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

Abstract

AbstractWe give different integral representations of the Lommel function $$s_{\mu ,\nu }(z)$$ s μ , ν ( 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 μ , ν ( z ) for certain regions in the plane $$(\mu ,\nu )$$ ( μ , ν ) . Further, thanks to a well known relation between the functions $$s_{\mu ,\nu }(z)$$ s μ , ν ( 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.