Le Roy, Lerch and Legendre chi functions and generalised Borel-Le Roy transform

TL;DR

This paper develops a unified framework for special functions like the Le Roy, Lerch transcendent, and Legendre chi, using Indicial Umbral Theory and Borel-Le Roy transforms to analyze their properties and extensions to divergent series.

Contribution

It introduces a reformulated Indicial Umbral Theory that unifies the study of these functions and incorporates Borel-Le Roy transforms for divergent series resummation.

Findings

Unified framework for special functions established.

Extension of formalism to divergent series via resummation.

Enhanced understanding of properties and generalisations of these functions.

Abstract

The Le Roy function has been the focus of intensive research in recent years, owing both to its relevance in analysis and its versatility in applications involving fractional differential operators. Other special functions - such as the Lerch transcendent and the Legendre chi function - have found applications ranging from Bose-Einstein and Fermi-Dirac statistics in physics to pure mathematical investigations involving polylogarithms and Dirichlet L-series. In this article, we present a unified framework based on a recent reformulation of Indicial Umbral Theory (IUT) grounded in the formal theory of power series. Within this setting, we study the properties and generalisations of these special functions. In particular, we build upon the revised formulation of IUT to incorporate the role of the Borel-Le Roy transform, and to explore the extension of the formalism to divergent series via appropriate resummation techniques.