The discrete Chebyshev–Meckler–Mermin–Schwarz polynomials and spin algebra

The polynomials discovered by Chebyshev and subsequently related to spin transition probabilities by Meckler [Meckler, Phys. Rev. 111, 1447 (1958)] and Mermin and Schwarz [Found. Phys. 12, 101 (1982)] are studied, and their application to phase space representations of spin states and operators is examined. In particular, a formula relating the end-point value of the polynomials to scale factors relating different phase space representations of spherical harmonic operators is found. This formula is applied to illustrative calculations of Wigner functions for a single spin and the singlet state of a pair of spins.