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 $\mathrm{d}^{m}[P_{n}(z)\ln(z\pm1)]/\mathrm{d}z^{m}$, where $P_{n}(z)$ is the Legendre polynomial.