Integral Representations and Zeros of the Lommel Function and the Hypergeometric $$_1F_2$$ Function

Abstract

We give different integral representations of the Lommel function \(s_{\mu ,\nu }(z)\) involving trigonometric and hypergeometric \(_2F_1\) functions. By using classical results of Pólya, we give the distribution of the zeros of \(s_{\mu ,\nu }(z)\) for certain regions in the plane \((\mu ,\nu )\). Further, thanks to a well known relation between the functions \(s_{\mu ,\nu }(z)\) and the hypergeometric \( _1F_2\) function, we describe the distribution of the zeros of \(_1F_2\) for specific values of its parameters.