Fractional integral operators and the generalized hypergeometric functions (Q1089866)

The authors study the following fractional integral operators associated with a generalized hypergeometric function \({}_ pF_ q\) defined as \[ (1)\quad I^ h_{\alpha}f(x)=x^{-h-\alpha}R_{\alpha}x^ hf(x)\quad and\quad (2)\quad K_{\beta}f(x)=x^{\lambda}W_{\beta}x^{-\lambda - \beta}f(x) \] where \[ R_{\alpha}f(x)=1/\Gamma (\alpha)\int^{x}_{0}(x-s)^{\alpha -1}_ AF_ B[(a);(b);z(1- s/x)]f(s)ds \] and \[ W_{\beta}f(x)=1/\Gamma (\beta)\int^{\infty}_{x}(s-x)^{\beta -1}_ CF_ D[(c);(d);z(1- x_ s)]f(s)ds. \] The authors obtain expressions for the composition of the operators (1) and (2). The inversion formulas and relations of these operators with the generalized Hankel transform are also developed. Some of the results given by the reviewer and \textit{R. K. Kumbhat} [Vijnana Parishad Anusandhan Patrika 18, 139-150 (1975)] are also generalized. It is interesting to observe that the operators studied by the authors are special cases of the operators of fractional integration defined and studied earlier by the reviewer and \textit{R. K. Kumbhat} [Indian J. Pure Appl. Math. 5, 1-6 (1974; Zbl 0305.44003]. (The assertion made by the authors in the Remark on page 252 of the paper does not appear to be correct.)