Dual $-1$ Hahn polynomials: “Classical” polynomials beyond the Leonard duality

Abstract:

We introduce the $-$1 dual Hahn polynomials through an appropriate $q \to -1$ limit of the dual $q$-Hahn polynomials. These polynomials are orthogonal on a finite set of discrete points on the real axis, but in contrast to the classical orthogonal polynomials of the Askey scheme, the $-1$ dual Hahn polynomials do not exhibit the Leonard duality property. Instead, these polynomials satisfy a 4th-order difference eigenvalue equation and thus possess a bispectrality property. The corresponding generalized Leonard pair consists of two matrices $A,B$ each of size $N+1 \times N+1$. In the eigenbasis where the matrix $A$ is diagonal, the matrix $B$ is 3-diagonal; but in the eigenbasis where the matrix $B$ is diagonal, the matrix $A$ is 5-diagonal.