Series representations of the Volterra function and the Fransén–Robinson constant

Abstract

The Volterra function μ(t,β,α) was introduced by Vito Volterra in 1916 as the solution to certain integral equations with a logarithmic kernel. Despite the large number of applications of the Volterra function, the only known analytic representations of this function are given in terms of integrals. In this paper we derive several convergent expansion of μ(t,β,α) in terms of incomplete gamma functions. These expansions may be used to implement numerical evaluation techniques for this function. As a particular application, we derive a numerical series representation of the Fransén–Robinson constant F≔μ(1,1,0)=∫0∞1Γ(x)dx. Some numerical examples illustrate the accuracy of the approximations.