  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.
