Integrals of products of four modified Bessel functions

TL;DR

This work addresses the evaluation of definite integrals of products of four modified Bessel functions with a power, for general parameter choices. It develops a Mellin-transform framework that yields general formulas expressed in terms of the Meijer $G$-function, regularized hypergeometric functions, and Lauricella $F_C$ functions. The authors also demonstrate reductions to simpler special functions and elementary forms, and apply the results to derive new quartic Airy-integral formulas. These results extend the theory of Bessel moment integrals and provide new tools for applications in mathematical physics, including novel identities for Airy-function products.

Abstract

We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized hypergeometric and Lauricella $F_C$ functions, and study a number of special cases in which the integrals can be evaluated in terms of simpler special functions or indeed take an elementary form. As a consequence, we deduce some new formulas for definite integrals of products of four Airy functions.

Theorems & Definitions (22)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Corollary 2.4
• proof
• Theorem 2.5
• Remark 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9