Very special special functions: Chebyshev polynomials of a discrete variable and their physical applications

TL;DR

This paper explores the properties and diverse applications of Chebyshev polynomials of a discrete real variable in spin physics, magnetic resonance, and tensor operator expansions, highlighting their role as a unifying mathematical tool.

Contribution

It provides a detailed analysis of these polynomials and demonstrates their novel applications in magnetic resonance and angular momentum coupling, establishing their significance in physical sciences.

Findings

Chebyshev polynomial operators form an orthonormal basis for expansion.

They serve as Clebsch-Gordan coupling coefficients.

They can be recoupled as rank-zero composite tensors.

Abstract

Over nearly six decades, the Chebyshev polynomials of a discrete real variable have found applications in spin physics, spin tomography, in the development of operator expansions, and in defining tensor operator equivalents. The properties of these polynomials are discussed in detail, and then examples are provided to illustrate the diversity of their applications in magnetic resonance. These examples include the use of the Chebyshev polynomial operators as an orthonormal basis to expand rotation operators, projection operators, and the Stratonovich-Weyl operator. The duality of the Chebyshev polynomials of a discrete real variable as Clebsch-Gordan coupling coefficients is noted and exploited, and it is shown that the Chebyshev polynomial operators can be recoupled as a rank-zero composite tensor defined by the product of a spin and spatial tensor. These application examples demonstrate that the Chebyshev polynomials of a discrete real variable are a unique nexus for spin physics, special functions, angular momentum (re)coupling, and irreducible representations of the rotation group.