Some differentiation formulas for Legendre polynomials

TL;DR

This paper derives new explicit formulas for derivatives of Legendre polynomials multiplied by logarithmic functions, expanding the mathematical tools available for special functions analysis.

Contribution

It introduces novel explicit formulas for derivatives of Legendre polynomials combined with logarithmic functions, building on previous work on derivatives with respect to degree.

Findings

New explicit formulas for derivatives of Legendre polynomials with logarithmic factors

Extension of previous derivative expressions for associated Legendre functions

Potential applications in mathematical physics and computational methods

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.