Series representations of the Volterra function and the Fransén-Robinson constant (Q2054270)

This paper discusses the Volterra function \(\mu(t,\beta,\alpha)\), and presents several convergent expansions of this function in terms of incomplete gamma functions. These expansions may be used to implement numerical evaluation techniques for this function. As an application the paper contains a numerical series representation of the Fransén-Robinson constant. Some numerical examples demonstrate the accuracy of approximation. The paper is organized as follow: some preliminary results needed for the analysis are introduced, given is an integral representation of the Volterra function which is different from the original definition. The expansion of the function by incomplete gamma functions is defined. The main result is derived, which is a family of convergent series representation of the Volterra function. The numerical series representation of the Fransén-Robinson constant and numerical experiments that show the accuracy of the expansion follow. Proofs are given throughout the sections, for the stated theorems and claimed results. The paper ends with acknowledgments and 15 references.