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A Note on an Expansion of Hypergeometric Functions of Two Variables

Arun Verma

1966
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Mathematics of Computation
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1. In 1927 Fox [5] obtained the expansions of a product of two Bessel functions in a series of product of a Bessel function and a Gaussian hypergeometric function. Later on Rice [8] and Bailey [2] discovered a number of results of this type, some of them contained the result of Fox as a particular case. Recently, Srivastava [9] gave four general expansions of products of generalised hypergeometric functions in a series of product of generalised hypergeometric functions of two variables and a
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... ssian hypergeometric function, which incorporated as a special case the results of Fox, Rice and Bailey mentioned above. The aim of the present note is to derive an expansion of a generalised hypergeometric function of two variables in a series of product of generalised hypergeometric functions of two variables and a generalised hypergeometric function. The result deduced is further generalised to an expansion of a Meijer's G-function of two variables (defined recently by Agarwal) in a series of product of a Meijer's G-function and a hypergeometric function of two variables. The results obtained are very general and contain as special cases the expansions of Jerry L. Fields and Jet Wimp [6], L. Carlitz and W. Alsalam [4], Meijer [7] and many other results. The results are obtained by the use of the Laplace transform and the inverse Laplace transform as has been done by the author elsewhere also [11], [12]. The following notation due to Chaundy [3] shall be used to represent the hypergeometric function of higher order and of two variables y-y^ [iap)U+n [jbq)]m [(r,)]. m " ¿-> ¿-< Ml Ml \l A M M Z? M \(T> M X y ' \ap): (&,); (r,); liAP):iBQ);iRs)X'y_ [l]m[l]n[iAP)Un[iBQ)]m[iRs)]n where (am,") shall mean [n -m -f-1] parameters am , am+í, ■ ■ ■ an . But when m = 1, we shall write (a") instead of (ai,"). Also as usual T[(ap); ißq)] denotes [IlJ-i-rWñHllUm]}-1-2. The result to be deduced is: F\M, if*): (a,), icp); ' *Lig,),ihs):ih);idQ) XZ>yZ_ /j-, _ -y [2\U(er)]" , /lN|» F [~X 4-n + |, (er) 4-n; z n=o [1]« [X]" [ig,)]n r s \_ X F 2\ + 2n + 1, (g.) + n. -n, 2X 4-n, ifR): (ap); (cP); \ + i,ihs):ibt);idQ) X,V provided r + R + p<l + s + S + q,r + R + P<l + s + S + Q,r<s+l or r = s 4-1 and | z | < 1 and the series on the right-hand side has a meaning. To prove (2.1) we start from the result of Srivastava [9]

doi:10.2307/2003596
fatcat:2qftesuzoreyhkbqicn2jcvvea