Lommel polynomial

A Lommel polynomial R_m,ν(z) 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

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

See also

• Lommel function
• Neumann polynomial