Convolution integral equations with Gegenbauer function kernel (Q1334799)

A solvability theory for Mellvin convolution equations with Gegenbauer functions in their kernel is developed. It is based on some properties of fractional operators and a number of formulas involving special functions. This theory is applied to the construction of explicit solutions of the equations: \[ f(x) = \int^ b_ x (t^ 2-x^ 2)^{\lambda - 1/2} {\mathcal G}^ \lambda_ \nu (x/t)t^{\mu + 1} g(t)dt,\;0<x<b, \] \[ f(x) = x^{\mu + 1} \int^ b_ x (t^ 2-x^ 2)^{\lambda - 1/2} {\mathcal G}^ \lambda_ \nu (x/t)t^{\mu + 1} g(t)dt,\;0<x<b. \] The method of investigation is highly connected with the contents of the monograph by \textit{S. G. Samko}, \textit{A. A. Kilbas}, \textit{O. I. Marichev} [``Fractional integrals and derivatives and some of their applications'' (1987; Zbl 0617.26004)].