# Some Ultraspheroidal Monogenic Clifford Gegenbauer Jacobi Polynomials and Associated Wavelets

## TL;DR

This paper introduces new wavelet functions derived from Clifford analysis using orthogonal polynomials that generalize Jacobi and Gegenbauer polynomials, with proven reconstruction and Fourier rules.

## Contribution

It presents novel classes of wavelets based on Clifford analysis and orthogonal polynomials, extending classical polynomial families and establishing their mathematical properties.

## Key findings

- New classes of orthogonal polynomials based on 2-parameter weight functions.
- Introduction of new wavelet functions derived from these polynomials.
- Proof of reconstruction and Fourier-Plancherel formulas for the wavelets.

## Abstract

In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well known ones of Jacobi and Gegenbauer polynomials when relaxing one of the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rules have been proved.