Satoru Odake

Multi-indexed Orthogonal Polynomials of a Discrete Variable

and Exactly Solvable Birth and Death Processes

It is well known that orthogonal polynomials of discrete variables belonging
to the Askey scheme satisfy second-order difference equations, and that on this basis it is possible to construct exactly solvable birth and death processes. However, it has been thought that this construction method cannot be applied to new types of orthogonal polynomials, namely exceptional and multi-indexed orthogonal polynomials.

The case-(1) multi-indexed orthogonal polynomials of a discrete variable have
been constructed so far for Racah, q-Racah, Meixner, little q-Jacobi and little q-Laguerre types.In this talk, I present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types (Hahn, dual Hahn, q-Hahn, dual q-Hahn, affine q-Krawtchouk, quantum q-Krawtchouk, dual quantum q-Krawtchouk and q-Meixner). Based on these 12 types of case-(1) multi-indexed orthogonal polynomials (excluding q-Meixner), I present exactly solvable continuous time birth and death processes. Their discrete time versions (Markov chains) are also obtained for finite types.