Change of basis for the tridiagonal pairs of type II

Abstract

We study tridiagonal pairs of type II. These involve two linear transformations $A$ and $A^{\star }$. We define two bases. In the first one, $A$ acts as a diagonal matrix while $A^{\star }$ acts as a block tridiagonal matrix, and in the second one, $A$ acts as a block tridiagonal matrix while $A^{\star }$ acts as a diagonal matrix. We obtain the change of basis coefficients between these two bases. The coefficients are special functions that are written as a nested product of polynomials that resemble Racah polynomials but involve shift operators in their expression.

19 pages