A Derivation of Classical Orthogonal Polynomials using Generalized Vandermonde Determinants

TL;DR

This paper derives classical Hermite, Laguerre, and Jacobi orthogonal polynomials via Gram-Schmidt orthogonalization using generalized Vandermonde determinants involving Gamma and Beta functions, with a geometric perspective.

Contribution

It introduces a novel derivation method for classical orthogonal polynomials using generalized Vandermonde determinants and a geometric formulation of Gram-Schmidt orthogonalization.

Findings

Explicit derivation of Hermite, Laguerre, and Jacobi polynomials

Use of generalized Vandermonde determinants with Gamma and Beta functions

Geometric interpretation of Gram-Schmidt process

Abstract

We present a derivation of classical Hermite, Laguerre, and Jacobi orthogonal polynomials directly through the Gram-Schmidt orthogonization process. The derivation uses certain generalized Vandermonde determinants with entries defined by Gamma and Beta functions. We also provide a geometric formulation of Gram-Schmidt orthogonalization using the Hodge star operator.