Volterra-type convolution of classical polynomials

TL;DR

This paper develops a general method to compute Volterra-type convolutions of polynomial sequences, providing new series representations for classical orthogonal polynomials like Jacobi and Laguerre, with potential applications in mathematical analysis.

Contribution

It introduces a unified framework for convolutions of polynomial sequences and derives new series formulas for classical orthogonal polynomials.

Findings

Series representations for Jacobi and Laguerre polynomial convolutions

Unified framework applicable to arbitrary polynomial sequences

New formulas enhance computational and theoretical analysis

Abstract

We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence $\{P_k(x)\}_{k \geqslant 0}$ with $\deg P_k(x) = k$. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas.