Series representations of the Volterra function and the Fransen-Robinson constant

[摘要] The Volterra function mu(t, beta, alpha) 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 mu(t, beta, alpha) 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 Fransen-Robinson constant F:= mu(1, 1, 0) = integral(infinity)(0) 1/Gamma(x) dx.. Some numerical examples illustrate the accuracy of the approximations. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Volterra function;Fransen-Robinson constant;Convergent series representation;Special functions