Lommel polynomial

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

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

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

They are given explicitly by

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

[edit] See also

Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II, McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
Ivanov, A. B.. (2001), “Lommel polynomial”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104