Integral Representations and Zeros of the Lommel Function and the Hypergeometric ₁F₂ Function

We give different integral representations of the Lommel function sμ ,ν(z) s μ , ν ( z ) involving trigonometric and hypergeometric ₂F₁ 2 F 1 functions. By using classical results of Pólya, we give the distribution of the zeros of sμ ,ν(z) s μ , ν ( z ) for certain regions in the plane (μ ,ν ) ( μ , ν ) . Further, thanks to a well known relation between the functions sμ ,ν(z) s μ , ν ( z ) and the hypergeometric ₁F₂ 1 F 2 function, we describe the distribution of the zeros of ₁F₂ 1 F 2 for specific values of its parameters.