Lommel polynomial

A Lommel polynomial R_m,ν(z) is a polynomial in 1/z giving the recurrence relation

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

where J_ν(z) is a Bessel function of the first kind.^[1]

They are given explicitly by

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

• Lommel function
• Neumann polynomial

• 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 

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Lommel polynomial. Read more