The orthogonal Shmaliy polynomials are Hahn polynomials

TL;DR

This paper demonstrates that the so-called orthogonal Shmaliy polynomials are actually a subclass of Hahn polynomials, providing their hypergeometric representation and analyzing their application in unbiased FIR filter design.

Contribution

It clarifies the relationship between Shmaliy polynomials and Hahn polynomials, including their hypergeometric representation and application in filter design.

Findings

Shmaliy polynomials are Hahn polynomials.

Hypergeometric function representation of these polynomials is established.

Transfer function of the unbiased FIR filter is determined.

Abstract

Morales-Mendoza et al. present in 2013 a new class of discrete orthogonal polynomials. They use these polynomials to design an unbiased FIR filter. In their paper they make the statement that a representation of the polynomials via hypergeometric functions is unknown. However Shakibaei Asli et al. found in 2017 a $_3F_2$ hypergeometric function representation. In this paper it is shown that the "new" orthogonal polynomials belong to the class of the Hahn polynomials. The transfer function of the unbiased FIR filter has been determined. In the second part, an orthogonal unbiased FIR filter is designed following the method of the orthogonal derivative described in the thesis of the author. In the Appendix a table with a number of possible transformations of the Hahn polynomials is given.