Integral Representations and Zeros of the Lommel Function and the Hypergeometric \(_1F_2\) Function

We give different integral representations of the Lommel function \(s_{\mu ,\nu }(z)\) $s_{\mu ,\nu }(z)$ involving trigonometric and hypergeometric \(_2F_1\) $_2^{}{F}_1$ functions. By using classical results of Pólya, we give the distribution of the zeros of \(s_{\mu ,\nu }(z)\) $s_{\mu ,\nu }(z)$ for certain regions in the plane \((\mu ,\nu )\) $(\mu ,\nu )$ . Further, thanks to a well known relation between the functions \(s_{\mu ,\nu }(z)\) $s_{\mu ,\nu }(z)$ and the hypergeometric \( _1F_2\) $_1^{}{F}_2$ function, we describe the distribution of the zeros of \(_1F_2\) $_1^{}{F}_2$ for specific values of its parameters.