More on Rotations as Spin Matrix Polynomials

TL;DR

This paper explores expressing rotations as spin matrix polynomials, using biorthogonal systems and factorial numbers to derive explicit coefficients, analyzing their structure and behavior for large spin values.

Contribution

It introduces a new framework utilizing biorthogonal systems and factorial numbers to explicitly determine polynomial coefficients for spin rotations.

Findings

Explicit coefficients for rotation polynomials are derived.

Structural features of the polynomial coefficients are analyzed.

Large j limits of the coefficients are examined.

Abstract

Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined.