General integral-identities involving Legendre polynomials and their derivatives

Abstract

Integrals involving derivatives of Legendre polynomials frequently arise in applications ranging from multipole expansions for processes involving electromagnetic probes to spectral methods in numerical physics. Despite their practical relevance, closed-form expressions for such integrals - particularly involving arbitrary derivative orders - are not readily accessible in standard references or symbolic tools. In this note, we derive and present general analytic expressions for integrals of the form $\int_{-1}^{+1} dx P^{(q)}_{n} (x) P^{(k)}_{m} (x)$, where $P_{n} (x)$ and $P_{m} (x)$ are Legendre polynomials and $q$, $k$ denote their order of differentiation. Using repeated integration by parts, parity arguments, and closed-form boundary evaluations, we obtain explicit binomial and Gamma-function representations valid for all non-negative integers $n$, $m$, $q$, $k$. These results unify and extend known orthogonality relations and provide ready-to-use tools for analytic and computational contexts.