Lommel polynomial

The polynomial $R_{m, ν} (z)$ of degree $m$ in $z^{-} 1$ which for $m = 0, 1, \dots$ and any $ν$ is defined by

$R_{m, ν} (z) =$

$= \frac{π z}{2 \sin ν π} [J_{ν + m} (z) J_{- ν + 1} (z) + (- 1)^{m} J_{- ν - m} (z) J_{ν - 1} (z)]$

or

$R_{m, ν} (z) = \frac{Γ (ν + m)}{Γ (ν)} {(\frac{2}{z})}^{m} \times$

$\times_{2} F_{3} (1 - \frac{m}{2}, - \frac{m}{2}; ν, - m, 1 - ν - m; - z^{2}) .$

Here $J_{μ} (z)$ is the Bessel function (cf. Bessel functions) and $_{2} F_{3}$ is the hypergeometric series. The Lommel polynomials satisfy the relations

$J_{ν + m} (z) = J_{ν} (z) R_{m, ν} (z) - J_{ν - 1} (z) R_{m - 1, ν + 1} (z),$

$R_{0, ν} (z) = 1, m = 1, 2, \dots .$

References[edit]

[1]	W. Magnus, F. Oberhettinger, R.P. Soni, "Formulas and theorems for the special functions of mathematical physics" , Springer (1966)