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A note on parameter derivatives of classical orthogonal polynomials

Coefficients in the expansions of the form $\partial P_{n}(λ;z)}/\partialλ=\sum_{k=0}^{n}a_{nk}(λ)P_{k}(λ;z)$, where $P_{n}(λ;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi) orthogonal polynomial of variable $z$ and $λ$ is a parameter, are evaluated. A method we adopt in the present paper differs from that used by Fröhlich [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.