By Semen B. Yakubovich, Yurii F. Luchko (auth.)

The goal of this booklet is to advance a brand new strategy which we known as the hyper geometric one to the idea of assorted necessary transforms, convolutions, and their purposes to ideas of integro-differential equations, operational calculus, and overview of integrals. we are hoping that this easy procedure, to be able to be defined under, permits scholars, put up graduates in arithmetic, physicists and technicians, and critical mathematicians and researchers to discover during this publication new fascinating leads to the speculation of fundamental transforms, specific services, and convolutions. the assumption of this process are available in quite a few papers of many authors, yet systematic dialogue and improvement is learned during this ebook for the 1st time. allow us to clarify in short the elemental issues of this procedure. because it is understood, within the idea of designated services and its functions, the hypergeometric capabilities play the most position. in addition to identified easy features, this classification comprises the Gauss's, Bessel's, Kummer's, services et c. normally case, the hypergeometric services are outlined as a linear mixtures of the Mellin-Barnes integrals. those ques tions are greatly mentioned in bankruptcy 1. additionally, the Mellin-Barnes kind integrals might be understood as an inversion Mellin remodel from the quotient of goods of Euler's gamma-functions. therefore we're resulted in the overall construc tions just like the Meijer's G-function and the Fox's H-function.

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**Example text**

1) 1 g(z) = [00 k(zy)f(y)dy = -2. *(l - s)z-·ds. 45) 1I'l1", Since f(z) E M;'~(L), then e,,"cl'l I s 1"1' /*(s) E L(u). (L). 47) show that the H-transform is a one-to-one transform from M;'~(L) onto M;)"'-Y+I'(L). Now we will find the inversion formula for H-transform. Let us consider the case K, = 0, p > 1 first. 7). 45) now and we obtain 00 k d k Hq-m,p-n+l ( (Op+t. 1J*(s)x-Sds = f(x). 1rZ q In order to obtain the inversion formula for H-transform in the case first consider the Laplace transform on the real line 1 00 g(x) = L{f(u)jx} = e-XUf(u)du, x> o.

9, it is sufficient to show that the function h(min(z 0 1)) = h(min(zt> ... 114). A. Brychkov et al. (1992)) of the function f(z) at the point 8 E en M n{f(Z);8} = /*(s) = f f(z)z·-ldz. 49)), where 8 E U, 8·1 = L:i=18j and z- l h(zl/n) E L 2(R+). 1. Note that the function z- l h(zl/n) belongs to L 2(R+) if and only if z-(n+1)/2h(z) E L 2(R+). (r) exists when = 1/2. (8 . 1) = n/2. It is clear that Mdh(z); -8' I} E u. Define Ar. = {z E R+, zr. = min(z 0 I)}. Since 8 E u, we have 8r. 1}. 124) and relationship U~=l Ale = R+.

76) in the case c· > 0 or q = P that u- 1/2 k(u) E L(E,oo). 77) + 6]dy = f£(q - p) roo y-l/2-"Y·+ it(q-p) cos[(q - p)y + 6]dy. 65) that ,. > -1/2. This means that the integral oo y-1 / 2-"Y· cos[(q - p)y + 6]dy L is convergent and the function yit(q-p) is uniformly bounded with respect to t. 77) converges uniformly. 72) in the case p ~ q. Let us consider the case p > q. 72) has been proved. 3 • k (s) = TIi:l r((3j + s) TIl=l r(1 - aj - s) q TIPj=n+! r (aj + S )TI j=m+! 79) + sgnh· -1) > O. 80) if 2sgn(c·) H- and G-transforms 53 :s We have to obtain the same relation in the case c* = 0, -1/2 < '1* 1.