Lommel functions, Pad\'e approximants and hypergeometric functions

TL;DR

This paper explores explicit representations of Lommel functions for specific parameter values, linking them to polynomials, trigonometric functions, and hypergeometric functions, with implications for approximation and zero distribution.

Contribution

It provides explicit formulas for Lommel functions with half-integer and integer parameters, connecting them to Padé approximants and hypergeometric functions, advancing their analytical understanding.

Findings

Lommel functions with half-integer parameters can be expressed using polynomials and trigonometric functions.

Polynomials derived serve as Padé approximants for trigonometric functions.

Explicit integral and hypergeometric formulas are obtained for specific Lommel functions.

Abstract

We consider the Lommel functions $s_{\mu,\nu}(z)$ for different values of the parameters $(\mu,\nu)$. We show that if $(\mu,\nu)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Pad\'e approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, when $\mu$ is an integer, $s_{\mu,\nu}(z)$ can be written as an integral involving an explicit combination of trigonometric functions. A closed formula for $_2F_1\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\sin(\frac{\theta}{2})^2\right)$ with $\mu$ an integer is given.