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

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

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

They are given explicitly by

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

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