Lommel polynomial

A Lommel polynomial R_m,ν(z), introduced by Eugen von Lommel (1871), is a polynomial in 1/z giving the recurrence relation

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

where J_ν(z) is a Bessel function of the first kind.

They are given explicitly by

${\displaystyle R_{m,\nu }(z)=\sum _{n=0}^{[m/2]}{\frac {(-1)^{n}(m-n)!\Gamma (\nu +m-n)}{n!(m-2n)!\Gamma (\nu +n)}}(z/2)^{2n-m}.}$