NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR INTEGRAL REPRESENTATIONS

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1 SARAJEVO JOURNAL OF MATHEMATICS Vol.4 7, No., 8, DOI:.5644/SJM.4.-5 NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR INTEGRAL REPRESENTATIONS MAGED G. BIN-SAAD, JIHAD A. YOUNIS, RABIA AKTAS ABSTRACT. While investigating the Exton s list of hypergeometric functions of four variables namely K,K,...,K and the Sharma s and Parihar s list of eighty three hypergeometric functions of four variables namely F 4,F 4,...,F 4 8, we noticed the existence of new hypergeometric series of four variables. In the present work, we first introduce five new quadruple series and then we give integral representations of Euler type and Laplace type for these new hypergeometric functions, which we denote by j., j,,,4,5.. INTRODUCTION The theory of hypergeometric function of a single variable is more than years old. There are many papers on this subject in the literature for example [,, 4, 5, 7, 8,,,, 4]. The great success of the theory of one-variable hypergeometric function has motivated the development of the theory of hypergeometric functions in two and more variables by the fact that the solutions of partial differential equations arising in many applied problems of mathematical physics are given in terms of such hypergeometric functions see [,, 8] and the references cited therein. It has seen an important increase in the study of hypergeometric functions in the last two decades. This interest results from the relationships between hypergeometric functions and many areas of mathematics such as representation theory, algebraic geometry, combinatorics, number theory, mirror symmetry, etc. Furthermore, hypergeometric functions of several variables have many applications in physical and chemical problems [9,, 7]. For example, the solutions of degenerate second-order partial differential equations seen in many problems in gas dynamics are connected with multiple hypergeometric functions. Also, such hypergeometric functions are seen in the problem of adiabatic flat-parallel gas flow without whirlwind and in many other problems connected with gas flow see [6]. Mathematics Subject Classification. Primary C; Secondary C65. Key words and phrases. Beta and Gamma functions, Gauss hypergeometric function, triple hypergeometric functions, Exton s hypergeometric functions, Appell functions, Horn function.

2 46 MAGED G. BIN-SAAD, JIHAD A. YOUNIS, RABIA AKTAS Appell defined four hypergeometric functions of two variables denoted by F,F,F,F 4 see [8]. Horn [8] introduced ten hypergeometric functions of two variables namely G,G,G,H,...,H 7. Then, Lauricella gave fourteen complete triple hypergeometric functions denoted by F,F,...,F 4 see [8, section.4 and.5]. F,F,F 5 and F 9 of these hypergeometric functions correspond to the Lauricella hypergeometric functions of three variables F A,F B,F C and F D,respectively. Saran [] presented systematic study of the triple hypergeometric functions of Lauricella given by the symbols F E,F F,...,F T. Srivastava s triple hypergeometric functions are given by H A,H B and H C which aren t included among Lauricella s hypergeometric functions [4, section.5]. Exton [5] introduced twenty distinct triple hypergeometric functions namely X i i,...,. By the motivation of double and triple hypergeometric functions, Exton [4] defined twenty one complete hypergeometric functions of four variables by symbols K,K,...,K. In [4], Sharma and Parihar introduced eighty three complete hypergeometric F 4,F 4,...,F 4 8 of four variables. It is seen that the following functions of these 8 functions had already introduced in Exton s work [4]: K F 4 9,K F 4,K F 4 6,K 4 F 4,K 5 F 4,K 6 F 4 59, K 7 F 4 9,K 8 F 4,K 9 F 4,K F 4,K F 4 6,K F 4 4, K F 4,K 4 F 4 77,K 5 F 4 78,K 6 F 4 79,K 9 F 4 8, K F 4 8,K F 4 8. Each quadruple hypergeometric series is of the form. m,n,p,q Λm,n, p,q xm y n z p u q m! n! p! q!, where Λm,n, p,q is a certain sequence of complex parameters and there are twelve parameters in each series. eight a s and four c s. The st, nd, rd and 4th parameters in. are connected with the integers m,n, p and q, respectively. Each repeated parameter in the series. points out a term with double parameters in Λm,n, p,q. For example, a,a,a,a,a,a,a 4,a 5 means that Λm,n, p,q includes the term a m+n a p+q a m+n a 4 p a 5 q. Similarly, a,a,a,a,a,a,a,a points out the term a m+n+p a p+q a q and a,a,a,a,a,a,a,a shows the existence of the term a m+n+p a p+n+q. Thus, it is possible to form various combinations of indices. Here, a m denotes the Pochhammer symbol given by a m Γa+m aa+...a+m m N:{,,...} and a. Γa By using the conventions and notations above, we introduce the following five quadruple functions apart from the hypergeometric function of four variables defined by Exton [4] and Sharma and Parihar [4] :

3 NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u a m+n+p+q a p+q x m y n z p c n+p c m c q m! n! p! m,n,p,q a,a,a,a,a,a,a,a ;c,c,c,c 4 ;x,y,z,u a m+n+q a q+n+p x m y n z p c m c n c p c 4 q m! n! p! m,n,p,q a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u a m+n+q a q+n+p x m y n z p c m+n c p c q m! n! p! m,n,p,q 4 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u a m+n+q a q+n+p x m y n z p c m+p c n c q m! n! p! m,n,p,q 5 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u a m+n+q a q+n+p x m y n z p c m+n+p c q m! n! p! m,n,p,q u q q!,. u q q!,. u q q!,. u q q!,.4 u q q!,.5 where the numerator and the denominator parameters are separated via a semicolon. This paper is devoted to obtain several integral representations for the new quadruple functions defined above. In section, we present five integral representations of Euler-type for each series j j,,,4,5 in terms of Appell s functions of two variables F and F 4, the Horn s function H 4 of two variables, the Gaussian hypergeometric function F, the Exton s triple series X,X 9,X,X and X see [5], and the quadruple series and F 4 C. In the next section, Laplace -type integrals are obtained for each series j j,,,4,5.. INTEGRAL REPRESENTATIONS OF EULER-TYPE We recall that the Gaussian hypergeometric function is defined by see [9] F a,b;c;x n a n b n c n x n, x <. n! Appell s hypergeometric functions F and F 4 of two variables and the Horn s series H 4 of two variables are given by F a,b,c,d;e;x,y m,n a m b n c m d n e m+n x m y n, max { x, y }<, m! n!

4 48 MAGED G. BIN-SAAD, JIHAD A. YOUNIS, RABIA AKTAS and F 4 a,b;c,d;x,y m,n H 4 a,b;c,d;x,y a m+n b m+n c m d n m,n x m y n m! n!, x + y <, a m+n b n c m d n x <r, y <s,4r s, x m y n m! n!, respectively see [9]. The Exton s triple functions X,X 9,X,X and X see [5] are defined by as follows: X a,b;c,d;x,y,z X 9 a, b; c ;x,y,z X a,b;c,d;x,y,z m,n,p m,n,p m,n,p a m+n+p b p c m d n+p a m+n b n+p c m+n+p a m+n b n+p c m+n d p x m y n m! n! x m y n m! n! x m y n m! n! z p p!, z p p!, z p p!, and X a,b;c,d;x,y,z X a,b;c,d,e;x,y,z m,n,p m,n,p a m+n b n+p c m+p d n a m+p b n+p c m d n e p x m y n m! n! x m y n m! n! z p p! z p p!. Lauricella hypergeometric function of four variables F 4 C F 4 C a,b;c,c,c,c 4 ;x,y,z,u m,n,p,q is as below see [9] a m+n+p+q b m+n+p+q x m c m c n c p c 4 q m! x + y + z + u <. z p n! p Now, by means of the Gauss hypergeometric function F, Appell hypergeometric functions F and F 4, Horn s function H 4 of two variables, the Exton s triple series X,X 9,X,X and X, and the quadruple series and F 4 C, we investigate some further integral representations of Euler-type for j j,,,4,5 as follows: y n u q q!,

5 NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u π π 4Γa + a Γc sin α a cos α c a sin β a Γa Γa ΓaΓc a cos β a 4 a + a F C, a + a + ;c,a,c a,c ;λ x,λ y,λ z,λ 4 u dαdβ λ 4sin 4 β,λ 4sin α sin 4 β,λ cos α sin β,λ 4 sin β, Rea >,Rea >,Rea>,Rec a>,. a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc S T a R T c a Γa Γc a S R c a S R α R a S α c a α T a c X a,+a c ;c,c ;λ x,λ y,λ zdα S R α T λ [S Rα T S Tα Ru], S RS Tα Tα R λ, R TS α[s Rα T S Tα Ru] Rea >,Rec a >,T < R<S,. a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc α a +α a c [+α αu] a Γa Γc a +α X a x +α y,+a c ;c,c ; [+α αu], [+α αu], α+αz dα [+α αu] Rea >,Rec a >,. a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc e α a e α c a ue α a Γa Γc a x y X a,+a c ;c,c ; ue α, ue α, ze α e α ue α dα Rea >,Rec a >,.4

6 5 MAGED G. BIN-SAAD, JIHAD A. YOUNIS, RABIA AKTAS a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc α a [ α+α y] c a αz a Γa Γc a α H 4 +a x c,a ;c,c ; [ α+α y], αu [ α+α dα y] αz Rea >,Rec a >,.5 a,a,a,a,a,a,a,a ;c,c,c,c 4 ;x,y,z,u Γa + a S T a R T a S α R a S α a Γa Γa S R a +a R α T a +a F 4 a + a C, a + a + ;c,c,c,c 4 ;λ x,λ y,λ z,λ u dα λ 4S T α R S R α T,λ 4S TR Tα RS α S R α T, Rea >,Rea >,T < R<S,.6 a,a,a,a,a,a,a,a ;c,c,c,c 4 ;x,y,z,u π π 4Γc Γc sin α a cosαsin β a Γa Γa Γc a Γc a cosβcos α+xsin 4 α c a cos β+zsin 4 β c a F 4 +a c,+a c ;c,c 4 ;λy,λu dαdβ sin α sin β λ cos α+xsin 4 αcos β+zsin 4, β Rea >,Rea >,Rec a >,Rec a >,.7 a,a,a,a,a,a,a,a ;c,c,c,c 4 ;x,y,z,u Γc Γc α a +α +a c β a Γa Γa Γc a Γc a +β +a c [+α+α x] c a [+β+β z] c a F 4 +a c,+a c ;c,c ;λy,λu dαdβ αβ+α+β λ [+α+α x][+β+β, z] Rea >,Rea >,Rec a >,Rec a >,.8

7 NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR... 5 a,a,a,a,a,a,a,a ;c,c,c,c 4 ;x,y,z,u 4Γc Γc 4 π/ π/ sin α a cos α c a sin β a Γa Γa Γc a Γc 4 a cos β c 4 a +utan αsin β c a +ysin αtan β c 4 a +a c F, a c 4xtan 4 α + ;c ; +utan αsin β +a c 4 F, a c 4 4ztan 4 β + ;c ; +ysin αtan β dαdβ Rea >,Rea >,Rec a >,Rec 4 a >,.9 a,a,a,a,a,a,a,a ;c,c,c,c 4 ;x,y,z,u Γc Γc 4 α a β a [ α+αβ u] c a Γa Γa Γc a Γc 4 a [ β+αβ y] c 4 a +a c F, a c 4α x + ;c ; [ α+αβ u] +a c 4 F, a c 4 4β z + ;c ; [ β+αβ y] dαdβ Rea >,Rea >,Rec a >,Rec 4 a >,. a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc α a α c a a,a,a,a,a,a,a,a ; ΓaΓc a a,c a,c,c ;αx, αy,z,u dα a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc S T a R T c a Γa Γc a S R c a Rea>,Rec a>,. S R α R a S α c a α T a c [S Rα T S Tα Ru] a X a,+a c ;c,c ;λ x,λ y,λ zdα S R α T λ [S Rα T S Tα Ru], S RS Tα Tα R λ R TS α[s Rα T S Tα Ru], λ S T α R R T S α,

8 5 MAGED G. BIN-SAAD, JIHAD A. YOUNIS, RABIA AKTAS Rea >,Rec a >,T < R<S,. a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u π Γc sin α a cos α c a u sin α a Γa Γc a x X a,+a c ;c,c ; y tan α u sin α, u sin α, z tan 4 α dα Rea >,Rec a >,. a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc α a [ α+α x] c a αy a Γa Γc a z αu H 4 a,+a c ;c,c ; αy, [ α+α dα x] αy Rea >,Rec a >,.4 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc +α +a +a c [α+α+x] c a Γa Γc a [+α y] a +α H 4 a z,+a c ;c,c ; [+α y], +α u dα [α+α+x][+α y] Rea >,Rec a >,.5 4 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u π Γc sin α a cos α c a u sin α a Γa Γc a u sin α, z u sin α dα Rea >,Rec a >,.6 4 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u X +a c,a ;c,c ;x tan 4 α, y tan α Γc Γa Γc a α a +α a c [+α αu] a X +a c,a ;c,c ;α x, α+αy [+α αu], +α z [+α αu] dα Rea >,Rec a >,.7

9 NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc Γc α a α c a β a β c a ΓaΓa Γc aγc a βy a X a,+a c ;a,c a,c ; αx z βy, αβ β, βu dαdβ β βy Rea>,Rea >,Rec a>,rec a >,.8 4 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u π π 4Γc Γc sin α a cos α c a ΓaΓa Γc aγc a sin β a cos β c a ysin β a X a,+a c ;a,c a,c ; x sin α ysin β,z cos α tan 4 u tan β β, ysin dαdβ β Rea>,Rea >,Rec a>,rec a >,.9 4 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc Γc α a β a Γa Γa Γc a Γc a [ β+αβy] c a [ α+αβu] c a F +a c, +a c, a c +, a c + ;c ; 4α x 4β z [ α+αβu], [+β+αβy] dαdβ Rea >,Rea >,Rec a >,Rec a >,. 5 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc Γa Γc a π sin α a cos α c a u sin α a X 9 +a c,a ;c ;x tan 4 α, y tan α u sin α, z u sin α dα Rea >,Rec a >,.

10 54 MAGED G. BIN-SAAD, JIHAD A. YOUNIS, RABIA AKTAS 5 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc Γa Γc a α a +α a c [+α αu] a X 9 +a c,a ;c ;α x, α+αy [+α αu], +α z [+α αu] 5 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc S T a R T c a Γa Γc a S R c a dα Rea >,Rec a >,. S R α R a S α c a α T a c [S Rα T S Tα Ru] a X 9 +a c,a ;c ;λ x,λ y,λ zdα λ S T α R R T S α, S RS Tα Tα R λ R TS α[s Rα T S Tα Ru], S R α T λ [S Rα T S Tα Ru], Rea >,Rec a >,T < R<S,. 5 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc α a α c a αu a Γa Γc a α X 9 +a x αy c,a ;c ; α, α αu, z αu dα Rea >,Rec a >,.4 5 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u Γc e α a e α c a ue α a Γa Γc a xe X 9 +a α ye α c,a ;c ; e α, e α ue α, z ue α dα Rea >,Rec a >..5 Proof. Once substituting the series definition of the special function in each integrand and then, changing the order of the integral and the summation, and finally

11 NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR taking into account the following integral representations of the Beta function and their various associated Eulerian integrals see, for example, [, p. 9-], [5, 6, Section.] and [9, p. 6 and p. 86, Problem ], we derive each of the integral representations from. to.5. ta t b dt Rea>,Reb>, Ba,b.6 ΓaΓb Γa+b a,b C\Z, Ba,b α a α b dα e α a e α b dα Rea>,Reb>,.7 π Ba,b sinα a cosα b α a dα dα +α a+b Rea>,Reb>,.8 Ba,b S Ta R T b S α R a S α b S R a+b R α T a+b dα T < R<S,Rea>,Reb>..9. INTEGRALS OF LAPLACE-TYPE We represent the quadruple series,x4,x4, 4,X4 5 in terms of binary integrals by means of Laplace transform. The Laplace integral representations of these quadruple series are given as follows: a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u e s+t s a t a F ;c ;s y+stz Γa Γa F ;c ;s x F ;c ;stu dsdt, Rea >,Rea >,. a,a,a,a,a,a,a,a ;c,c,c,c 4 ;x,y,z,u e s+t s a t a F ;c ;s x F ;c ;sty Γa Γa F ;c ;t z F ;c 4 ;stu dsdt, Rea >,Rea >,. a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u e s+t s a t a F ;c ;s x+sty Γa Γa F ;c ;t z F ;c ;stu dsdt, Rea >,Rea >,.

12 56 MAGED G. BIN-SAAD, JIHAD A. YOUNIS, RABIA AKTAS 4 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u e s+t s a t a F ;c ;s x+t z Γa Γa F ;c ;sty F ;c ;stu dsdt, Rea >,Rea >,.4 5 a,a,a,a,a,a,a,a ;c,c,c,c ;x,y,z,u e s+t s a t a F ;c ;s x+sty+t z Γa Γa F ;c ;stu dsdt, Rea >,Rea >,.5 where F is the confluent hypergeometric function defined by see [9] F ;c;x m x m c m m!..6 Proof To prove each of the integral representations from. to.5, it is enough to substitute the expression of the hypergeometric function F ;c;x given by.6 in each integrand and then to change the order of the integral and the summation, and finally to use Gamma function [] Γz e t t z dt, Rez>. REFERENCES [] S. Ahmad, Hypergeometric functions of three variables in terms of integral representations, IOSR Journal of Mathematics, 85, [] J. Choi, A. Hasanov, H.M. Srivastava and M. Turaev, Integral representations for Srivastava s triple hypergeometric functions, Taiwanese Journal of Mathematics, 56, [] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 95. [4] H. Exton, Multiple hypergeometric functions and applications, Halsted Press, New York, London, Sydney and Toronto, 976. [5] H. Exton, Hypergeometric functions of three variables, J. Indian Acad. Math., 4 98, 9. [6] F.I. Frankl, Selected Works in Gas Dynamics, Nauka, Moscow, 97 in Russian. [7] A. Hasanov, H.M. Srivastava and M. Turaev, Decomposition formulas for some triple hypergeometric functions, J. Math. Anal. Appl. 4 6, [8] G. Lauricella, Sull funzioni ipergeometric a più variabili, Rend. Cric. Mat. Palermo 7 89, 58. [9] G. Lohöfer, Theory of an electromagnetically deviated metal sphere. I: Absorbed power, SIAM J. Appl. Math., , [] A.W. Niukkanen, Generalised hypergeometric series N Fx,...,x N arising in physical and quantum chemical applications, J. Phys. A: Math. Gen., 6 98, [] S.B. Opps, N. Saad and H.M. Srivastava, Some reduction and transformation formulas for the Appell hypergeometric function F, J. Math. Anal. Appl., 5, [] P.A. Padmanabham and H.M. Srivastava, Summation formulas associated with the Lauricella function F r A, Appl. Math. Lett., 65 7.

13 NEW QUADRAPLE HYPERGEOMETRIC SERIES AND THEIR [] S. Saran, Hypergeometric functions of three variables, Ganita 5 954, [4] C. Sharma and C.L. Parihar, Hypergeometric functions of four variables I, J. Indian Acad. Math., 989,. [5] H.M. Srivastava, and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London,. [6] H.M. Srivastava, and J. Choi, Zeta and q-zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York,. [7] H.M. Srivastava, A class of generalised multiple hypergeometric series arising in physical and quantum chemical applications, J. Phys. A: Math. Gen., 8 985, L7 L4. [8] H.M. Srivastava and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press, Bristone, London, New York and Toronto, 985. [9] H.M. Srivastava and H.L. Manocha, A treatise on generating functions, Ellis Horwood Lt., Chichester, 984. Received: May 9, 7 Maged G. Bin-Saad Aden University Department of Mathematics Aden, Khormaksar, P.O.Box 64,Yemen mgbinsaad@yahoo.com Jihad A. Younis Aden University Department of Mathematics Aden, Khormaksar, P.O.Box 64,Yemen jihadalsaqqaf@gmail.com Rabia Aktas Ankara University Department of Mathematics Faculty of Science Ankara, Turkey raktas@science.ankara.edu.tr

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