Lommel polynomial

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A Lommel polynomial Rm(z), introduced by Eugen von Lommel (1871), is a polynomial in 1/z giving the recurrence relation

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

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

They are given explicitly by

R_{m,\nu} = \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[edit]