A recurrence relation for generalised connection coefficients

TL;DR

This paper introduces a general recurrence relation for integrals involving orthogonal polynomials and related functions, unifying various specific cases such as connection coefficients and Legendre function integrals.

Contribution

It formulates and proves a broad recurrence relation applicable to multiple types of integrals involving orthogonal functions, advancing theoretical understanding.

Findings

Derived a general recurrence relation for integrals involving orthogonal polynomials.

Unified treatment of connection coefficients and Legendre function integrals.

Provides a tool for simplifying complex integral calculations.

Abstract

We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example is integrals of products of Legendre functions.