Lommel function

Remarks

The hypothesis on $\mu \pm \nu$ says exactly that no factor $(\mu + 2j + 1)^2 - \nu^2$ vanishes. The function $s_{\mu,\nu}$ is a particular solution of $$z^2 w'' + z w' + (z^2 - \nu^2)\, w = z^{\mu + 1}.$$ Check: the operator $w \mapsto z^2 w'' + z w' - \nu^2 w$ multiplies $z^p$ by $p^2 - \nu^2$, and $z^2 w$ shifts each term up by two powers. So the coefficients $c_k$ of $z^{\mu + 1 + 2k}$ must satisfy $\left( (\mu + 1)^2 - \nu^2 \right) c_0 = 1$ and $\left( (\mu + 2k + 1)^2 - \nu^2 \right) c_k = -c_{k-1}$, which the series does. Example: for $\mu = 1$ and $\nu = 0$ the first factor is $4$, so $s_{1,0}$ starts with $\frac{z^2}{4}$. In fact $s_{1,0}$ is $1 - J_0(z)$, where $J_0$ is the Bessel function of the first kind. A non-example: for $\mu = -2$ and $\nu = 1$ one has $\mu + \nu = -1$, so the factor with $j = 0$ is $(-2 + 1)^2 - 1 = 0$ and $s_{-2,1}$ is not defined by the series. The second Lommel function $S_{\mu,\nu}$ is obtained from $s_{\mu,\nu}$ by adding a suitable combination of $J_\nu$ and $Y_\nu$ so that it has a simple asymptotic expansion at infinity.