Derivatives with respect to the order of the Legendre Polynomials

TL;DR

This paper derives explicit formulas for derivatives of Legendre polynomials with respect to their order, including a detailed expression for the fourth derivative, advancing mathematical understanding of these functions.

Contribution

It provides new explicit formulas for derivatives of Legendre polynomials with respect to their order, including the fourth derivative, which was not previously available.

Findings

Explicit formulas for derivatives of Legendre polynomials with respect to order

An explicit form for the fourth derivative of Legendre polynomials

Enhanced mathematical tools for Legendre polynomial analysis

Abstract

Expressions for the derivatives of the Legendre polynomials of the first kind with respect to the order of these polynomials are given. An explicit form for the fourth derivative is presented.