FINITE DOUBLE SUMS OF KAMPE DE FE RIET S DOUBLE HYPERGEOMETRIC FUNCTION OF HIGHER ORDER

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1 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 FINITE DOUBLE SUMS OF KAMPE DE FE RIET S DOUBLE HYPERGEOMETRIC FUNCTION OF HIGHER ORDER Mohaad Idis Queshi Deepak Kua Kaa Yaseen Depatent of Applied Sciences and Huanities, Depatent of Matheatics, Depatent of Matheatics Faculty of Engineeing and Technology, Govt. College Kota, Govt. College Kota, Jaia Millia Islaia (A Cental Univesity) Univesity of Kota Univesity of Kota New Delhi , India Kota, Rajasthan , India Kota, Rajasthan , India E-ail: dkaa20@gail.co May 11, 2016 Astact In this pape we otain fou finite doule sus of Kape de Fe iet s doule hypegeoetic function of highe ode using Gauss s suation theoe. Associated and contiguous elations fo Kue s confluent hypegeoetic function, Gauss s odinay hypegeoetic function, Saalschu tz s suation theoe, doule sus involving Appell, Huet, Kalsson, Bessel functions, ae otained as special cases of ou ain doule suations. Keywods and Phases: Gauss s suation theoe, Saalschu tz s suation theoe, Kalsson s quaduple hypegeoetic function, Bessel s function, Kape de Fe iet s doule hypegeoetic function MSC(AMS) Piay:33C45 Seconday:33C65, 33C70 1 Intoduction and Peliinaies Thoughout in pesent pape, we use the following standad notations: N : {1, 2, 3,...}, N0 : {0, 1, 2, 3,...} N {0} and Z : { 1, 2, 3,...} Z 0 \{0}. Hee, as usual, Z denotes the set of integes, R denotes the set of eal nues, R+ denotes the set of positive eal nues and C denotes the set of coplex nues. The Pochhae syol (o the shifted factoial) (λ)ν (λ, ν C) is defined, in tes of the 1 Page 104

2 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 failia Gaa function, y 1 (λ)ν : Γ(λ + ν) Γ(λ) λ(λ + 1)... (λ + n 1) (ν 0 λ C\{0}) (ν n N λ C) it eing undestood conventionally that (0)0 1 and assued tacitly that the Gaa quotient exists. Kape de Fe iet s doule hypegeoetic function of highe ode in the odified notation of Sivastava and Panda [8,pp. 423(26), 424(27)], is given y p:qk Fj:n p (ap ) : (q ) (dk ) (gj ) : (e ) (hn ),s0 (ai )+s q (i ) k i1 j i1 i1 n i1 i1 i1 (gi )+s (ei ) (di )s x y s! s! (1.1) (hi )s whee (ap ) aeviates the aay of p paaetes given y a1, a2,..., ap with siila intepetations fo (q ), (dk ) et cetea and fo convegence of doule hypegeoetic seies(1.1), we have (i)p + q < j + + 1,p + k < j + n + 1, x < and y < o (ii)p + q j + + 1,p + k j + n + 1, and 1 1 x p j + y p j < 1 if ax{ x, y } < 1, if p j p>j The Appell s doule hypegeoetic functions F1, F2, F3 and F4 [3,p.224( ,..., )] 1:11 1:11 0:22 2:00 ae denoted y F1:00, F0:11, F1:00 and F0:11 espectively. In ou investigation we shall use the following esults: (i) Algeaic popeties of Pochhae s syol (a)p+q (a)p (a + p)q (a)q (a + q)p () k ( 1)k, (1 )k ( 0, ±1, ±2, ±3,... (n k)! ( 1)k (n)!, ( n)k and (0 k n) 2 k 1, 2, 3,...) (1.2) (1.3) (1.4) Page 105

3 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 (ii) Gauss s suation theoe [7,p.49(Th. 18)] Γ(c)Γ(c a ) 1 Γ(c a)γ(c ) c a, (1.5) whee Re(c a ) > 0 and c 0, 1, 2, 3, 4,... (4) (iii) Kalsson s quaduple hypegeoetic function Hc Hc(4) [a1, a2, a3, a4 c x1, x2, x3, x4 ] k1,k2,k3,k4 0 [5,p. 37(2.1)] (a1 )k4 +k1 (a2 )k1 +k2 (a3 )k2 +k3 (a4 )k3 +k4 xk11 xk22 xk33 xk44 (c)k1 +k2 +k3 +k4 (k1 )! (k2 )! (k3 )! (k4 )! (1.6) (iv) Binoial coefficient ( ) n (n)! ()!(n )! (1.7) Any values of paaetes and vaiales leading to the esults given in sections 2 and 4 which do not ake sense, ae tacitly excluded. 2 Fou Finite Doule Suations In this section we otain the following fou finite doule suation foulae with the help of seies anipulations. If the values of a and ae adjusted in such a way that (1 ), (1 + a ) ae not integes c is neithe zeo no a negative intege then n (pp ), a + : (dd ) (ee ) C(, n,, s, a,, c)fqp +1:DE :GH+1 (qq ) : (gg ) (hh ), c + n + s ()+n P +2:DE+1 (pp ), a, + n : (dd ) (ee ), + + n FQ+1:GH+2 (c)+n (qq ), : (gg ) (hh ), + n, c + + n n P :D+1E C(, n,, s, a,, c)fq+1: G H (pp ) : (dd ), a + (ee ) (qq ), c + n + s : (gg ) (hh ) : (dd ), a, + n (ee ) ()+n P +1:D+2E (pp ), + + n FQ+2:G+1H (c)+n (qq ), + n, c + + n : (gg ), (hh ) (2.1) 3 (2.2) Page 106

4 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 n (pp ) : (dd ), a + (ee ) P :D+1E C(, n,, s, a,, c)fq: G H+1 (qq ) : (gg ) (hh ), c + n + s ()+n P :D+2E+1 (pp ) : (dd ), a, + n (ee ), + + n FQ :G+1H+2 (c)+n (qq ) : (gg ), (hh ), + n, c + + n n P +1:DE C(, n,, s, a,, c)fq+1:gh (pp ), a + : (dd ) (ee ) whee (qq ), c + n + s : (gg ) (hh ) (2.3) ()+n P +2:DE (pp ), + + n, a : (dd ) (ee ) F (c)+n Q+2:GH (qq ), c + + n, : (gg ) (hh ) (2.4) ( )( ) n ( 1)n++s (a) (c )s C(, n,, s, a,, c) s (a + 1) n (c)n+s (2.5) and Pochhae syols and Gaa functions ae well defined denoinato paaetes in hypegeoetic notations ae neithe zeo no negative integes. 3 Poofs of Finite Doule Suations On expessing each Kape de Fe iet s doule hypegeoetic function of left hand sides of (2.1), (2.2), (2.3) and (2.4) in powe seies fos with the help of (1.1) and (2.5), intechanging the ode of suations with the help of (1.2), (1.3) and (1.4) and successive applications of Gauss s suation theoe (1.5), we get the ight hand sides of (2.1), (2.2), (2.3) and (2.4). 4 Special Cases Making suitale adjustent of paaetes in (2.1), (2.2), (2.3) and (2.4) we can find a nue of finite doule sus involving doule hypegeoetic functions of Appell and Huet [3,pp ]. Putting P E 1, Q D G H 0 in (2.2) and using a esult of Calson [1,p. 222(4)], we have following doule sus 4 Page 107

5 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 )( ) n++s n ( 1) (a) (c )s 2:10 p, a + + e : a + F 1:10 x y, y s (a + 1) n (c)n+s c + n + s : a + + e n ( : a, + n e ()+n 2:21 p, + + n F2:10 (c)+n + n, c + + n : (4.1) When y x and e 0, (4.1) educes to a known esult of Pathan [6,p.58(2.1)] which was otained with the help of opeational calculus technique )( ) n ( 1)n++s (a) (c )s a +, p ()+n a, + + n, p x x 3 F2 (c) s (a + 1) n (c)n+s +n c + n + s, c + + n n ( (4.2) whee 2 F1 and 3 F2 ae Gauss s odinay hypegeoetic function [7,p.45(1)] and Clausen s hypegeoetic function [7,p.73(2)], espectively. In (4.2) putting n 0, 1, a and c d 1, we get a known contiguous elation [7,p.71,Ex. 21(2,13)] a, p a + 1, p a, p (a d + 1) x a x (d 1) x d d d 1 (4.3) When 0, x 1 and p in (4.2) and using Gauss s suation theoe (1.5), we get Saalschu tz s suation theoe[7,p.87(th. 29)] in the following fo n, 3 F2 a, 1+a c n 1 + a n, 1 + a + c n 1 (c )n ()n ( a)n (c a )n (4.4) ecause su of its denoinato paaetes exceeds the su of its nueato paaetes y unity. Putting P Q H D 0, G E 1 and setting e1 + n 1, g1 a, c 1 + in (2.3), and using a esult of Kalsson [4,p.197(8)] fo 2 F2, we get a esult involving the poduct 5 Page 108

6 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 of two Kue s confluent hypegeoetic functions 1 F1 [7,p.123(1)] n ( )( ) n++s n ( 1) (a) s! a + +n 1 x 1 F1 y 1 F1 s (a + 1) n (1 + )n+s +1 a n + s (1 n) + n 1 + n ++n x 1 F1 y + x 1 F1 y 1 F1 ( + 1)( + + n) +n n + n 1 F1 (4.5) When P Q E H D G 0 in (2.3) and y z2 we get 4 ( ) n ( )( ) n ( 1)n++s (a) (c )s 2 s Jc+n+s 1 (z) s (a + 1) n (1 x) z (1 x)a ( z )c+n 1 2 ()+n z2 a, + n ++n x 1 F2 (4.6) Γ(c + + n) 4 + n, c + + n whee J (z) is Bessel function of fist kind of ode [7,p. 108(1)]. When 0, (4.2) educes to n ( 0 ) n ( 1)n+ (a) a +, p x ()n (a + 1) n c+n a, + n, p 3 F2, c + n x (4.7) In (4.5), eplacing x and y y xt and yt espectively, ultiplying oth the sides y e t tc 1 and integating te y te with espect to t fo 0 to, we have n ( )( ) n ( 1)n++s (a) s! F2 [c a +, + n 1 a, n + s x, y] s (a + 1) n (1 + )n+s (1 n) F2 [c + n, + + n, n x, y] ( + 1)( + + n) + F2 [c + n, + n 1, + n x, y] +1 (4.8) whee F2 is Appell s doule hypegeoetic function of second kind [3,p. 224(7)]. In (4.8), eplacing x and y y xt and y(1 t) espectively, ultiplying oth the sides y 6 Page 109

7 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 tk 1 (1 t)c k 1 and integating te y te with espect to t fo 0 to 1, we have n ( )( ) n ( 1)n++s (a) s! (a + 1) n (1 + )n+s s c k, + n 1 x y 1++n+s a k, + n c k, + n 1 x y +1 +n + k, a + (1 n) c k, + + n k, + n y x ( + 1)( + + n) 1+++n (4.9) When P E 1, G D Q H 0 and p1 p2 + p3 in (2.2) and using a eduction foula of Kalsson [5,p. 40(4.5)], we get n ( )( ) n ( 1)n++s (a) (c )s s (a + 1) n (c)n+s Hc(4) [a +, p2, e, p3 c + n + s x, y, y, x] ()+n 2:21 p2 + p3, + + n : a, + n e F (c)n+ 2:10 + n, c + + n : (4.10) In all equations the values of a and ae adjusted in such a way that (1 ), (1 + a ) ae not integes c is neithe zeo no a negative intege Pochhae syols and Gaa functions ae well defined and denoinato paaetes in hypegeoetic notations ae neithe zeo no negative integes. Refeences [1] Calson, B.C. The need fo a new classification of doule hypegeoetic seies, Poc. Ae. Math. Soc., 56(1976), [2] Ede lyi,a. Hypegeoetic functions of two vaiales, Acta. Math., 83(1)(1950), [3] Ede lyi, A., Magnus, W., Oehettinge, F. and Ticoi, F. G. Highe Tanscendental Functions, Vol.I (Batean Manuscipt Poject), McGaw-Hill Book Co. Inc., New Yok, Toonto and London, Page 110

8 Intenational Jounal of Matheatics Tends and Technology (IJMTT) - Volue 33 Nue 2 - May 2016 [4] Kalsson, Pe W. Reduction of hypegeoetic functions with integal paaetes diffeences, Nedel. Akad. Wetensch. Poc.,Se.A,77Indag.Math.,36(1974), [5] Kalsson, Pe W. A class of hypegeoetic functions, Nedel. Akad. Wetensch. Poc.,Se.A,79Indag.Math.,38(1976), [6] Pathan, M.A. On a suation of 2 F1, Math. Chonicle,7, (1978), [7] Rainville, E.D. Special Functions, The Macillan, New Yok,1960, Repinted y Chelsea Pulishing Copany, Bonx,New Yok [8] Sivastava, H.M. and Panda, R. An integal epesentation fo the poduct of two Jacoi polynoials, J. London Math.Soc., (2), 12(1976), Page 111