From Wikipedia, the free encyclopedia
A Lommel polynomial Rm,ν(z) 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4aafa015cfa644459ff383cd9fde516f955308)
where Jν(z) is a
Bessel function of the first kind.
[1]
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c01d46ac1e0813de47d962ee9ba94be163df843a)
See also
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