A summation formula ramified with hypergeometric function and involving recurrence relation

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1 South Asian Journal of Mathematics 017, Vol. 7 ( 1): ISSN RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin 1 1 Mewar University,Gangrar, Chittorgarh, Rajasthan, India vsludn@gmail.com Received: Nov ; Accepted: Jan *Corresponding author Abstract The main of the present paper is to develop a summation formula ramified with Hypergeometric function and recurrence relation. Key Words Contiguous relation, Gauss second summation theorem, Recurrence relation MSC C05, 33C0 1 Introduction Generalized Gaussian Hypergeometric function of one variable is defined by AF B a 1, a,, a A ; b 1, b,, b B ; z = k=0 (a 1 ) k (a ) k (a A ) k z k (b 1 ) k (b ) k (b B ) k k! or AF B (a A ) ; (b B ) ; z AF B (a j ) A j=1 ; (b j ) B j=1 ; z = k=0 ((a A )) k z k ((b B )) k k! (1) where the parameters b 1, b,, b B are neither zero nor negative integers and A, B are non-negative integers and z = 1. Contiguous Relation is defined by [ Andrews p.363(9.16)] (a b) F 1 a, b ; c ; z = a F 1 a + 1, b ; c ; z b F 1 a, b + 1 ; c ; z () Citation: Salahuddin, A summation formula ramified with hypergeometric function and involving recurrence relation, South Asian J Math, 017, 7(1), 1-4.

2 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation Gauss second summation theorem is cited by [Prudnikov., 491( )] F 1 a, b ; a+b+1 ; 1 = Γ( a+b+1 Γ( a+1 ) Γ( 1 ) b+1 ) Γ( ) (3) = (b 1) Γ( b a+b+1 ) Γ( ) Γ(b) Γ( a+1 ) (4) In a monograph of Prudnikov et al., a summation theorem is given in the form [Prudnikov., p.491( )] F 1 a, b ; a+b 1 ; 1 = [ π Γ( a+b+1 ) Γ( a+1 b+1 ) Γ( ) + Γ( ] a+b 1 ) Γ(a) Γ(b) (5) Now using Legendre s duplication formula and Recurrence relation for Gamma function, the above theorem can be written in the form F 1 a, b ; a+b 1 ; 1 = (b 1) Γ( a+b 1 ) Γ(b) [ Γ( b ) Γ( a 1 ) + (a b+1) a a+1 Γ( ) Γ( ) {Γ(a)} + Γ( b+ ) Γ( a+1 ) ] (6) Recurrence relation is defined by Γ(z + 1) = z Γ(z) (7) Main summation formula Proceeding on the same way of Ref[7],we get the main result. F 1 a, b ; a+b+40 ; 1 = b Γ( a+b+40 ) (a b) Γ(b) [ Γ( b ) { 5488( a a ) Γ( a ) ( a a 4 ) ( a a a 7 )

3 South Asian J. Math. Vol. 7 No ( a a a 10 ) ( a a a a 14 ) ( a a a 17 34a 18 + a 19 ) ( b a b) ( a3 b a 4 b) ( a5 b a 6 b) ( a7 b a 8 b) ( a9 b a 10 b a 11 b) ( a1 b a 13 b a 14 b a 15 b) ( a4 b a 5 b ) ( a16 b 67488a 17 b + 703a 18 b b ) ( ab a 3 b ) ( a6 b a 7 b ) 3

4 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation ( a8 b a 9 b a 10 b ) ( a11 b a 1 b a 13 b ) ( a14 b a 15 b a 16 b a 17 b ) ( b ab 3 ) ( a b a 4 b 3 ) ( a5 b a 6 b 3 ) ( a7 b a 8 b 3 ) ( a9 b a 10 b a 11 b 3 ) ( a1 b a 13 b a 14 b a 15 b 3 ) (760681a16 b b ab 4 ) ( a b a 3 b 4 ) ( a5 b a 6 b 4 ) ( a7 b a 8 b a 9 b 4 ) 4

5 South Asian J. Math. Vol. 7 No ( a10 b a 11 b a 1 b 4 ) ( a13 b a 14 b a 15 b 4 ) ( b ab 5 ) ( a b a 3 b 5 ) ( a4 b a 6 b 5 ) ( a7 b a 8 b a 9 b 5 ) ( a10 b a 11 b a 1 b 5 ) ( a13 b a 14 b b 6 ) ( ab a b 6 ) ( a3 b a 4 b 6 ) ( a5 b a 7 b a 8 b 6 ) ( a9 b a 10 b a 11 b 6 ) ( a1 b a 13 b b 7 ) 5

6 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation ( ab a b 7 ) ( a3 b a 4 b 7 ) ( a5 b a 6 b a 8 b 7 ) ( a9 b a 10 b a 11 b 7 ) ( a1 b b ab 8 ) ( a b a 3 b 8 ) ( a4 b a 5 b a 6 b 8 ) ( a7 b a 9 b a 10 b 8 ) ( a11 b b ab 9 ) ( a b a 3 b a 4 b 9 ) ( a5 b a 6 b a 7 b 9 ) ( a8 b a 10 b b 10 ) ( ab a b a 3 b 10 ) 6

7 South Asian J. Math. Vol. 7 No ( a4 b a 5 b a 6 b 10 ) ( a7 b a 8 b a 9 b 10 ) ( b ab a b 11 ) ( a3 b a 4 b a 5 b 11 ) ( a6 b a 7 b a 8 b 11 ) ( b ab a b 1 ) ( a3 b a 4 b a 5 b 1 ) ( a6 b a 7 b b ab 13 ) ( a b a 3 b a 4 b 13 ) ( ab a b a 3 b a 4 b b 16 ) ( a5 b a 6 b b ab 14 ) ( a b a 3 b a 4 b a 5 b 14 ) ( b ab a b a 3 b b 17 ) 7

8 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation (67488ab a b b ab 18 + b 19 ) b( a) b( a a 3 ) b( a a 5 ) b( a a a 8 ) b( a a a 11 ) b( a a a a 15 ) b(457368a a a b) b( ab a 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 1 b) 8

9 South Asian J. Math. Vol. 7 No b( a13 b a 14 b a 15 b a 16 b + 418a 17 b) b( b ab ) b( a 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 1 b a 13 b ) b( a14 b a 15 b a 16 b b 3 ) b( ab a b 3 ) b( a3 b a 4 b 3 ) b( a5 b a 6 b 3 ) b( a7 b a 8 b a 9 b 3 ) b( a10 b a 11 b a 1 b 3 ) 9

10 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation b( a13 b a 14 b a 15 b 3 ) b( b ab 4 ) b( a b a 3 b 4 ) b( a4 b a 5 b 4 ) b( a6 b a 7 b a 8 b 4 ) b( a9 b a 10 b a 11 b 4 ) b( a1 b a 13 b a 14 b 4 ) b( b ab 5 ) b( a b a 3 b 5 ) b( a4 b a 5 b 5 ) b( a6 b a 7 b a 8 b 5 ) b( a9 b a 10 b a 11 b 5 ) b( a1 b a 13 b b 6 ) 10

11 South Asian J. Math. Vol. 7 No b( ab a b 6 ) b( a3 b a 4 b 6 ) b( a5 b a 6 b a 7 b 6 ) b( a8 b a 9 b a 10 b 6 ) b( a11 b a 1 b b 7 ) b( ab a b 7 ) b( a3 b a 4 b 7 ) b( a5 b a 6 b a 7 b 7 ) b( a8 b a 9 b a 10 b 7 ) b( a11 b b ab 8 ) b( a b a 3 b a 4 b 8 ) b( a5 b a 6 b a 7 b 8 ) b( a8 b a 9 b a 10 b 8 ) 11

12 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation b( a8 b a 9 b a 10 b 8 ) b( b ab a b 9 ) b( a3 b a 4 b a 5 b 9 ) b( a6 b a 7 b a 8 b 9 ) b( a9 b b ab 10 ) b( a b a 3 b a 4 b 10 ) b( a5 b a 6 b a 7 b 10 ) b( a8 b b ab 11 ) b( a b a 3 b a 4 b 11 ) b( a5 b a 6 b a 7 b 11 ) b( b ab a b 1 ) b( a3 b a 4 b a 5 b 1 ) b( a6 b b ab a b 13 ) 1

13 South Asian J. Math. Vol. 7 No b( a3 b a 4 b a 5 b b 14 ) b( ab a b a 3 b a 4 b 14 ) b( b ab a b a 3 b b 16 ) b( 15118ab a b b ab b 18 } ) [ 19 { } ][ 18 { } ] b+1 Γ( ) { a( a) Γ( a+1 ) a( a a 3 ) a( a a 5 ) a( a a a 8 ) a( a a a 11 ) a( a a a a 15 ) a(457368a a a b) a( ab a b) a( a3 b a 4 b) 13

14 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation a( a5 b a 6 b) a( a7 b a 8 b a 9 b) a( a10 b a 11 b a 1 b) a( a13 b a 14 b a 15 b 15118a 16 b) a(418a17 b b ab ) a( a b a 3 b ) a( a4 b a 5 b ) a( a6 b a 7 b ) a( a8 b a 9 b a 10 b ) a( a11 b a 1 b a 13 b ) a( a14 b a 15 b a 16 b b 3 ) a( ab a b 3 ) a( a3 b a 4 b 3 ) 14

15 South Asian J. Math. Vol. 7 No a( a5 b a 6 b 3 ) a( a7 b a 8 b a 9 b 3 ) a( a10 b a 11 b a 1 b 3 ) a( a13 b a 14 b a 15 b 3 ) a( b ab 4 ) a( a b a 3 b 4 ) a( a4 b a 5 b 4 ) a( a6 b a 7 b a 8 b 4 ) a( a9 b a 10 b a 11 b 4 ) a( a1 b a 13 b a 14 b 4 ) a( b ab 5 ) a( a b a 3 b 5 ) a( a4 b a 5 b 5 ) 15

16 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation a( a6 b a 7 b a 8 b 5 ) a( a9 b a 10 b a 11 b 5 ) a( a1 b a 13 b b 6 ) a( ab a b 6 ) a( a3 b a 4 b 6 ) a( a5 b a 6 b a 7 b 6 ) a( a8 b a 9 b a 10 b 6 ) a( a11 b a 1 b b 7 ) a( ab a b 7 ) a( a3 b a 4 b 7 ) a( a5 b a 6 b a 7 b 7 ) a( a8 b a 9 b a 10 b 7 ) a( a11 b b ab 8 ) 16

17 South Asian J. Math. Vol. 7 No a( a b a 3 b a 4 b 8 ) a( a5 b a 6 b a 7 b 8 ) a( a8 b a 9 b a 10 b 8 ) a( b ab a b 9 ) a( a3 b a 4 b a 5 b 9 ) a( a6 b a 7 b a 8 b 9 ) a( a9 b b ab 10 ) a( a b a 3 b a 4 b 10 ) a( a5 b a 6 b a 7 b 10 ) a( a8 b b ab 11 ) a( a b a 3 b a 4 b 11 ) a( a5 b a 6 b a 7 b 11 ) a( b ab a b 1 ) 17

18 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation a( a3 b a 4 b a 5 b 1 ) a( a6 b b ab a b 13 ) a( a3 b a 4 b a 5 b b 14 ) a( ab a b a 3 b a 4 b 14 ) a( b ab a b a 3 b b 16 ) a(15118ab a b b ab b 18 ) ( a a ) ( a a 4 ) ( a a a 7 ) ( a a a 10 ) ( a a a 13 ) ( a a a a a 18 + a 19 ) ( b a b) 18

19 South Asian J. Math. Vol. 7 No ( a3 b a 4 b) ( a5 b a 6 b) ( a7 b a 8 b) ( a9 b a 10 b a 11 b) ( a1 b a 13 b a 14 b a 15 b) (133343a16 b a 17 b + 703a 18 b b ) ( ab a 3 b ) ( a4 b a 5 b ) ( a6 b a 7 b ) ( a8 b a 9 b a 10 b ) ( a11 b a 1 b a 13 b ) ( a14 b a 15 b a 16 b a 17 b ) ( b ab 3 ) 19

20 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation ( a b a 4 b 3 ) ( a5 b a 6 b 3 ) ( a7 b a 8 b a 9 b 3 ) ( a10 b a 11 b a 1 b 3 ) ( a13 b a 14 b a 15 b a 16 b 3 ) ( b ab 4 ) ( a b a 3 b 4 ) ( a5 b a 6 b 4 ) ( a7 b a 8 b a 9 b 4 ) ( a10 b a 11 b a 1 b 4 ) ( a13 b a 14 b a 15 b 4 ) ( b ab 5 ) ( a b a 3 b 5 ) 0

21 South Asian J. Math. Vol. 7 No ( a4 b a 6 b 5 ) ( a7 b a 8 b a 9 b 5 ) ( a10 b a 11 b a 1 b 5 ) ( a13 b a 14 b b 6 ) ( ab a b 6 ) ( a3 b a 4 b 6 ) ( a5 b a 7 b a 8 b 6 ) ( a9 b a 10 b a 11 b 6 ) ( a1 b a 13 b b 7 ) ( ab a b 7 ) ( a3 b a 4 b 7 ) ( a5 b a 6 b a 8 b 7 ) ( a9 b a 10 b a 11 b 7 ) 1

22 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation ( a1 b b ab 8 ) ( a b a 3 b 8 ) ( a4 b a 5 b a 6 b 8 ) ( a7 b a 9 b a 10 b 8 ) ( a11 b b ab 9 ) ( a b a 3 b a 4 b 9 ) ( a5 b a 6 b a 7 b 9 ) ( a8 b a 10 b b 10 ) ( ab a b a 3 b 10 ) ( a4 b a 5 b a 6 b 10 ) ( a7 b a 8 b a 9 b 10 ) ( b ab a b 11 ) ( a3 b a 4 b a 5 b 11 )

23 South Asian J. Math. Vol. 7 No ( a6 b a 7 b a 8 b 11 ) ( b ab a b 1 ) ( a3 b a 4 b a 5 b 1 ) ( a6 b a 7 b b 13 ) ( ab a b a 3 b 13 ) ( a4 b a 5 b a 6 b b 14 ) ( ab a b a 3 b a 4 b 14 ) ( a5 b b ab a b 15 ) ( a3 b a 4 b b ab 16 ) ( a b a 3 b b ab a b 17 34b 18 ) + [ (703ab 18 + b 19 ) }] { } ][ 18 { } ] (8) References 1 Andrews, L.C.(199), 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(008),

24 Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation 3 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 pf q with applications to Bateman s Jn u,v Bull. Amer. Math. Soc., 51(1945), and Rice s H n (ζ, p, ν), 5 Salahuddin, Chaudhary, M.P, Development of some summation formulae using Hypergeometric function, Global journal of Science Frontier Research, 10(010), Salahuddin, Chaudhary, M.P, Certain summation formulae associated to Gauss second summation theorem, Global journal of Science Frontier Research, 10(010), Salahuddin, On certain summation formulae based on half argument associated to hypergeometric function, International Journal of Mathematical Archive, (011), Salahuddin, Two summation formulae based on half argument involving contigious relation, Elixir App. Math., 33(011), Salahuddin, Two summation formulae based on half argument associated to Hypergeometic function, Global journal of Science Frontier Research, 10(010),