Computation of fractional integrals via functions of hypergeometric and Bessel type

The paper is devoted to computation of the fractional integrals of power exponential functions. It is considered a function lambda(gamma,sigma)((beta))(z) defined by lambda(gamma,sigma)((beta))(Z)=beta/Gamma(gamma + 1 - 1/beta) integral(1)(infinity)(t(beta) - 1)(gamma-1/beta)t(sigma)e(-2t) dt with positive beta and complex gamma, sigma and z such that Re(gamma) > (1/beta)- 1 and Re(z) > 0. The special cases are discussed when lambda(gamma,sigma)((beta))(z) is expressed in terms of the Tricomi confluent hypergeometric function Psi(a,c;x) and of modified Bessel function of the third kind K-y(x). Representations of these functions via fractional integrals are proved. The results obtained apply to compute fractional integrals of power exponential functions in terms of lambda(gamma,sigma)((beta))(x), Psi(a,c;x) and K-y(x). Examples are considered. (C) 2000 Elsevier Science B.V. All rights reserved. MSC 26A33; 33C15; 33C10.