# Chebyshev, Legendre, Hermite and other orthonormal polynomials in D-dimensions

## TL;DR

This paper introduces a comprehensive method for constructing symmetric tensor polynomials in D-dimensional space that are orthonormal under various weights, generalizing classical polynomials and enabling new applications in physics and mathematics.

## Contribution

It presents a unified framework for generating orthonormal tensor polynomials in multiple dimensions, including new polynomials for non-standard weights like Fermi-Dirac and Bose-Einstein distributions.

## Key findings

- Derived explicit formulas for polynomials up to fifth order.
- Validated the orthonormalization process through matching equations and coefficients.
- Extended classical polynomials to new weights relevant in physics.

## Abstract

We propose a general method to construct symmetric tensor polynomials in the D-dimensional Euclidean space which are orthonormal under a general weight. The D-dimensional Hermite polynomials are a particular case of the present ones for the case of a gaussian weight. Hence we obtain generalizations of the Legendre and of the Chebyshev polynomials in D dimensions that reduce to the respective well-known orthonormal polynomials in D=1 dimensions. We also obtain new D-dimensional polynomials orthonormal under other weights, such as the Fermi-Dirac, Bose-Einstein, Graphene equilibrium distribution functions and the Yukawa potential. We calculate the series expansion of an arbitrary function in terms of the new polynomials up to the fourth order and define orthonormal multipoles. The explicit orthonormalization of the polynomials up to the fifth order (N from 0 to 4) reveals an increasing number of orthonormalization equations that matches exactly the number of polynomial coefficients indication the correctness of the present procedure.