Some differentiation formulas for Legendre polynomials

Abstract

In a series of recent works, we have provided a number of explicit expressions for the derivative of the associated Legendre function of the first kind with respect to its degree, $[\partial P_{\nu }^m(z)/\partial \nu {]}_{\nu =n}$, with $m,n\in \mathbb{N}$. In this communication, we use some of those expressions to obtain several, we believe new, explicit formulas for the derivatives ${\mathrm{d}}^m[P_n(z)\ln (z\pm 1)]/\mathrm{d}{z}^m$, where $P_n(z)$ is the Legendre polynomial.

LaTeX, 6 pages

References in corpus (1)

• On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)