The sign of Lommel’s function

Abstract:

Lommel’s function ${s_{\mu ,\nu }}(x)$ is a particular solution of the differential equation ${x^2}y'' + xy’ + ({x^2} - {\nu ^2})y = {x^{\mu + 1}}$. It is shown here that ${s_{\mu ,\nu }}(x) > 0$ for $x > 0$, if $\mu = \tfrac {1}{2}$ and $|\nu | < \tfrac {1}{2}$, or if $\mu > \tfrac {1}{2}$ and $|\nu | \leqq \mu$. This includes earlier results of R. G. Cooke’s. The sign of ${s_{\mu ,\nu }}(x)$ for other values of $\mu$ and $\nu$ is also discussed.