Lommel polynomial

The polynomial $R_{m,\nu }(z)$ of degree $m$ in $z^{-}1$ which for $m=0,1,\dots$ and any $\nu$ is defined by

$$R_{m,\nu }(z)=$$

$$=\frac{\pi z}{2\sin \nu \pi }[J_{\nu +m}(z)J_{-\nu +1}(z)+(-1{)}^m{J}_{-\nu -m}(z)J_{\nu -1}(z)]$$

or

$$R_{m,\nu }(z)=\frac{\mathrm{\Gamma }(\nu +m)}{\mathrm{\Gamma }(\nu )}{\left(\frac{2}{z}\right)}^m\times$$

$$\times {}_2{F}_3(1-\frac{m}{2},-\frac{m}{2};\nu ,-m,1-\nu -m;-z^2).$$

Here $J_{\mu }(z)$ is the Bessel function (cf. Bessel functions) and ${}_2{F}_3$ is the hypergeometric series. The Lommel polynomials satisfy the relations

$$J_{\nu +m}(z)=J_{\nu }(z)R_{m,\nu }(z)-J_{\nu -1}(z)R_{m-1,\nu +1}(z),$$

$$R_{0,\nu }(z)=1,m=1,2,\dots .$$

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