The quantum q-Krawtchouk and q-Meixner polynomials and their related D-functions for the quantum groups SUq(2) and SUq(1,1)

Abstract

The complete comparative analysis of the quantum q-Krawtchouk and q-Meixner polynomials of a discrete variable on a nonuniform grid ( x( s)= q 2 s ) and the D-function for the quantum groups SU q (2) and SU q (1,1) is done. The complete set of characteristics of these polynomials (i.e., orthogonality relations, normalization factors, recurrent relations, the explicit analytical expressions, the Rodrigues formulas, the formulas of difference derivatives, various particular values and cases) are calculated. The correlations between the properties of the polynomials mentioned above and the D-functions for the quantum groups SU q (2) and SU q (1,1) are established. In the case of SU q (1,1) only D-functions for the positive discrete series of the unitary irreducible representations are considered. It is known that on the nonuniform grid x( s)= q 2 s there are two kinds of Krawtchouk and Meixner polynomials. Also the properties of the second kind of these polynomials which are not connected to the D-functions are discussed.