Lommel functions, Pad\'e approximants and hypergeometric functions

We consider the Lommel functions $s_{\mu,\nu}(z)$ for different values of the parameters $(\mu,\nu)$. We show that if $(\mu,\nu)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Pad\'e approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, when $\mu$ is an integer, $s_{\mu,\nu}(z)$ can be written as an integral involving an explicit combination of trigonometric functions. A closed formula for $_2F_1\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\sin(\frac{\theta}{2})^2\right)$ with $\mu$ an integer is given.