Integral representations and zeros of the Lommel function and the hypergeometric ${}_1{F}_2$ function

Abstract

We give different integral representations of the Lommel function $s_{\mu ,\nu }(z)$ involving trigonometric and hypergeometric ${}_2{F}_1$ functions. By using classical results of Polya, 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 ${}_1{F}_2$ function, we describe the distribution of the zeros of ${}_1{F}_2$ for specific values of its parameters.

15 pages, 3 figures, 1 Table