A note on an asymptotic formula for integrals of products of Jacobi polynomials

TL;DR

This paper derives an asymptotic formula for integrals of products of Jacobi polynomials by extending a known formula for Legendre polynomials, suggesting a general behavior for orthogonal polynomial integrals.

Contribution

It adapts Byerly's formula to Jacobi polynomials and derives a new asymptotic expression, highlighting its potential universality for orthogonal polynomial products.

Findings

Derived an asymptotic formula for Jacobi polynomial integrals.

Showed the formula's similarity to previous results for different orthogonal polynomials.

Indicated the potential generality of such asymptotic behaviors.

Abstract

We recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi polynomials. The asymptotic formula is similar to an analogous one recently obtained by the first author and Jeff Geronimo for a different case. Thus, it suggests that such an asymptotic behavior is rather generic for integrals of products of orthogonal polynomials.