On Rotations as Spin Matrix Polynomials

TL;DR

This paper derives polynomial expressions for rotations using spin matrices through elementary differential equations and biorthogonal systems, highlighting the role of central factorial numbers in these formulations.

Contribution

It introduces new derivations of rotation polynomials in spin matrices using elementary methods and biorthogonal systems, emphasizing the significance of central factorial numbers.

Findings

Polynomial rotation expressions derived via differential equations

Alternative derivation using biorthogonal systems

Central factorial numbers are fundamental in the formulations

Abstract

Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an alternate derivation. The central factorial numbers play key roles in both derivations.