#### 8.57 Lommel functions

8.570 Definitions of the Lommel functions sμν(z) and Sμν(z):

1.12

$\begin{array}{ll}{s}_{\mu ,v}\left(z\right)\hfill & =\sum _{m=0}^{\infty }\frac{\left(-1\right){m}_{z}\mu +1+2m}{\left[{\left(\mu +1\right)}^{2}-{v}^{2}\right]\left[{\left(\mu +3\right)}^{2}-{v}^{2}\right]\dots \left[{\left(\mu +2m+1\right)}^{2}-{v}^{2}\right]}\hfill \\ \begin{array}{l}\begin{array}{l}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\left[\mu ±v\text{?}\text{is}\text{?}\text{not}\text{?}\text{a}\text{?}\text{negative}\text{?}\text{odd}\text{?}\text{integer}\right]\hfill \end{array}\hfill \end{array}\hfill \\ ={z}^{\mu -1}\sum _{m=0}^{\infty }\frac{{\left(-1\right)}^{m}{\left(\frac{z}{2}\right)}^{2m+2}\Gamma \left(\frac{1}{2}\mu -\frac{1}{2}v+\frac{1}{2}\right)\Gamma \left(\frac{1}{2}\mu +\frac{1}{2}v+\frac{1}{2}\right)}{\Gamma \left(\frac{1}{2}\mu -\frac{1}{2}v+m+\frac{3}{2}\right)\Gamma \left(\frac{1}{2}\mu +\frac{1}{2}v+m+\frac{3}{2}\right)}\hfill \\ \begin{array}{l}\begin{array}{l}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\left[\mu ±v\text{?}\text{is}\text{?}\text{not}\text{?}\text{a}\text{?}\text{negative}\text{?}\text{odd}\text{?}\text{integer}\right]\hfill \end{array}\hfill \end{array}\hfill \end{array}$ EH II 40(69), WA 377(2)

2.12

EH II 40(71), WA 379(2)

$\begin{array}{ll}=\hfill & {s}_{\mu ,v}\left(z\right)+{2}^{\mu -1}\Gamma \left(\frac{1}{2}\mu -\frac{1}{2}v+\frac{1}{2}\right)\Gamma \left(\frac{1}{2}\mu +\frac{1}{2}v+\frac{1}{2}\right)\hfill \\ ×\left\{\mathrm{sin}\left[\frac{1}{2}\left(\mu -\hfill \end{array}$

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