ON THE MELLIN TRANSFORM OF A PRODUCT OF HYPERGEOMETRIC FUNCTIONS

Transcription

1 J. Autral. Math. Soc. Ser. B 40(998), 37 ON THE MELLIN TRANSFORM OF A PRODUCT OF HYPERGEOMETRIC FUNCTIONS ALLEN R. MILLER H. M. SRIVASTAVA (Received 7 June 996) Abtract We obtain repreentation for the Mellin tranform of the product of generalized hypergeometric function 0 F [ a x ] F [ b x ] for a; b > 0. The later tranform i a generalization of the dicontinuou integral of Weber Schafheitlin; in addition to reducing to other known integral (for example, integral involving product of power, Beel Lommel function), it contain numerou integral of interet that are not readily available in the mathematical literature. A a by-product of the preent invetigation, we deduce the econd fundamental relation for 3 F []. Furthermore, we give the ine coine tranform of F [ b x ].. Introduction Although definite integral of product of two generalized hypergeometric function have numerou application in pure applied mathematic (ee, for example, [3]), not all uch integral have been collected in table or are readily available in the mathematical literature. In what follow, we hall conider for a > 0 b > 0 one uch rather general improper integral, namely the Mellin tranform: F./ :D Z x 0 I 0F C ¼I a x Þ I F þ; C ¹I b x dx: (.) Convergence of thi integral will be dicued in Section. If Þ D þ, the above Mellin tranform reduce to the hypergeometric formulation of the dicontinuou integral of Weber Schafheitlin (cf. [, p. 398] [6, Equation 66 Eighteenth Street NW, Wahington, D. C , USA Department of Mathematic Statitic, Univerity of Victoria, Victoria, B. C. V8W 3P4, Canada. c Autralian Mathematical Society, 998, Serial-fee code /98

2 [] On the Mellin tranform of a product of hypergeometric function 3 (.3) to (.5)]) which, ince we hall need it later, we record below: Z x I 0F a x I 0 C ¼I 0F b x dx C ¹I D b 0. C ¹/0. / 0. C ¹ / F ; ¹I a (.a) C ¼I b.0 < a < bi 0 < <./ <<. C ¼ C ¹// D a 0. /0. C ¼/0. C ¹/0. C ¼ C ¹ / 0. C ¼ /0. C ¹ /0. C ¼ C ¹ / (.b).b D ai 0 < <./ <<. C ¼ C ¹// D a 0. C ¼/0. / 0. C ¼ / F ; ¼I C ¹I b a.0 < b < ai 0 < <./ <<. C ¼ C ¹//: (.c) The above integral i, in fact, continuou at a D b. It i called dicontinuou becaue the repreentation on the right-h ide of Equation (.a) (.c) are not analytic continuation of each other. Thu dicontinuou refer to a dicontinuity in repreentation acro a D b. It hould be noted alo that when a D b, if¼ ¹ i an odd poitive integer, then Equation (.b) actually hold true for 0 < <./ < <. C ¼ C ¹/ (ee Waton treatie [, p. 403] for further detail). In addition, we record for later ue the well-known Mellin tranform (ee, for example, [7, Section (), p. 77]) Z x 0 I 0F C ¼I a x dx D a 0. C ¼/0. / 0. C ¼ / ; (.3) where a > 0 0 < <./ <<. 3 C ¼/. Since we may write Beel, Struve Lommel function, repectively, a (ee, for example, [0, p. 44]) ¼;¹.z/ D J ¼.z/ D. z /¼ I 0. C ¼/ 0 F C ¼I. z /C¼ H ¼.z/ D 0. 3 /0. 3 C ¼/ F 3 ; 3 z ; (.4) 4 I ; 4 C ¼I z z C¼. C ¼ ¹/. C ¼ C ¹/ F.3 C ¼ ¹/; I z.3 C ¼ C ¹/I 4 once a repreentation in term of generalized hypergeometric function i deduced for the Mellin tranform F./ (which will be done in Section 5), not only will we be able ;

3 4 Allen R. Miller H. M. Srivatava [3] to obtain immediately all the known pecial cae (for example, integral containing the product of two Beel function or the integral of Weber Schafheitlin [, pp ], product of Beel Lommel function [7, Section.9.5.(), p. 0], product of Beel Struve function [7, Section.7.4.(), p. 88]), but we hall alo be able to pecialize, for example, to product of Beel function the Frenel ine coine integral S.z/ D 3 r " 3 ³ z3= F I # 4 z 3 ; 7 I 4 4 C.z/ D r " ³ z= F I # 4 z ; 5 I ; 4 4 every other pecial cae of F that i of interet. Recently, M. A. Chaudhry [] reconidered F./ pecialized to Beel Lommel function ¼;¹.z/ dicued the importance of thee integral in many application. Unfortunately, Chaudhry reult [, Equation (4)] hold true for the Lommel function S ¼;¹.z/ not for ¼;¹.z/ the right ide of [, Equation (6)] i in error by a factor of. Aide from numerou typographical other overight in [], the latter two mentioned equation produced further erroneou reult. Prudnikov et al. [7, Section..4.(), p. 337] record the Mellin tranform of product of Beel generalized hypergeometric function p F q [z], but the reult i obviouly not correct when p D q, ince the integral mut be dicontinuou.. Convergence of the Integral (.) We hall need the following aymptotic reult: for jxj! j arg.x/j < ³, I 0F C ¼I x D 0. C ¼/ 0. / x ¼ co x ³ C ¼ (.) ÞI F þ; C ¹I x D 0.þ/0. C ¹/ 0. /0.Þ/ x ¹CÞ þ co x C ³ C ¹ C þ Þ 0.þ/0. C ¹/ 0.þ Þ/0. C ¹ Þ/ x Þ : (.)

4 [4] On the Mellin tranform of a product of hypergeometric function 5 The above reult are further implification of more precie aymptotic formula which we hall record in Section 4 (ee Equation (4.) (4.)). Note that Equation (.) reduce to Equation (.) if we et Þ D þ. Replacing x by bx (b > 0) in Equation (.) uing Equation (.), it i eay to ee that we may alo write for jxj! j arg.x/j < ³ ÞI 0.þ/0. C ¹/ F b x I D þ; C ¹I 0.Þ/0. C ¹ C þ Þ/ 0 F C ¹ C þ ÞI b x 0.þ/0. C ¹/ C 0.þ Þ/0. C ¹ Þ/.bx/ Þ ; (.3) which now yield an identity when a D þ. Next, by multiplying Equation (.3) by x ð 0F I C ¼I a x Ł ;, for z > 0 ufficiently large, integrating the reult over.z; /, wehavefor a; b > 0 the aymptotic reult Z x I 0F a x ÞI z C ¼I F b x dx þ; C ¼I Z 0.þ/0. C ¹/ D x I 0F a x 0.Þ/0. C ¹ C þ Þ/ z C ¼I (.4) I 0 F b x dx C ¹ C þ ÞI Þ Z 0.þ/0. C ¹/b C x Þ I 0F a x dx: 0.þ Þ/0. C ¹ Þ/ C ¼I It i obviou that, in order for the Mellin tranform F./ defined by equation (.) to converge at it lower limit of integration, we mut have <./ >0. Further, from the latter aymptotic reult the convergence criteria of equation (.) (.3), we ee that, for b 6D a, convergence at the upper limit of integration i attained, provided that <./ <<. C ¼ C ¹ C þ Þ/ <./ << 3 C Þ C ¼ : If b D a, F./ converge, provided that 0 < <./ <<. C ¼ C ¹ C þ Þ/ 0 < <./ << 3 C Þ C ¼ I but if b D a ¼ ¹ C Þ þ i an odd poitive integer, then F./ converge, provided that 0 < <./ <<. C ¼ C ¹ C þ Þ/ 0 < <./ << 3 C Þ C ¼ : z

5 6 Allen R. Miller H. M. Srivatava [5] We note that if Þ D þ, the condition <./ <<. 3 C Þ C ¼/ become uperfluou, ince the econd term in equation (.4) i no longer preent. Thi evidently complete the analyi of the convergence of F./. 3. The Incomplete Mellin Tranform If the upper limit of integration in equation (.) i replaced by z uch that jzj <, the integral i aid to be incomplete we define the Euler-type integral Z z F.I z/ :D x I 0F a x ÞI 0 C ¼I F b x dx; (3.) þ; C ¹I where, for convergence at the lower limit of integration, <./ >0. But now the parameter a b may be arbitrary complex number. The incomplete Mellin tranform F.I z/ i eaily evaluated by expreing the function 0 F F a hypergeometric um, interchanging the reulting double um integral then performing the required term-by-term integration. Or the integral in equation (3.) may be evaluated by uing a tabulated reult in Exton hbook [3, Equation A(..5), p. 7]. Either way we obtain F.I z/ in term of a ingle Kampé defériet function which i everywhere convergent in it two independent variable: F.I z/ D z F :0I : I ÞI :I C : C ¼Iþ; C ¹I a z ; b z ; (3.) where <./ >0. Region of convergence for Kampé defériet other generalized hypergeometric function in two variable may be determined by uing Horn theorem for double erie (ee Srivatava Karlon [9, p. 57]). Further, it i eay to how that the incomplete tranform F.I z/ in equation (3.) may be written in the following two different way for <./ >0: X F.I z/ D z.þ/ n n. b z /n nd0 C.þ/ n n. C ¹/ n n! F C ni C C n; C ¼I a z (3.3) X F.I z/ D z n. a z / n nd0 C. C ¼/ n n n! F C n;þi 3 C C n;þ; C ¹I b z : (3.4)

6 [6] On the Mellin tranform of a product of hypergeometric function 7 4. Aymptotic Formula for p F pc [ z ] We hall want to evaluate F.I z/ given by equation (3.3) (3.4) a z!, thereby obtaining the Mellin tranform F./ defined by equation (.). To thi end, we record below omewhat implified aymptotic formula for the generalized hypergeometric function p F pc [.a p /I.b pc /I z ]; p D 0; ; : Thee reult are pecial cae of a general formula due to C. S. Meijer (circa 946) given by Luke [4, p. 03, Eq. (4)] for jzj! j arg.z/j <³=: I 0F z D 0.a/ ai 0. / C O co ¾.z/; (4.) z z where ai F b; ci D a ¾.z/ D z ³ C O z I 0.b/0.c/ a z D 0.b a/0.c a/ z 3 F 0 a; C a b; C a ci I z C 0.b/0.c/ 0. /0.a/ C O co ¾.z/; (4.) z z where a; bi F 3 c; d; ei D.b C c a / ¾.z/ D z ³ C O. z /I z D 0.b a/ 0.b/ 0.c/0.d/0.e/ 0.c a/0.d a/0.e a/ a; C a c; C a d; C a ei 4 F C a bi 0.a b/ 0.c/0.d/0.e/ C 0.a/ 0.c b/0.d b/0.e b/ z b; C b c; C b d; C b ei 4 F C b ai C 0.c/0.d/0.e/ 0. /0.a/0.b/ C O z z z a z b z co ¾.z/; (4.3)

7 8 Allen R. Miller H. M. Srivatava [7] where D.c C d C e a b / ¾.z/ D z ³ C O. z /: Noting equation (.4), we ee that equation (4.) i eentially the well-known aymptotic formula for Beel function. We remark alo that the right member of equation (4.) to (4.3) may be written in further abbreviated, yet ueful, approximate form (cf. [5, p. 46]) I 0F z ¾ Az a co.z C B/; (4.4) ai ai F z ¾ Az a C Bz Ca b c co.z C C/ (4.5) b; ci a; bi F 3 c; d; ei z ¾ Az a C Bz b C Cz CaCb c d e co.z C D/; where jzj!; j arg.z/j <³=, A; B; C; D are dependent on the parameter of the function p F pc [ z ].p D 0; ; /. 5. Evaluation of the Mellin Tranform (.) Now that we have noted developed ome preliminarie, we are ready to derive our main reult for the Mellin tranform F./ defined in equation (.) a Z F./ :D x I 0F a x ÞI C ¼I F b x dx; þ; C ¹I 0 where a > 0b > 0. Thu we hall how that F./ D a 0. /0. C ¼/ 0. C ¼ / 3 F Þ; ; ¼I þ; C ¹I b a.b < a/ (5.a) F./ D b 0. /0.Þ /0.þ/0. C ¹/ 0.Þ/0.þ /0. C ¹ / 3 F ; C þ; ¹I a C Þ; C ¼I b a Þ C a 0. C ¼/0. C ¹/0.þ/0. Þ/ b 0. C ¹ Þ/0. C ¼ C Þ /0.þ Þ/ (5.b) Þ; C Þ þ;þ ¹I 3 F a C Þ ; C ¼ C Þ I.a < b/; b

8 [8] On the Mellin tranform of a product of hypergeometric function 9 where, for convergence of the integral (ee Section ), 0 < <./ <<. C ¼ C ¹ C þ Þ/ 3 0 < <./ << C Þ C ¼ : If Þ D þ, the latter conditional inequality i uperfluou equation (5.a) (5.b) reduce, repectively, to equation (.c) (.a). The cae a D b will be dicued in Section 6. In Section 7 we hall ue equation (5.) to derive the ine coine tranform of F [ b x ]. To derive equation (5.a), we approximate F [ a z ] in equation (3.3) for large poitive z by uing equation (4.), thu obtaining F C ni C C n; C ¼I a z D 0. C C n/0. C ¼/ 0. C ¼ n/ a z Cn 3 F 0 C n; 0; ¼ C ni C 0. C C n/0. C ¼/ 0. /0. C n/ a z h C O co az z I a z. 3 C¼/ ³ i. 3 C ¼/ C O. / : z Noting that 3 F 0 [ =.a z /] D for all integer n 0 uing the identity we may then write F C ni C C n; C ¼I a z 0.Þ n/ D. / n 0.Þ/. Þ/ n (5.) D 0. C ¼/0. C / 0. C ¼ / C n C 3 C¼ 0. C ¼/. C / n az 0. /. / n where D i obviouly dependent on ¼ O. z /. ¼ n az C O n a z co.az C D/; z (5.3)

9 30 Allen R. Miller H. M. Srivatava [9] Now, ubtituting the reult in equation (5.3) for F [ a z ] into equation (3.3), we obtain F.I z/ D a 0. /0. C ¼/ Þ; 0. C ¼ / 3 F ; ¼I b þ; C ¹I a C a ¼ 3 0. C ¼/ ÞI F b z (5.4) C O 0. / z ¼ 3 z co.az C D/; þ; C ¹I h b where, for convergence of 3 F i, we mut have b < a. Next, denoting the econd a term in the above equation (5.4) by S 0.z/ uing equation (4.5) to approximate F [ b z ], we have S 0.z/ D " Aa ¼ 3 b Þ 0. C ¼/ 0. / z Þ ¼ C Ba ¼ 3 b Þ þ ¹ 0. C ¼/ 0. / z ¼ ¹CÞ þ co.bz C C/ C O co.az C D/: z Finally, letting z!we ee that, ince 3 <. Þ ¼ 3 /<0 <. ¼ ¹ C Þ þ / <0 # (5.5) (both of which condition ecure the convergence of F.I z/ to F./), S 0.z/ or the econd term in equation (5.4) vanihe. Thu equation (5.4) yield equation (5.a) upon letting z!. The derivation of equation (5.b) i imilar to that of equation (5.a), but i omewhat more complex in it detail, becaue the aymptotic formula for F 3 [ z ]given by equation (4.3) ha three term. Thu, for large z, by uing the latter together with equation (3.4) employing equation (5.), we obtain, for a < b, F.I z/ D b 0. /0.Þ 0.Þ/0.þ /0.þ/0. C ¹/ /0. C ¹ C / 3 F C S.z/ C S.z/; ; C þ; ¹I C Þ; C ¼I a b (5.6)

10 [0] On the Mellin tranform of a product of hypergeometric function 3 where S.z/ D b 0 Þ 0.þ/0. C ¹/ Þ z Þ 0 C Þ 0.þ Þ/0. C ¹ Þ/ X Þ n nd0 C Þ a z n. C ¼/ n n n! Þ; C Þ þ;þ ¹; Þ ni 4 F ; C Þ ni b z (5.7) S.z/ D zcþ þ ¹ 3 C O co bz z I 0 F C ¼I ³ þ Þ C ¹ C 3 C O z a z : (5.8) Now, by uing Equation (4.4) we can rewrite Equation (5.8) a S.z/ D AzCÞ þ ¼ ¹ C O z co.az C B/ co.bz C C/; (5.9) where the definition of C i obviou. Further, by writing 4 F [ ] in Equation (5.7) b z a a hypergeometric m-ummation, interchanging it with it preceding n-ummation, then employing Equation (5.), we arrive at S.z/ D b 0 Þ 0.þ/0. C ¹/ Þ z Þ 0 C Þ 0.þ Þ/0. C ¹ Þ/ X m.þ/ m. C Þ þ/ m.þ ¹/ m Þ m b z md0 C Þ m! m Þ mi F C a z : Þ m; C ¼I (5.0) We ee from Equation (4.) (5.) that F [ a z ] in Equation (5.0) may be

11 3 Allen R. Miller H. M. Srivatava [] written a Þ mi F C Þ m; C ¼I Þ a z. a z / m C ¼ C Þ m m D zþ 0. C ¼/0 C Þ a Þ 3 F 0 Þ m; 0; Þ ¼ mi I C z 3 ¼ 0. C ¼/0 C Þ a 3 C¼ 0 0 Þ C Þ Þ m ³ 3 co az C ¼ C O : z Upon noting in Equation (5.) that 3 F 0 [ find from Equation (5.0) (5.) that md0 b Þ C a 3 C¼ z ¼ Þ C ¼ C Þ m (5.) a z C O =a z ] D for all integer m 0, we a Þ S.z/ D a 0. C ¼/0. C ¹/0.þ/0 Þ b 0. C ¹ Þ/0 C ¼ C Þ m a X.Þ/ m. C Þ þ/ m.þ ¹/ m b C Þ C ¼ C Þ m! m m co az ³ 3 C ¼ 0.þ Þ/ 0. C ¼/0. C ¹/0.þ/ C O 0.þ Þ/0. C ¹ Þ/ C O 3 F 0 Þ; C Þ þ;þ ¹I I z b z : z z (5.) Equation (5.) may be written more imply a a Þ S.z/ D a 0. C ¼/0. C ¹/0.þ/0 Þ b 0. C ¹ Þ/0 C ¼ C Þ 0.þ Þ/ Þ; C Þ þ;þ ¹I 3 F a C Þ ; C ¼ C Þ I b C A0 C O z ¼ Þ 3 z co.az C B 0 / 3 F 0 Þ; C Þ þ;þ ¹I I ; b z (5.3)

12 [] On the Mellin tranform of a product of hypergeometric function 33 where a < b the definition of A 0 B 0 are obviou. Now, combining Equation (5.6), (5.9), (5.3), we find that F.I z/ D b 0 0 Þ 0.þ/0. C ¹/ 0.Þ/0 þ 0 C ¹ 3 F ; C þ; ¹I a C Þ; C ¼I b a Þ C a 0. C ¼/0. C ¹/0.þ/0 Þ b 0. C ¹ Þ/0 C ¼ C Þ 0.þ Þ/ Þ; C Þ þ;þ ¹I 3 F a C Þ ; C ¼ C Þ I b C AzCÞ þ ¼ ¹ C O co.az C B/ co.bz C C/ z C A0 z ¼ Þ 3 C O co.az C B 0 / (5.4) z 3 F 0 Þ; C Þ þ;þ ¹I I : b z Finally, recalling that 0 < <./ <<. C ¼ C ¹ C þ Þ/ 0 < <./ << 3 C Þ C ¼ ; which ecure the convergence of F.I z/ to F./, upon letting z!, we ee that the third fourth term in Equation (5.4) vanih, we are left with Equation (5.b). Thi evidently complete the derivation of Equation (5.). 6. The Cae a D b The expreion on the right-h ide of Equation (5.a) (5.b) are not analytic continuation of each other. In particular, for Þ D þ we noted thi in Section mentioned alo that the dicontinuou nature of F./ refer to the dicontinuity in it repreentation acro a D b. However,F./ i continuou when a D b. To ee thi, ince the firt integral on the right-h ide of Equation (.4) i continuou when a D b (ee [, p. 403]), then o i the integral on the left-h ide. Thu alo it i evident that F./ i continuou when a pae through b. In addition, we howed in Section that when a D b, a neceary condition that the integral F./ converge i that 0 < <./ <<. C ¼ C ¹ C þ Þ/; a fortiori all three 3 F function in equation (5.) converge abolutely when a D b, provided that the latter condition hold true. Thu, for a > 0, 0 < <./ <<. C ¼ C ¹ C þ Þ/ 0 < <./ <<. 3 C Þ C ¼/;

13 34 Allen R. Miller H. M. Srivatava [3] we deduce, repectively, from equation (5.a) (5.b) that Z x 0 Z x 0 I 0F C ¼I D a 0. /0. C ¼/ 0. C ¼ / 3 F I 0F C ¼I D a 0. /0.Þ 0.Þ/0.þ a x ÞI F þ; C ¹I Þ; ; ¼I þ; C ¹I a x ÞI F þ; C ¹I /0.þ/0. C ¹/ /0. C ¹ / 3 F a x dx C a 0. C ¼/0. C ¹/0.þ/0. Þ/ 0. C ¹ Þ/0. C ¼ C Þ /0.þ Þ/ Þ; C Þ þ;þ ¹I 3 F C Þ ; C ¼ C Þ I : a x dx (6.) ; C þ; ¹I C Þ; C ¼I (6.) When Þ D þ, the econd right member in equation (6.) vanihe it i eay to ee that the right-h ide of equation (6.) (6.) reduce, repectively, via Gau theorem to equation (.b). However, the 3 F [] function in equation (6.) (6.) are not, in general, reducible, ince their parameter are not interrelated []. If we equate the right member of equation (6.) (6.), divide the reult by a =, then et a D ¼; b D ; c D Þ; e D þ; f D C ¹; (6.3) we deduce the econd fundamental relation for 3 F [a; b; ci e; f I ], which i a; b; ci 3F e; f I D 0. a/0.e/0. f /0.c b/ 0.e b/0. f b/0. C b a/0.c/ 3 F C 0. a/0.e/0. f /0.b c/ 0.e c/0. f c/0. C c a/0.b/ 3 F b; b e C ; b f C I C b c; C b ai c; c e C ; c f C I C c b; C c ai : The conditional inequality 0 < <./ <<. 3 C Þ C ¼/ with the relevant ubtitution of equation (6.3) may now be waived by appealing to the principle of analytic continuation. That equation (6.4) manifet itelf a a corollary i both urpriing intereting, epecially ince Bailey [] derive it by conidering a certain contour

14 [4] On the Mellin tranform of a product of hypergeometric function 35 integral, deforming it contour in two different way, then computing the integral via reidue [, p. 5]. Finally, we recall that in Section we howed, in the cae a D b, that F./ converge alo when 0 < <./ <<. C ¼ C ¹ C þ Þ/ 0 < <./ << 3 C Þ C ¼ ; provided that ¼ ¹ C Þ þ i an odd poitive integer. It i evident that equation (6.) doe not hold true in thi cae ince 3 F [] doe not converge. It hould be remarked alo that, in an earlier work, Srivatava Exton [8] conidered a generalization of the Weber-Schafheitlin integral given in equation (.) for the product of everal Beel (or 0 F ) function. 7. The Sine Coine Tranform of F [ b x ] For brevity, we define S.a; b/ :D Z 0 in.ax/ F [ÞI þ;i b x ] dx C.a; b/ :D Z 0 co.ax/ F [ÞI þ;i b x ] dx; which are, repectively, the ine coine tranform of F [ b x ]. Since in z D z 0 F I 3 I 4 z co z D 0 F I I 4 z ; it i eay to deduce from equation (5.) that 8 " a 3 F ; ;ÞI # b a þ;i " # >< a.þ /. / ; þ; I a S.a; b/ D b Þ 3F C p ³ 3 b ; ÞI a " 0.þ/0./0. Þ/ C Þ þ; C Þ I >: 0.þ Þ/0. Þ/0. CÞ/ F a b Þ a b C ÞI #.0 < b < a/.0 < a < b/;

15 36 Allen R. Miller H. M. Srivatava [5] 8 0 >< C.a; b/ D >: p ³ b 0.Þ /0.þ/0. / 0.Þ/0.þ /0. / F 0. Þ/0.þ/0./ 0.Þ/0.þ Þ/0. Þ/ F " 3 þ; 3 I 3 ÞI a b # C p ³ a " C Þ þ; C Þ I a b C ÞI # a b Þ.0 < b < a/.0 < a < b/; where 0 < <.Þ/ < <.þ C /: 8. Concluding Remark It hould be mentioned that, by uing a general reult for the Mellin tranform of a product of generalized hypergeometric function in [7, Section., p. 333], F./ defined by equation (.) may be repreented by Meijer G-function. Thu, for a > 0 b > 0, we have where F./ D 0. C ¼/0. C ¹/0.þ/ a G ; 3;3 0.Þ/ b þ a ; Þ; C ¼ ; 0; ¹; þ (8.) 0 < <./ <<. C ¼ C ¹ C þ Þ/ 0 < <./ << 3 C Þ C ¼ : Furthermore, by uing formula for reducing the G-function to generalized hypergeometric function (ee, for example, [4, Section 6.5, p. 30]), we may obtain equation (5.) from equation (8.). Nonethele, the derivation of equation (5.) preented herein i elementary in the ene that it doe not require knowledge of the G-function it propertie. In addition, the reult given by equation (6.), (6.), (6.4) require a detailed analyi of the continuity convergence criteria for F./ when a D b,, therefore, may not be deduced directly from equation (8.). Acknowledgement The preent invetigation wa upported, in part, by the Natural Science Engineering Reearch Council of Canada under Grant OGP

16 [6] On the Mellin tranform of a product of hypergeometric function 37 Reference [] W. N. Bailey, Generalized Hypergeometric Serie, Cambridge Math. Tract No. 3 (Cambridge Univerity Pre, Cambridge, 935; Reprinted by Stechert-Hafner, New York, 964). [] M. A. Chaudhry, On an integral of Lommel Beel function, J. Autral. Math. Soc. Ser. B 35 (994) [3] H. Exton, Hbook of Hypergeometric Integral (Halted Pre (Elli Horwood Limited, Chicheter), John Wiley Son, New York, Bribane, Chicheter Toronto, 978). [4] Y. L. Luke, The Special Function Their Approximation, Vol. I (Academic Pre, New York London, 969). [5] A. M. Mathai, A Hbook of Generalized Special Function for Statitical Phyical Science (Clarendon Pre, Oxford, 993). [6] A. R. Miller H. Exton, Sonine-Gegenbauer-type integral, J. Comput. Appl. Math. 55 (994) [7] A. P. Prudnikov, Yu. A. Brychkov O. I. Marichev, Integral Serie, Vol. 3 (Gordon Breach, New York, 990). [8] H. M. Srivatava H. Exton, A generalization of the Weber-Schafheitlin integral, J. Reine Angew. Math. 309 (979) 6. [9] H. M. Srivatava Per W. Karlon, Multiple Gauian Hypergeometric Serie (Halted Pre (Elli Horwood Limited, Chicheter), John Wiley Son, New York, Bribane, Chicheter Toronto, 985). [0] H. M. Srivatava H. L. Manocha, A Treatie on Generating Function (Halted Pre (Elli Horwood Limited, Chicheter), John Wiley Son, New York, Bribane, Chicheter Toronto, 984). [] G. N. Waton, A Treatie on the Theory of Beel Function (Second ed. Cambridge Univerity Pre, Cambridge, 944). [] J. Wimp, Irreducible recurrence repreentation theorem for 3 F./, Comput. Math. Appl. 9 (983)