South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin 1 1 Mewar University,Gangrar, Chittorgarh, Rajasthan, India E-mail: vsludn@gmail.com Received: Nov-18-016; Accepted: Jan-8-017 *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 010 33C05, 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.
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation Gauss second summation theorem is cited by [Prudnikov., 491(7.3.7.3)] 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(7.3.7.3)] 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(16783438571436083000a 9399657880654315500a ) Γ( a ) + 5488(90678938076577790a3 9935668513495008563a 4 ) + 5488(9410447001061791536a5 61861140341809608a 6 + 963840016609734464a 7 )
South Asian J. Math. Vol. 7 No. 1 + 5488( 11463603808354304a8 + 1048757665104179a 9 7531905745368576a 10 ) + 5488(4536786937088a11 18887770077696a 1 + 655803985408a 13 175935704a 14 ) + 5488(35718115a15 530368a 16 + 5464a 17 34a 18 + a 19 ) + 5488(16783438571436083000b+ 113000753873834083040a b) + 5488( 319665691581146545408a3 b + 334059160754895544a 4 b) + 5488( 508040803080987840a5 b + 186538357910351773696a 6 b) + 5488( 1767737337179870848a7 b + 37301697937071104a 8 b) + 5488( 1953084596375168a9 b + 0659570551856896a 10 b 7803314469470a 11 b) + 5488(495989540640a1 b 114870090304a 13 b + 4390181440a 14 b 5667644a 15 b) + 5488( 95516331081971008a4 b + 1095567086594743091a 5 b ) + 5488(+133343a16 b 67488a 17 b + 703a 18 b + 9399657880654315500b ) + 5488(113000753873834083040ab + 97176070133097938a 3 b ) + 5488( 77790443397565840a6 b + 30807469015600531a 7 b ) 3
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 5488( 1545098188013858a8 b + 300077375008156416a 9 b 10060768071383040a 10 b ) + 5488(1109596414387a11 b 305434530880a 1 b + 15055663867a 13 b ) + 5488( 17947533760a14 b + 686083008a 15 b 3513594a 16 b + 73815a 17 b ) + 5488(90678938076577790b3 + 319665691581146545408ab 3 ) + 5488(97176070133097938a b 3 + 940849603717870387a 4 b 3 ) + 5488( 113344455768535859a5 b 3 + 119849846790553083904a 6 b 3 ) + 5488( 49157435818540368a7 b 3 + 1909960033818368a 8 b 3 ) + 5488( 5576097505313800a9 b 3 + 1077391468519840a 10 b 3 0553881878835a 11 b 3 ) + 5488(534398707a1 b 3 46687761600a 13 b 3 + 1560704990a 14 b 3 7571536a 15 b 3 ) + 5488(760681a16 b 3 + 9935668513495008563b 4 + 334059160754895544ab 4 ) + 5488(95516331081971008a b 4 + 940849603717870387a 3 b 4 ) + 5488(33898545104011304a5 b 4 60575956903954579a 6 b 4 ) + 5488(619460936418863616a7 b 4 145080464930590a 8 b 4 + 5444416679733760a 9 b 4 ) 4
South Asian J. Math. Vol. 7 No. 1 + 5488( 908103786596160a10 b 4 + 169671845105504a 11 b 4 171867617840a 1 b 4 ) + 5488(17883858830a13 b 4 81505800a 14 b 4 + 4890349a 15 b 4 ) + 5488(9410447001061791536b5 + 508040803080987840ab 5 ) + 5488(1095567086594743091a b 5 + 113344455768535859a 3 b 5 ) + 5488(33898545104011304a4 b 5 + 1099048603633706784a 6 b 5 ) + 5488( 160484043558944a7 b 5 + 15468863995166771a 8 b 5 057164674867456a 9 b 5 ) + 5488(7335864685748a10 b 5 649317066048a 11 b 5 + 11578751904a 1 b 5 ) + 5488( 48138915a13 b 5 + 47733756a 14 b 5 + 61861140341809608b 6 ) + 5488(186538357910351773696ab6 + 77790443397565840a b 6 ) + 5488(119849846790553083904a3 b 6 + 60575956903954579a 4 b 6 ) + 5488(1099048603633706784a5 b 6 + 58668485783347a 7 b 6 109590737176384a 8 b 6 ) + 5488(18988774304864a9 b 6 13314407198400a 10 b 6 + 4461919043904a 11 b 6 ) + 5488( 1644850888a1 b 6 + 707475148a 13 b 6 + 963840016609734464b 7 ) 5
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 5488(1767737337179870848ab7 + 30807469015600531a b 7 ) + 5488(49157435818540368a3 b 7 + 619460936418863616a 4 b 7 ) + 5488(160484043558944a5 b 7 + 58668485783347a 6 b 7 + 3033919388640a 8 b 7 ) + 5488( 191397389900a9 b 7 + 10791677413440a 10 b 7 309457310a 11 b 7 ) + 5488(9669554100a1 b 7 + 11463603808354304b 8 + 37301697937071104ab 8 ) + 5488(1545098188013858a b 8 + 1909960033818368a 3 b 8 ) + 5488(145080464930590a4 b 8 + 15468863995166771a 5 b 8 + 109590737176384a 6 b 8 ) + 5488(3033919388640a7 b 8 + 16693063160400a 9 b 8 8781143380a 10 b 8 ) + 5488(+39974430a11 b 8 + 1048757665104179b 9 + 1953084596375168ab 9 ) + 5488(300077375008156416a b 9 + 5576097505313800a 3 b 9 + 5444416679733760a 4 b 9 ) + 5488(057164674867456a5 b 9 + 18988774304864a 6 b 9 + 191397389900a 7 b 9 ) + 5488(16693063160400a8 b 9 + 33578000610a 10 b 9 + 7531905745368576b 10 ) + 5488(0659570551856896ab10 + 10060768071383040a b 10 + 1077391468519840a 3 b 10 ) 6
South Asian J. Math. Vol. 7 No. 1 + 5488(908103786596160a4 b 10 + 7335864685748a 5 b 10 + 13314407198400a 6 b 10 ) + 5488(10791677413440a7 b 10 + 8781143380a 8 b 10 + 33578000610a 9 b 10 ) + 5488(4536786937088b11 + 7803314469470ab 11 + 1109596414387a b 11 ) + 5488(0553881878835a3 b 11 + 169671845105504a 4 b 11 + 649317066048a 5 b 11 ) + 5488(4461919043904a6 b 11 + 309457310a 7 b 11 + 39974430a 8 b 11 ) + 5488(18887770077696b1 + 495989540640ab 1 + 305434530880a b 1 ) + 5488(534398707a3 b 1 + 171867617840a 4 b 1 + 11578751904a 5 b 1 ) + 5488(1644850888a6 b 1 + 9669554100a 7 b 1 + 655803985408b 13 + 114870090304ab 13 ) + 5488(15055663867a b 13 + 46687761600a 3 b 13 + 17883858830a 4 b 13 ) + 5488(5667644ab15 + 686083008a b 15 + 7571536a 3 b 15 + 4890349a 4 b 15 + 530368b 16 ) + 5488(48138915a5 b 13 + 707475148a 6 b 13 + 175935704b 14 + 4390181440ab 14 ) + 5488(17947533760a b 14 + 1560704990a 3 b 14 + 81505800a 4 b 14 + 47733756a 5 b 14 ) + 5488(35718115b15 + 133343ab 16 + 3513594a b 16 + 760681a 3 b 16 + 5464b 17 ) 7
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 5488(67488ab17 + 73815a b 17 + 34b 18 + 703ab 18 + b 19 ) 1048576b(16783438571436083000+ 559005839506947000a) + + 1048576b(3141644375353511116800a + 5511699835081005056a 3 ) + 1048576b(5376615565844483a4 + 5705777169097044787a 5 ) + 1048576b(05704078504073656a6 + 150354794700067a 7 + 96336688095791104a 8 ) + 1048576b(135011130748544a9 + 1485778949043a 10 + 43056378888a 11 ) + 1048576b(88047549696a1 + 5151176896a 13 + 05958368a 14 + 0868384a 15 ) + 1048576b(457368a16 + 1938a 17 + 19a 18 559005839506947000b) + 1048576b(74594557741084367360ab+ 703573885454640744a b) + 1048576b(31563945104706369536a3 b + 7007485000916701184a 4 b) + 1048576b(414040675953041408a5 b + 1454397937340683059a 6 b) + 1048576b(53478987770794803a7 b + 043367804105470a 8 b + 417557144459315a 9 b) + 1048576b(1149678716784a10 b + 1383933883008a 11 b + 168658418a 1 b) 8
South Asian J. Math. Vol. 7 No. 1 + 1048576b(13743305811a13 b + 1367194400a 14 b + 5048104a 15 b + 15118a 16 b + 418a 17 b) + 1048576b(3141644375353511116800b 703573885454640744ab ) + 1048576b(5367608455754056755a b + 51777604071368944a 3 b ) + 1048576b(85615830641864933376a4 b + 40178446500509958144a 5 b ) + 1048576b(375986364570007a6 b + 11804561473496816a 7 b ) + 1048576b(40633364638814184a8 b + 10837808355510a 9 b + 184061905584896a 10 b ) + 1048576b(30459189780096a11 b + 309718559384a 1 b + 983543480a 13 b ) + 1048576b(1770660160a14 b + 745744a 15 b + 50971a 16 b 5511699835081005056b 3 ) + 1048576b(31563945104706369536ab3 51777604071368944a b 3 ) + 1048576b(1809694141738530864a3 b 3 + 756544917014691840a 4 b 3 ) + 1048576b(91084505646688640a5 b 3 + 4899917156335488a 6 b 3 ) + 1048576b(185185099833580544a7 b 3 + 40469618800540160a 8 b 3 + 194733987445760a 9 b 3 ) + 1048576b(19606689863040a10 b 3 + 379763784576a 11 b 3 + 9903696590a 1 b 3 ) 9
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 1048576b(8391390a13 b 3 + 115685680a 14 b 3 + 631018a 15 b 3 ) + 1048576b(5376615565844483b4 7007485000916701184ab 4 ) + 1048576b(85615830641864933376a b 4 756544917014691840a 3 b 4 ) + 1048576b(16931910361016430a4 b 4 + 15438700470633a 5 b 4 ) + 1048576b(44454516598779858a6 b 4 + 78766654679040a 7 b 4 + 49039931438586880a 8 b 4 ) + 1048576b(635049700676160a9 b 4 + 1878348708640a 10 b 4 + 1559754648480a 11 b 4 ) + 1048576b(4190973760a1 b 4 + 9336110a 13 b 4 + 8150580a 14 b 4 ) + 1048576b( 5705777169097044787b5 + 414040675953041408ab 5 ) + 1048576b( 40178446500509958144a b 5 + 91084505646688640a 3 b 5 ) + 1048576b( 15438700470633a4 b 5 + 591300036963008a 5 b 5 ) + 1048576b(4107064937046368a6 b 5 + 10674468578890880a 7 b 5 + 101331465875704a 8 b 5 ) + 1048576b(671766878313a9 b 5 + 4417891604896a 10 b 5 + 103388000a 11 b 5 ) + 1048576b(41168008a1 b 5 + 601661144a 13 b 5 + 05704078504073656b 6 ) 10
South Asian J. Math. Vol. 7 No. 1 + 1048576b( 1454397937340683059ab6 + 375986364570007a b 6 ) + 1048576b( 4899917156335488a3 b 6 + 44454516598779858a 4 b 6 ) + 1048576b( 4107064937046368a5 b 6 + 137888078614464a 6 b 6 + 546379013005056a 7 b 6 ) + 1048576b(17400883534304a8 b 6 + 6509913870a 9 b 6 + 3671973353664a 10 b 6 ) + 1048576b(108990059a11 b 6 + 707475148a 1 b 6 150354794700067b 7 ) + 1048576b(53478987770794803ab7 11804561473496816a b 7 ) + 1048576b(185185099833580544a3 b 7 78766654679040a 4 b 7 ) + 1048576b(10674468578890880a5 b 7 546379013005056a 6 b 7 + 1609754508046080a 7 b 7 ) + 1048576b(3374704317600a8 b 7 + 709188448300a 9 b 7 + 150164870a 10 b 7 ) + 1048576b(773564380a11 b 7 + 96336688095791104b 8 043367804105470ab 8 ) + 1048576b(40633364638814184a b 8 40469618800540160a 3 b 8 + 49039931438586880a 4 b 8 ) + 1048576b( 101331465875704a5 b 8 + 17400883534304a 6 b 8 3374704317600a 7 b 8 ) + 1048576b(8816118656400a8 b 8 + 7573985100a 9 b 8 + 14390571690a 10 b 8 ) 11
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 1048576b(8816118656400a8 b 8 + 7573985100a 9 b 8 + 14390571690a 10 b 8 ) + 1048576b( 135011130748544b9 + 417557144459315ab 9 10837808355510a b 9 ) + 1048576b(194733987445760a3 b 9 635049700676160a 4 b 9 + 671766878313a 5 b 9 ) + 1048576b( 6509913870a6 b 9 + 709188448300a 7 b 9 7573985100a 8 b 9 ) + 1048576b(1767631900a9 b 9 + 1485778949043b 10 1149678716784ab 10 ) + 1048576b(184061905584896a b 10 19606689863040a 3 b 10 + 1878348708640a 4 b 10 ) + 1048576b( 4417891604896a5 b 10 + 3671973353664a 6 b 10 150164870a 7 b 10 ) + 1048576b(14390571690a8 b 10 43056378888b 11 + 1383933883008ab 11 ) + 1048576b( 30459189780096a b 11 + 379763784576a 3 b 11 1559754648480a 4 b 11 ) + 1048576b(103388000a5 b 11 108990059a 6 b 11 + 773564380a 7 b 11 ) + 1048576b(88047549696b1 168658418ab 1 + 309718559384a b 1 ) + 1048576b( 9903696590a3 b 1 + 4190973760a 4 b 1 41168008a 5 b 1 ) + 1048576b(707475148a6 b 1 5151176896b 13 + 13743305811ab 13 983543480a b 13 ) 1
South Asian J. Math. Vol. 7 No. 1 + 1048576b(8391390a3 b 13 9336110a 4 b 13 + 601661144a 5 b 13 + 05958368b 14 ) + 1048576b( 1367194400ab14 + 1770660160a b 14 115685680a 3 b 14 + 8150580a 4 b 14 ) + 1048576b( 0868384b15 + 5048104ab 15 745744a b 15 + 631018a 3 b 15 + 457368b 16 ) + 1048576b( 15118ab16 + 50971a b 16 1938b 17 + 418ab 17 + 19b 18 } ) [ 19 { } ][ 18 { } ] b+1 Γ( ) { 1048576a(16783438571436083000 559005839506947000a) Γ( a+1 ) + 1048576a(3141644375353511116800a 5511699835081005056a 3 ) + 1048576a(5376615565844483a4 5705777169097044787a 5 ) + 1048576a(05704078504073656a6 150354794700067a 7 + 96336688095791104a 8 ) + 1048576a( 135011130748544a9 + 1485778949043a 10 43056378888a 11 ) + 1048576a(88047549696a1 5151176896a 13 + 05958368a 14 0868384a 15 ) + 1048576a(457368a16 1938a 17 + 19a 18 + 559005839506947000b) + 1048576a(74594557741084367360ab 703573885454640744a b) + 1048576a(31563945104706369536a3 b 7007485000916701184a 4 b) 13
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 1048576a(414040675953041408a5 b 1454397937340683059a 6 b) + 1048576a(53478987770794803a7 b 043367804105470a 8 b + 417557144459315a 9 b) + 1048576a( 1149678716784a10 b + 1383933883008a 11 b 168658418a 1 b) + 1048576a(13743305811a13 b 1367194400a 14 b + 5048104a 15 b 15118a 16 b) + 1048576a(418a17 b + 3141644375353511116800b + 703573885454640744ab ) + 1048576a(5367608455754056755a b 51777604071368944a 3 b ) + 1048576a(85615830641864933376a4 b 40178446500509958144a 5 b ) + 1048576a(375986364570007a6 b 11804561473496816a 7 b ) + 1048576a(40633364638814184a8 b 10837808355510a 9 b + 184061905584896a 10 b ) + 1048576a( 30459189780096a11 b + 309718559384a 1 b 983543480a 13 b ) + 1048576a(1770660160a14 b 745744a 15 b + 50971a 16 b + 5511699835081005056b 3 ) + 1048576a(31563945104706369536ab3 + 51777604071368944a b 3 ) + 1048576a(1809694141738530864a3 b 3 756544917014691840a 4 b 3 ) 14
South Asian J. Math. Vol. 7 No. 1 + 1048576a(91084505646688640a5 b 3 4899917156335488a 6 b 3 ) + 1048576a(185185099833580544a7 b 3 40469618800540160a 8 b 3 + 194733987445760a 9 b 3 ) + 1048576a( 19606689863040a10 b 3 + 379763784576a 11 b 3 9903696590a 1 b 3 ) + 1048576a(8391390a13 b 3 115685680a 14 b 3 + 631018a 15 b 3 ) + 1048576a(5376615565844483b4 + 7007485000916701184ab 4 ) + 1048576a(85615830641864933376a b 4 + 756544917014691840a 3 b 4 ) + 1048576a(16931910361016430a4 b 4 15438700470633a 5 b 4 ) + 1048576a(44454516598779858a6 b 4 78766654679040a 7 b 4 + 49039931438586880a 8 b 4 ) + 1048576a( 635049700676160a9 b 4 + 1878348708640a 10 b 4 1559754648480a 11 b 4 ) + 1048576a(4190973760a1 b 4 9336110a 13 b 4 + 8150580a 14 b 4 ) + 1048576a(5705777169097044787b5 + 414040675953041408ab 5 ) + 1048576a(40178446500509958144a b 5 + 91084505646688640a 3 b 5 ) + 1048576a(15438700470633a4 b 5 + 591300036963008a 5 b 5 ) 15
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 1048576a( 4107064937046368a6 b 5 + 10674468578890880a 7 b 5 101331465875704a 8 b 5 ) + 1048576a(671766878313a9 b 5 4417891604896a 10 b 5 + 103388000a 11 b 5 ) + 1048576a( 41168008a1 b 5 + 601661144a 13 b 5 + 05704078504073656b 6 ) + 1048576a(1454397937340683059ab6 + 375986364570007a b 6 ) + 1048576a(4899917156335488a3 b 6 + 44454516598779858a 4 b 6 ) + 1048576a(4107064937046368a5 b 6 + 137888078614464a 6 b 6 546379013005056a 7 b 6 ) + 1048576a(17400883534304a8 b 6 6509913870a 9 b 6 + 3671973353664a 10 b 6 ) + 1048576a( 108990059a11 b 6 + 707475148a 1 b 6 + 150354794700067b 7 ) + 1048576a(53478987770794803ab7 + 11804561473496816a b 7 ) + 1048576a(185185099833580544a3 b 7 + 78766654679040a 4 b 7 ) + 1048576a(10674468578890880a5 b 7 + 546379013005056a 6 b 7 + 1609754508046080a 7 b 7 ) + 1048576a( 3374704317600a8 b 7 + 709188448300a 9 b 7 150164870a 10 b 7 ) + 1048576a(773564380a11 b 7 + 96336688095791104b 8 + 043367804105470ab 8 ) 16
South Asian J. Math. Vol. 7 No. 1 + 1048576a(40633364638814184a b 8 + 40469618800540160a 3 b 8 + 49039931438586880a 4 b 8 ) + 1048576a(101331465875704a5 b 8 + 17400883534304a 6 b 8 + 3374704317600a 7 b 8 ) + 1048576a(8816118656400a8 b 8 7573985100a 9 b 8 + 14390571690a 10 b 8 ) + 1048576a(135011130748544b9 + 417557144459315ab 9 + 10837808355510a b 9 ) + 1048576a(194733987445760a3 b 9 + 635049700676160a 4 b 9 + 671766878313a 5 b 9 ) + 1048576a(6509913870a6 b 9 + 709188448300a 7 b 9 + 7573985100a 8 b 9 ) + 1048576a(1767631900a9 b 9 + 1485778949043b 10 + 1149678716784ab 10 ) + 1048576a(184061905584896a b 10 + 19606689863040a 3 b 10 + 1878348708640a 4 b 10 ) + 1048576a(4417891604896a5 b 10 + 3671973353664a 6 b 10 + 150164870a 7 b 10 ) + 1048576a(14390571690a8 b 10 + 43056378888b 11 + 1383933883008ab 11 ) + 1048576a(30459189780096a b 11 + 379763784576a 3 b 11 + 1559754648480a 4 b 11 ) + 1048576a(103388000a5 b 11 + 108990059a 6 b 11 + 773564380a 7 b 11 ) + 1048576a(88047549696b1 + 168658418ab 1 + 309718559384a b 1 ) 17
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 1048576a(9903696590a3 b 1 + 4190973760a 4 b 1 + 41168008a 5 b 1 ) + 1048576a(707475148a6 b 1 + 5151176896b 13 + 13743305811ab 13 + 983543480a b 13 ) + 1048576a(8391390a3 b 13 + 9336110a 4 b 13 + 601661144a 5 b 13 + 05958368b 14 ) + 1048576a(1367194400ab14 + 1770660160a b 14 + 115685680a 3 b 14 + 8150580a 4 b 14 ) + 1048576a(0868384b15 + 5048104ab 15 + 745744a b 15 + 631018a 3 b 15 + 457368b 16 ) + 1048576a(15118ab16 + 50971a b 16 + 1938b 17 + 418ab 17 + 19b 18 ) + 5488(16783438571436083000a+ 9399657880654315500a ) + 5488(90678938076577790a3 + 9935668513495008563a 4 ) + 5488(9410447001061791536a5 + 61861140341809608a 6 + 963840016609734464a 7 ) + 5488(11463603808354304a8 + 1048757665104179a 9 + 7531905745368576a 10 ) + 5488(4536786937088a11 + 18887770077696a 1 + 655803985408a 13 ) + 5488(175935704a14 + 35718115a 15 + 530368a 16 + 5464a 17 + 34a 18 + a 19 ) + 5488(16783438571436083000b+ 113000753873834083040a b) 18
South Asian J. Math. Vol. 7 No. 1 + 5488(319665691581146545408a3 b + 334059160754895544a 4 b) + 5488(508040803080987840a5 b + 186538357910351773696a 6 b) + 5488(1767737337179870848a7 b + 37301697937071104a 8 b) + 5488(1953084596375168a9 b + 0659570551856896a 10 b + 7803314469470a 11 b) + 5488(495989540640a1 b + 114870090304a 13 b + 4390181440a 14 b + 5667644a 15 b) + 5488(133343a16 b + 67488a 17 b + 703a 18 b 9399657880654315500b ) + 5488(113000753873834083040ab + 97176070133097938a 3 b ) + 5488(95516331081971008a4 b + 1095567086594743091a 5 b ) + 5488(77790443397565840a6 b + 30807469015600531a 7 b ) + 5488(1545098188013858a8 b + 300077375008156416a 9 b + 10060768071383040a 10 b ) + 5488(1109596414387a11 b + 305434530880a 1 b + 15055663867a 13 b ) + 5488(17947533760a14 b + 686083008a 15 b + 3513594a 16 b + 73815a 17 b ) + 5488(90678938076577790b3 319665691581146545408ab 3 ) 19
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 5488(97176070133097938a b 3 + 940849603717870387a 4 b 3 ) + 5488(113344455768535859a5 b 3 + 119849846790553083904a 6 b 3 ) + 5488(49157435818540368a7 b 3 + 1909960033818368a 8 b 3 + 5576097505313800a 9 b 3 ) + 5488(1077391468519840a10 b 3 + 0553881878835a 11 b 3 + 534398707a 1 b 3 ) + 5488(46687761600a13 b 3 + 1560704990a 14 b 3 + 7571536a 15 b 3 + 760681a 16 b 3 ) + 5488( 9935668513495008563b4 + 334059160754895544ab 4 ) + 5488( 95516331081971008a b 4 + 940849603717870387a 3 b 4 ) + 5488(33898545104011304a5 b 4 + 60575956903954579a 6 b 4 ) + 5488(619460936418863616a7 b 4 + 145080464930590a 8 b 4 + 5444416679733760a 9 b 4 ) + 5488(908103786596160a10 b 4 + 169671845105504a 11 b 4 + 171867617840a 1 b 4 ) + 5488(17883858830a13 b 4 + 81505800a 14 b 4 + 4890349a 15 b 4 ) + 5488(9410447001061791536b5 508040803080987840ab 5 ) + 5488(1095567086594743091a b 5 113344455768535859a 3 b 5 ) 0
South Asian J. Math. Vol. 7 No. 1 + 5488(33898545104011304a4 b 5 + 1099048603633706784a 6 b 5 ) + 5488(160484043558944a7 b 5 + 15468863995166771a 8 b 5 + 057164674867456a 9 b 5 ) + 5488(7335864685748a10 b 5 + 649317066048a 11 b 5 + 11578751904a 1 b 5 ) + 5488(48138915a13 b 5 + 47733756a 14 b 5 61861140341809608b 6 ) + 5488(186538357910351773696ab6 77790443397565840a b 6 ) + 5488(119849846790553083904a3 b 6 60575956903954579a 4 b 6 ) + 5488(1099048603633706784a5 b 6 + 58668485783347a 7 b 6 + 109590737176384a 8 b 6 ) + 5488(18988774304864a9 b 6 + 13314407198400a 10 b 6 + 4461919043904a 11 b 6 ) + 5488(1644850888a1 b 6 + 707475148a 13 b 6 + 963840016609734464b 7 ) + 5488( 1767737337179870848ab7 + 30807469015600531a b 7 ) + 5488( 49157435818540368a3 b 7 + 619460936418863616a 4 b 7 ) + 5488( 160484043558944a5 b 7 + 58668485783347a 6 b 7 + 3033919388640a 8 b 7 ) + 5488(191397389900a9 b 7 + 10791677413440a 10 b 7 + 309457310a 11 b 7 ) 1
Salahuddin: A summation formula ramified with hypergeometric function and involving recurrence relation + 5488(9669554100a1 b 7 11463603808354304b 8 + 37301697937071104ab 8 ) + 5488( 1545098188013858a b 8 + 1909960033818368a 3 b 8 ) + 5488( 145080464930590a4 b 8 + 15468863995166771a 5 b 8 109590737176384a 6 b 8 ) + 5488(3033919388640a7 b 8 + 16693063160400a 9 b 8 + 8781143380a 10 b 8 ) + 5488(39974430a11 b 8 + 1048757665104179b 9 1953084596375168ab 9 ) + 5488(300077375008156416a b 9 5576097505313800a 3 b 9 + 5444416679733760a 4 b 9 ) + 5488( 057164674867456a5 b 9 + 18988774304864a 6 b 9 191397389900a 7 b 9 ) + 5488(16693063160400a8 b 9 + 33578000610a 10 b 9 7531905745368576b 10 ) + 5488(0659570551856896ab10 10060768071383040a b 10 + 1077391468519840a 3 b 10 ) + 5488( 908103786596160a4 b 10 + 7335864685748a 5 b 10 13314407198400a 6 b 10 ) + 5488(10791677413440a7 b 10 8781143380a 8 b 10 + 33578000610a 9 b 10 ) + 5488(4536786937088b11 7803314469470ab 11 + 1109596414387a b 11 ) + 5488( 0553881878835a3 b 11 + 169671845105504a 4 b 11 649317066048a 5 b 11 )
South Asian J. Math. Vol. 7 No. 1 + 5488(4461919043904a6 b 11 309457310a 7 b 11 + 39974430a 8 b 11 ) + 5488( 18887770077696b1 + 495989540640ab 1 305434530880a b 1 ) + 5488(534398707a3 b 1 171867617840a 4 b 1 + 11578751904a 5 b 1 ) + 5488( 1644850888a6 b 1 + 9669554100a 7 b 1 + 655803985408b 13 ) + 5488( 114870090304ab13 + 15055663867a b 13 46687761600a 3 b 13 ) + 5488(17883858830a4 b 13 48138915a 5 b 13 + 707475148a 6 b 13 175935704b 14 ) + 5488(4390181440ab14 17947533760a b 14 + 1560704990a 3 b 14 81505800a 4 b 14 ) + 5488(47733756a5 b 14 + 35718115b 15 5667644ab 15 + 686083008a b 15 ) + 5488( 7571536a3 b 15 + 4890349a 4 b 15 530368b 16 + 133343ab 16 ) + 5488( 3513594a b 16 + 760681a 3 b 16 + 5464b 17 67488ab 17 + 73815a b 17 34b 18 ) + [ 19 5488(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), 335-34. 3
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, 1986. Translated from the Russian by G.G. Gould, Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, 1990. 4 Rainville, E. D., The contiguous function relations for pf q with applications to Bateman s Jn u,v Bull. Amer. Math. Soc., 51(1945), 714-73. 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),36-48. 6 Salahuddin, Chaudhary, M.P, Certain summation formulae associated to Gauss second summation theorem, Global journal of Science Frontier Research, 10(010),30-35. 7 Salahuddin, On certain summation formulae based on half argument associated to hypergeometric function, International Journal of Mathematical Archive, (011),55-57. 8 Salahuddin, Two summation formulae based on half argument involving contigious relation, Elixir App. Math., 33(011),316-318. 9 Salahuddin, Two summation formulae based on half argument associated to Hypergeometic function, Global journal of Science Frontier Research, 10(010),08-19. 4