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 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)^{n}(m-n)!\Gamma (\nu +m-n)}{n!(m-2n)!\Gamma (\nu +n)}}(z/2)^{{2n-m}}.

References

Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
Ivanov, A. B. (2001), "l/l060810", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Lommel, Eugen von (1871), "Zur Theorie der Bessel'schen Functionen", Mathematische Annalen, Berlin / Heidelberg: Springer, 4 (1): 103–116, doi:10.1007/BF01443302

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