Representations of the q-deformed algebras Uq(so3) and Uq(so5) and q-orthogonal polynomials

Abstract

Orthogonal polynomials related to irreducible representations of the classical type of the q-deformed algebras Uq(so3) and Uq(so5) are investigated. The main method consists in the diagonalization of corresponding infinitesimal operators (generators) of representations. For the algebra Uq(so3) this method leads to q-analogs of Krawtchouk polynomials. The properties of these polynomials are considered, the q-difference equation, the recurrence and explicit formulas. For the algebra Uq(so5), the diagonalization process of generators of representations leads to the connection with some class of orthogonal polynomials in two discrete variables. These variables are the so-called q-numbers [n], where [n]=(qn−q−n)∕(q−q−1). The introduced polynomials can be considered as two-dimensional q-analogs of Krawtchouk polynomials. The q-difference equation of the Sturm-Liouville type for these polynomials is constructed. The corresponding eigenvalues are investigated including the explicit formulas for their multiplicities. The structure of polynomial solutions is described.