A New Algebraization of the Lame Equation

TL;DR

This paper introduces a novel algebraic framework for the Lame equation, enabling explicit formulas for its solutions using Chebyshev and orthogonal polynomials, thus advancing the analytical understanding of these special functions.

Contribution

It presents a new Lie-algebraic formulation of the Lame Hamiltonian, providing explicit expressions for Lame polynomials and functions in terms of classical and orthogonal polynomials.

Findings

Explicit formulas for Lame polynomials derived

Representation of Lame functions via Chebyshev and orthogonal polynomials

Enhanced algebraic understanding of Lame equation solutions

Abstract

We develop a new way of writing the Lame Hamiltonian in Lie-algebraic form. This yields, in a natural way, an explicit formula for both the Lame polynomials and the classical non-meromorphic Lame functions in terms of Chebyshev polynomials and of a certain family of weakly orthogonal polynomials