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

[math]\displaystyle{ \displaystyle J_{m+\nu}(z) = J_\nu(z)R_{m,\nu}(z) - J_{\nu-1}(z)R_{m-1,\nu+1}(z) }[/math]

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

They are given explicitly by

[math]\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}. }[/math]

See also

• Lommel function
• Neumann polynomial

References

• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II, McGraw-Hill Book Company, Inc., New York-Toronto-London, http://apps.nrbook.com/bateman/Vol2.pdf 
• Hazewinkel, Michiel, ed. (2001), "l/l060810", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=l/l060810 
• Lommel, Eugen von (1871), "Zur Theorie der Bessel'schen Functionen", Mathematische Annalen (Berlin / Heidelberg: Springer) 4 (1): 103–116, doi:10.1007/BF01443302