<* - i ^ x o.2) I n r (*-z^'*)n r o- c >+z c / v,w:(p',q');...;(p«\qv) [(a): A',..., AW] : [(6') : '] :... ; : 5«] ; ^ ^{itwir f... - PDF

İstanbul Üniv. Fen Fak. Mec. Seri A, 47 ( , CONTOUR INTEGRALS INVOLVING FOX'S H-FUNCTION AND THE MULTIVARIABLE H-FUNCTION V. B. L. CHAURASIA The author presents two contour integrals

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İstanbul Üniv. Fen Fak. Mec. Seri A, 47 ( , CONTOUR INTEGRALS INVOLVING FOX'S H-FUNCTION AND THE MULTIVARIABLE H-FUNCTION V. B. L. CHAURASIA The author presents two contour integrals for the multivariable //-function. Each of these contour integral formulae involves a product of Fox's li-function the multivariable.//-function. On account of the general nature of Fox's //-function the multivariable //-function a large number of known as well as new interesting contour integral formulae follow as special cases of these results. 1. Introduction The multivariable //-function has been recently introduced by H.M. Srivastava R. Pa is defined represented in the following form 1 v,w:(p',q';...;(p«\qv [(a: A',..., AW] : [(6' : '] :... ; : 5«] ; ^ [(c : C,..., C«] : [(d' : D'] ;... ; :^ ]. 1 ' ' ' ' ^{itwir f... f UM^UMVis, s r z\ l...z s ;ds y...ds rt where u* =n r w k} - D t k *â n r b * - i ^ x o.2 S=-k j=k «r V x - 7 I n r (*-z^'*n r o- c +z c / J=H+1 = l j^-l k=l The parameters 112 V.BX. CHAURASIA (fly, J= I, -, v; j=\,...,pw (cj,j=l,... W ; dp tj= l,...,(2 ( * ;V/ce {l,...,r}, (1.4 are complex numbers, the associated coefficients I CP, j = 1,..w ; D/«, 1 G t t ; Vk e {1,..r}, (1.5 are positive real numbers such that A* - E ^ + E - E c t* - Z ^ 0 ( 1, 6 j=l /=1 J-l j=i = - E 4 W + E */* } - E E c / w + o- 7 + E ^ - E ^ 0, V e {1.., r} where the integers w, A/ i A , JV,i:, v, P (ft , w Q ik are constrained by the inequalities 0 w v, 1 M (fc g^, w 0, 0 7V (W , V&e {1,...,r}, the equality in (1.6 holds for suitable restricted values of the complex variables z i z r. The contours k are defined suitable all the poles of the integr are assumed to be simple. The multiple integral in (1.1 converges absolutely under the conditions (1.7, when arg(z A 7W2, V f t e {1,..., r}, (1.8 the points z k = 0, k = \,...,r, tactically excluded. various exceptional parameter values, being The following result is required in our investigation: The series representation of Fox's if-function (C 1 ]^ 2 ] if' - p. i y {e p, E p ( ^ * *,/ (1.9 {f a,f q \ ^s\f where CONTOUR INTEGRALS INVOLVING FOX'S JÏ-FUNCTION 113 /-U4-0 y=i x { fl rq-fj + Fjgjf[r ej-&jg \ ' g, = (fe + swa- (LID We remark in passing that, throughout the present work, we shall assume that the convergence ( existence conditions corresponding appropriately to the ones detailed above are satisfied by the multivariable //-functions involved. With a view to facilitating the derivation of our main contour integrals (2.1 (2.2 in the next section, we give here an elementary contour integral contained in the following Lemma. If h 0, c Re (z Re (p 0, then lit C + /00 - I ' f f -I- 7 1 H -. Vit -4-7.Y C JOO (e P Epï (f q F q dt (1.12 (~ iy l (g s (yhf s s\f a T(p + agj Proof. The assertion (1.12 of the above lemma follows at once by applying the definition (1.9 in conjunction with the following form of the well known Hankers contour integral for the Gamma function 1 r h p ~ l e~ hz / e*'(/ + z- p.ji = e, 2%i J T(p h 0, (1.13 C ( co where, for convergence, o Re (z./te (p The main contour integrals Our main results of the paper are the contour integrals contained in the following theorems: Theorem 1. With A k S\ defined by (1.6 (1.7, respectively, let A k 0 arg (z k j to k TU/2, V/C e {1,..., r}, (2.1 where each of the equalities holds for suitably restricted values of the complex variables z x,...,z r. Also let the function H q[y] be defined by (1.9, let h 0 c Re (z. 114 V.B.L. CHAURASIA Then c+i o (f g,f s X (2.2 X H{Z X (t + Z- P Z f (t + z- r dt '-22^ ( - i 5 Fr. 4 {g s (yhf*x XH 0, w:(m', JV ;... ;(M^,N(r v,w + i:(p' f G';...;(f ( r,g 0 [1 -p-c^ip!,..., P r ], [(a :A\..., A*y\ : [( 0 : P-'j;... ; [(#- ; jfr] ; [(c: C,..., C«J : [(rf': D'];...; [(a* : /* ]; provided that p y - 0, y = 1,..., r, a 0, j argy ] TTC/2 with H p HI p j=l j=n +1 J ~ 1 7=' +1 (2.4 Theorem 2. Under the hypothesis preceding the assertion (2.2 of Theorem I, J-1 7 «+*rj«r» E P X (2.5 XH( Zl (i + z p i,...,z p (/ + Z p rrfi / 7 p-i e-iu ( - i y i!p X r0, u : (M\ N' ;... ; (M ' , 7V«/[(a: /(',...,^' ] ; [p + ag s : P l p r ] v +1, w: (P', 20 ;...; (P«, gw \ [(c : C,..., C ' ]: [(6':^];-;[0 ( i :fi w ]; provided that py 0, j = 1,..., r, a 0, argy j Tn/2 CONTOUR INTEGRALS INVOLVING FOX'S //-FUNCTION 115 Re(p -cY i Pj (2-6 j=-i Proofs of Theorems 1 2. The contour integral formula (2.2 contained in Theorem 1, can be established if we first replace the multivariable //-function in the integr by its multiple contour integral (1.1, change the order of integration, evaluate the inner most integral by appealing to the assertion (1.12 of our lemma the interpret the resulting multiple contour integral as an //-function of r complex variables. Similar is the derivation of the contour integral (2.5 of Theorem Applications Each of contour integral formulae given by Theorem 1 2 of the preceding section possesses manifold generality. On specializing the parameters the multivariable //-function may be transformed into G-functions, P-functions, //-functions, Lauricella's functions, Appell's functions, Kampe de Feriet functions, Hypergeometric functions, Legendre functions, Bessel functions several other higher transcendental functions in one, two or more arguments, Therefore, the results established in this paper are of general nature hence encompass several cases of interest. A C K N O W L E D G E M E N T The author is thankful to the University Grants Commission, India, for providing necessary financial assistance for carrying out Che present work. ['] FOX, C. R E E F E R N C E S : The G H functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961, [! ] SKIBINSKI, P. : Some expansion theorems for the H-function, Ann. Polon. Math. 23 (1970, [ ] SR1VASTAVA, H.M. : Some bilateral generating function for a class of gene PANDA, R. ralized hypergeometric polynomials, I. reine angew. Math (1976, DEPARTMENT OF MATHEMATICS UNIVERSITY OF RAJASTHAN JAIPUR , INDIA ÖZET Bu çalışmada çok değişkenli //-fonksiyonları İçin iki çevre integrali sunulmaktadır.
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