On some integral transforms with generalized function of hypergeometric type
by
Nina Virchenko
National Technical University of Ukraine

Let us the integral transform with the generalized (according to Wright) confluent hypergeometric function as:

\tauFca(f(x))= ó õ \infty 0  t[ a/(\tau)]-1 1\Phi1\tau(a;c;-xt)f(t)dt, \tag1

(1)

where f(x) in L_p, 0 < [ 1/p] <= 1, x > 0, c > a > 0, a > [ 1/p], \tau in R, \tau > 0,  ₁\Phi₁^\tau(a;c;-xt) is the function of the type:

1\Phi1\tau(a;c;z)=  \Gamma(c) \Gamma(a) \infty å n=0   \Gamma(a+\taun) \Gamma(c+\taun) ·  zn n! , \tag2

(2)

here a, c can be complex, \Gamma(a) is gamma-function.

Theorem 1 (Inversion formula for (1)) If c-a > 0, x > 0, a > [ 1/p], 0 < [ 1/p] <= 1, \tau in R, \tau > 0, then

f(x)=  \Gamma(a) \Gamma(c) x1-[ a/(\tau)]L-1 ì í î x1-[ a/(\tau)]  d dx [ x[ 2/p]Ic-a, 1-[ 2/p]1-c+ax[ c/(\tau)]+a-c-1j(x)] ü ý þ , \tag3

(3)

where j(x)= _\tauF_c^a(f(x)), L^-1- is the inverse Laplace transform,

\tauI\nu, \beta\alphaf(x)=  x\beta-\nu-\alpha \Gamma(\alpha) ó õ x 0  ( x[ 1/(\tau)]-t[ 1/(\tau)])\alpha-1t\nuf(t)dt. \tag4

(4)

Date received: August 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-77.