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

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

See also

• Lommel function
• Neumann polynomial

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) [1994], "Lommel polynomial", Encyclopedia of Mathematics, EMS Press
• Lommel, Eugen von (1871), "Zur Theorie der Bessel'schen Functionen", Mathematische Annalen, Berlin / Heidelberg: Springer, 4 (1): 103–116, doi:10.1007/BF01443302