On the Solution of an Integral Equation of Convolution Type

Abstract

The solution $Q_\lambda ^\mu $ of the integral equation $xQ_\lambda ^\mu (x) = [\{ (\lambda - \mu ) + \mu f\} ^ * Q_\lambda ^\mu ](x)$ is considered as a transform of the function f; f is a function of the real variable x, and $\lambda $ and $\mu $ are real-valued parameters with $\lambda $ positive. The functions $Q_\lambda ^\mu [f]$ with f fixed form a group under the convolution operator. The functions $F_\lambda ^\mu (x) = \Gamma (\lambda )x^{1 - \lambda } Q_\lambda ^\mu [f](x)$, occurring in elementary particle physics, exhibit properties, such as their analyticity and behavior at the origin and at infinity, similar to those of the input function $f(x)$. Besides several applications concerning hypergeometric functions, a table is presented giving various input functions f and their transforms $F_\lambda ^\mu $.