A note on parameter derivatives of classical orthogonal polynomials

Radoslaw Szmytkowski

Published 2009-01-17, updated 2010-11-16Version 3

Coefficients in the expansions of the form $\partial P_{n}(\lambda;z)}/\partial\lambda=\sum_{k=0}^{n}a_{nk}(\lambda)P_{k}(\lambda;z)$, where $P_{n}(\lambda;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi) orthogonal polynomial of variable $z$ and $\lambda$ is a parameter, are evaluated. A method we adopt in the present paper differs from that used by Fr\"ohlich [Integral Transforms Spec. Funct. 2 (1994) 253] for the Jacobi polynomials and by Koepf [Integral Transforms Spec. Funct. 5 (1997) 69] for the generalized Laguerre and the Gegenbauer polynomials.