 [ 19 { } ][ a( a8 b a 9 b 2 [ 19 { } ][ a( a10 b a 11 b a 12 b 2 [ 19 { } ][ a( a13 b a 14 b a 15 b a 16 b 2 [ 19 { } ][ a(82251a17 b b ab 3 [ 19 { } ][ a( a2 b a 3 b 3 [ 19 { } ][ a( a4 b a 5 b 3 [ 19 { } ][ Global Journal of Science Frontier Research F Volume XII Issue X V ersion I a( a6 b a 7 b 3 [ 19 { } ][ 20 Global Journals Inc. (US
9 62 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I a( a8 b a 9 b 3 [ 19 { } ][ a( a10 b a 11 b a 12 b 3 [ 19 { } ][ a( a13 b a 14 b a 15 b a 16 b 3 [ 19 { } ][ a( b ab 4 [ 19 { } ][ a( a2 b a 3 b 4 [ 19 { } ][ a( a4 b a 5 b 4 [ 19 { } ][ a( a6 b a 7 b 4 [ 19 { } ][ a( a8 b a 9 b 4 [ 19 { } ][ a( a10 b a 11 b a 12 b 4 [ 19 { } ][ a( a13 b a 14 b a 15 b 4 [ 19 { } ][ a( b ab 5 [ 19 { } ][ a( a2 b a 3 b 5 [ 19 { } ][ a( a4 b a 5 b 5 [ 19 { } ][ 20 Global Journals Inc. (US
10 a( a6 b a 7 b 5 [ 19 { } ][ a( a8 b a 9 b a 10 b a( a11 b a 12 b a 13 b 5 [ 19 { } ][ a( a14 b b ab 6 [ 19 { } ][ a( a2 b a 3 b 6 [ 19 { } ][ a( a4 b a 5 b 6 [ 19 { } ][ a( a6 b a 7 b 6 [ 19 { } ][ a( a8 b a 9 b a 10 b 6 [ 19 { } ][ a( a11 b a 12 b a 13 b 6 [ 19 { } ][ a( b ab 7 [ 19 { } ][ a( a2 b a 3 b 7 [ 19 { } ][ a( a4 b a 5 b 7 [ 19 { } ][ a( a6 b a 7 b a 8 b 7 63 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Global Journals Inc. (US
11 a( a9 b a 10 b a 11 b 7 [ 19 { } ][ a( a1 2b b ab 8 [ 19 { } ][ Global Journal of Science Frontier Research F Volume XII Issue X V ersion I a( a2 b a 3 b 8 [ 19 { } ][ a( a4 b a 5 b a 6 b a( a7 b a 8 b a 9 b 8 [ 19 { } ][ a( a10 b a 11 b b 9 [ 19 { } ][ a( ab a 2 b a 3 b a( a4 b a 5 b a 6 b a( a7 b a 8 b a 9 b 9 [ 19 { } ][ a( a10 b b ab 10 [ 19 { } ][ a( a2 b a 3 b a 4 b a( a5 b a 6 b a 7 b 10 [ 19 { } ][ a( a8 b a 9 b b 11 [ 19 { } ][ 20 Global Journals Inc. (US
12 a( ab a 2 b a 3 b 11 [ 19 { } ][ a( a4 b a 5 b a 6 b 11 [ 19 { } ][ a( a7 b a 8 b b 12 [ 19 { } ][ a( ab a 2 b a 3 b 12 [ 19 { } ][ a( a4 b a 5 b a 6 b 12 [ 19 { } ][ a( a7 b b ab 13 [ 19 { } ][ a( a2 b a 3 b a 4 b 13 [ 19 { } ][ a( a5 b a 6 b b ab a( a2 b a 3 b a 4 b a 5 b 14 [ 19 { } ][ a( b ab a 2 b a 3 b 15 [ 19 { } ][ a( a4 b b ab a 2 b 16 [ 19 { } ][ a( a3 b b ab a 2 b 17 [ 19 { } ][ Global Journal of Science Frontier Research F Volume XII Issue X V ersion I + [ a(4199b ab b 19 { } ][ 20 { } ]+ Global Journals Inc. (US
13 524288b( a b( a a b( a a 5 66 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I b( a a b( a a a b( a a a b( a a a a a b(39a b ab b( a2 b a 3 b b( a4 b a 5 b b( a6 b a 7 b b( a8 b a 9 b a 10 b b( a11 b a 12 b a 13 b Global Journals Inc. (US
14 b( a14 b a 15 b a 16 b a 17 b b(9139a1 8b b ab b( a2 b a 3 b b( a4 b a 5 b b( a6 b a 7 b b( a8 b a 9 b b( a10 b a 11 b a 12 b b( a13 b a 14 b a 15 b a 16 b b(575757a17 b b ab b( a2 b a 3 b b( a4 b a 5 b b( a6 b a 7 b 3 67 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I b( a8 b a 9 b 3 Global Journals Inc. (US
15 b( a10 b a 11 b a 12 b b( a13 b a 14 b a 15 b a 16 b b( b ab 4 68 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I b( a2 b a 3 b b( a4 b a 5 b b( a6 b a 7 b b( a8 b a 9 b a 10 b b( a11 b a 12 b a 13 b b( a14 b a 15 b b b( ab a 2 b b( a3 b a 4 b b( a5 b a 6 b b( a7 b a 8 b a 9 b 5 Global Journals Inc. (US
16 b( a10 b a 11 b a 12 b b( a13 b a 14 b b b( ab a 2 b b( a3 b a 4 b b( a5 b a 6 b b( a7 b a 8 b a 9 b b( a10 b a 11 b a 12 b b( a13 b b ab b( a2 b a 3 b b( a4 b a 5 b b( a6 b a 7 b a 8 b b( a9 b a 10 b a 11 b 7 69 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I b( a12 b b ab 8 Global Journals Inc. (US
17 b( a2 b a 3 b b( a4 b a 5 b a 6 b b( a7 b a 8 b a 9 b 8 70 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I b( a10 b a 11 b b b( ab a 2 b b( a3 b a 4 b a 5 b b( a6 b a 7 b a 8 b b( a9 b a 10 b b b( ab a 2 b a 3 b b( a4 b a 5 b a 6 b b( a7 b a 8 b a 9 b b( b ab a 2 b b( a3 b a 4 b a 5 b 11 Global Journals Inc. (US
18 b( a6 b a 7 b a 8 b b( b ab a 2 b b( a3 b a 4 b a 5 b b( a6 b a 7 b b b( ab a 2 b a 3 b b( a4 b a 5 b a 6 b b b( ab a 2 b a 3 b a 4 b b( a5 b b ab a 2 b b( a3 b a 4 b b ab b( a2 b a 3 b b ab a 2 b 17 Γ(b+1 2 Γ( a 2 + } b( 361b ab 18 + b 19 { } ][ 19 { } ] [ 20 { ( a 71 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a a 3 [ 19 { } ][ 20 Global Journals Inc. (US
19 ( a a 5 [ 19 { } ][ ( a a 7 [ 19 { } ][ ( a a 9 [ 19 { } ][ Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a a a 12 [ 19 { } ][ ( a a a a 16 [ 19 { } ][ ( a a a b [ 19 { } ][ ( ab a2 b [ 19 { } ][ ( a3 b a 4 b [ 19 { } ][ ( a5 b a 6 b [ 19 { } ][ ( a7 b a 8 b [ 19 { } ][ ( a9 b a 10 b a 11 b [ 19 { } ][ ( a12 b a 13 b a 14 b [ 19 { } ][ ( a15 b a 16 b a 17 b a 18 b [ 19 { } ][ 20 Global Journals Inc. (US
20 ( b ab 2 [ 19 { } ][ ( a2 b a 3 b 2 [ 19 { } ][ ( a4 b a 5 b 2 [ 19 { } ][ ( a6 b a 7 b 2 [ 19 { } ][ ( a8 b a 9 b 2 [ 19 { } ][ ( a10 b a 11 b a 12 b ( a13 b a 14 b a 15 b a 16 b (575757a17 b b ab ( a2 b a 3 b 3 [ 19 { } ][ ( a4 b a 5 b 3 [ 19 { } ][ ( a6 b a 7 b 3 [ 19 { } ][ ( a8 b a 9 b 3 [ 19 { } ][ ( a10 b a 11 b a 12 b 3 [ 19 { } ][ Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Global Journals Inc. (US
21 74 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a13 b a 14 b a 15 b a 16 b ( b ab 4 [ 19 { } ][ ( a2 b a 3 b 4 [ 19 { } ][ ( a4 b a 5 b 4 [ 19 { } ][ ( a6 b a 7 b 4 [ 19 { } ][ ( a8 b a 9 b 4 [ 19 { } ][ ( a10 b a 11 b a 12 b 4 [ 19 { } ][ ( a13 b a 14 b a 15 b 4 [ 19 { } ][ ( b ab 5 [ 19 { } ][ ( a2 b a 3 b 5 [ 19 { } ][ ( a4 b a 5 b 5 [ 19 { } ][ ( a6 b a 7 b 5 [ 19 { } ][ ( a8 b a 9 b a 10 b 5 [ 19 { } ][ 20 Global Journals Inc. (US
22 ( a11 b a 12 b a 13 b 5 [ 19 { } ][ ( a14 b b ab ( a2 b a 3 b 6 [ 19 { } ][ ( a4 b a 5 b 6 [ 19 { } ][ ( a6 b a 7 b a 8 b ( a9 b a 10 b a 11 b 6 [ 19 { } ][ ( a12 b a 13 b b 7 [ 19 { } ][ ( ab a 2 b 7 [ 19 { } ][ ( a3 b a 4 b 7 [ 19 { } ][ ( a5 b a 6 b a 7 b ( a8 b a 9 b a 10 b 7 [ 19 { } ][ ( a11 b a 12 b b 8 [ 19 { } ][ ( ab a 2 b 8 [ 19 { } ][ Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Global Journals Inc. (US
23 ( a5 b a 6 b a 7 b ( a8 b a 9 b a 10 b 8 [ 19 { } ][ ( a11 b b ab 9 [ 19 { } ][ Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a2 b a 3 b a 4 b ( a5 b a 6 b a 7 b 9 [ 19 { } ][ ( a8 b a 9 b a 10 b 9 [ 19 { } ][ ( b ab a 2 b ( a3 b a 4 b a 5 b 10 [ 19 { } ][ ( a6 b a 7 b a 8 b 10 [ 19 { } ][ ( a9 b b ab 11 [ 19 { } ][ ( a2 b a 3 b a 4 b 11 [ 19 { } ][ ( a5 b a 6 b a 7 b 11 [ 19 { } ][ ( a8 b b ab 12 [ 19 { } ][ 20 Global Journals Inc. (US
24 ( a2 b a 3 b a 4 b 12 [ 19 { } ][ ( a5 b a 6 b a 7 b 12 [ 19 { } ][ ( b ab a 2 b 13 [ 19 { } ][ ( a3 b a 4 b a 5 b a 6 b ( b ab a 2 b a 3 b ( a4 b a 5 b b ab 15 [ 19 { } ][ ( a2 b a 3 b a 4 b b 16 [ 19 { } ][ ( ab a 2 b a 3 b b ab 17 [ 19 { } ][ (82251a2 b b ab 18 + b 19 [ 19 { } ][ 20 { } ] ( a ( a a ( a a 5 77 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a a 7 Global Journals Inc. (US
25 ( a a a ( a a a ( a a a a a 18 + a Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( b ab ( a2 b a 3 b ( a4 b a 5 b ( a6 b a 7 b ( a8 b a 9 b a 10 b ( a11 b a 12 b a 13 b ( a14 b a 15 b a 16 b a 17 b + 741a 18 b ( b ab ( a2 b a 3 b ( a4 b a 5 b 2 Global Journals Inc. (US
26 ( a6 b a 7 b ( a8 b a 9 b a 10 b ( a11 b a 12 b a 13 b ( a14 b a 15 b a 16 b a 17 b ( b ab ( a2 b a 3 b ( a4 b a 5 b ( a6 b a 7 b ( a8 b a 9 b ( a10 b a 11 b a 12 b ( a13 b a 14 b a 15 b a 16 b ( b ab 4 79 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a2 b a 3 b 4 Global Journals Inc. (US
27 ( a4 b a 5 b ( a6 b a 7 b ( a8 b a 9 b a 10 b 4 80 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a11 b a 12 b a 13 b ( a14 b a 15 b b ( ab a 2 b ( a3 b a 4 b ( a5 b a 6 b ( a7 b a 8 b a 9 b ( a10 b a 11 b a 12 b ( a13 b a 14 b b ( ab a 2 b ( a3 b a 4 b 6 Global Journals Inc. (US
28 ( a5 b a 6 b ( a7 b a 8 b a 9 b ( a10 b a 11 b a 12 b ( a13 b b ab ( a2 b a 3 b ( a4 b a 5 b ( a6 b a 7 b a 8 b ( a9 b a 10 b a 11 b ( a1 2b b ab ( a2 b a 3 b ( a4 b a 5 b a 6 b ( a7 b a 8 b a 9 b 8 81 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a10 b a 11 b b 9 Global Journals Inc. (US
29 ( ab a 2 b ( a3 b a 4 b a 5 b ( a6 b a 7 b a 8 b 9 82 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I ( a9 b a 10 b b ( ab a 2 b a 3 b ( a4 b a 5 b a 6 b ( a7 b a 8 b a 9 b ( b ab a 2 b ( a3 b a 4 b a 5 b ( a6 b a 7 b a 8 b ( b ab a 2 b ( a3 b a 4 b a 5 b ( a6 b a 7 b b 13 Global Journals Inc. (US
30 ( ab a 2 b a 3 b ( a4 b a 5 b a 6 b b ( ab a 2 b a 3 b a 4 b ( a5 b b ab a 2 b ( a3 b a 4 b b ab ( a2 b a 3 b b ab a 2 b 17 III ( 4199b ab b 19 }] [ 20 { } ][ 19 { } ] Derivation of the Main Summation Formula Proceeding on the same way of Ref[8],we get the main result. References Références Referencias 1. 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, Bells, Richard. Wong, Roderick; Special Functions, A Graduate Text. Cambridge Studies in Advanced Mathematics, Luke, Y.L; Mathematical Functions and their Approximations. Academic Press Inc, Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O.I.; Integrals and Series Vol. 3: More Special Functions. Nauka, Moscow, Translated from the Russian by G.G. Gould, Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, Rainville, E. D.; The contiguous function relations for p F q with applications to Bateman s Jn u,v and Rice s H n (ζ, p, ν, Bull. Amer. Math. Soc., 51(1945, (22 83 Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Global Journals Inc. (US
31 6. Salahuddin, Chaudhary, M.P ; Development of some summation formulae using Hypergeometric function, Global journal of Science Frontier Research, 10(2010, Salahuddin, Chaudhary, M.P ; Certain summation formulae associated to Gauss second summation theorem, Global journal of Science Frontier Research, 10(2010, Salahuddin ; On certain summation formulae based on half argument associated to hypergeometric function, International Journal of Mathematical Archive, 2(2011, Salahuddin ; Two summation formulae based on half argument involving contigious relation, Elixir App. Math., 33(2011, Global Journal of Science Frontier Research F Volume XII Issue X V ersion I Global Journals Inc. (US
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