On functions K and E generated by a sequence of moments

TL;DR

This paper investigates the asymptotic properties of two functions, E(z) and K(z), defined via moments and integrals, which are relevant in classical analysis problems.

Contribution

It provides a detailed analysis of the asymptotic behavior of functions generated by moment sequences and their integral representations.

Findings

Asymptotic formulas for E(z) and K(z) derived

Connections established between these functions and classical analysis problems

Insights into the growth and behavior of functions generated by moments

Abstract

We study the asymptotic behaviour of the entire function \[ E(z) = \sum_{n\ge 0} \frac{z^n}{\gamma (n+1)} \] and the analytic function \[ K(z) = \frac1{2\pi {\rm i}}\, \int_{c-{\rm i}\infty}^{c+{\rm i}\infty} z^{-s}\gamma (s)\, {\rm d}s\,, \] which naturally appear in various classical problems of analysis.