Operational Methods in the Study of Sobolev-Jacobi Polynomials

TL;DR

This paper develops a multivariate umbral calculus framework to derive new formulas for hypergeometric functions and applies it to analyze Sobolev-Jacobi polynomials, revealing deep links with normal-ordering and Hermite polynomials.

Contribution

It introduces a multivariate umbral calculus approach and series transforms, providing explicit formulas for Sobolev-Jacobi polynomials and connecting them with existing mathematical techniques.

Findings

Derived new formulas for generalized hypergeometric functions.

Explicitly calculated K-tuple L-shifted lacunary exponential generating functions.

Established links between Sobolev-Jacobi polynomials, Hermite polynomials, and normal-ordering techniques.

Abstract

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of SJ polynomials explicitly.