Some definite integrals involving Jacobi polynomials

TL;DR

This paper explores new definite integrals involving Jacobi polynomials, generalizing previous results with Gegenbauer polynomials, and employs six different methods to derive and analyze these integrals, revealing interesting hypergeometric function identities.

Contribution

It introduces novel integrals with Jacobi polynomials and demonstrates multiple derivation techniques, including connections to hypergeometric functions, expanding the mathematical understanding of these integrals.

Findings

Derived new integrals involving Jacobi polynomials

Established connections to hypergeometric functions like F3 and 2F1

Presented multiple methods for deriving these integrals

Abstract

Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough to prove the first integral, but by the other methods there arises remarkable formula such as par example a zero-balanced F3 Appell function which can be converted into a 2F1 hypergeometric function. Another three integrals complete the paper.