Integral representation for a product of two Jacobi functions of the second kind

TL;DR

This paper develops integral representations and sum formulas for products of Jacobi functions of the second kind, extending to related special functions and deriving Nicholson-type relations, enhancing analytical tools for these functions.

Contribution

It introduces new integral and sum representations for Jacobi functions of the second kind and related functions, generalizing classical exponential relations.

Findings

Derived integral representations for products of Jacobi functions

Established Nicholson-type integral relations for sums of squares

Extended results to related special functions like Legendre and Gegenbauer

Abstract

By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting functions such as associated Legendre functions of the first and second kind, Ferrers functions and also Gegenbauer functions of the first and second kind. By examining the behavior of one of these products near singularities of the relevant functions, we also derive integral representations for single functions, including a Laplace-type integral representation for the Jacobi function of the second kind. Finally, we use the product formulas for the functions of the second kind to derive Nicholson-type integral relations for the sums of squares of Jacobi functions of the first and second kinds, and in a confluent limit, Laguerre functions of the first and second kinds, which generalize the relation $\expe^{ix}\expe^{-ix}=1$ to those functions.