Discrete orthogonal polynomials associated with Macdonald function

TL;DR

This paper introduces new discrete orthogonal polynomials linked to the Macdonald function, exploring their properties, recurrence relations, and special cases, including connections to Meixner polynomials.

Contribution

It presents novel sequences of discrete orthogonal polynomials associated with the Macdonald function and analyzes their properties and recurrence relations.

Findings

Derived properties and recurrence relations for the new polynomials.

Established connections to Meixner polynomials in the limit case.

Investigated modified polynomial sequences with different weight functions.

Abstract

New sequences of discrete orthogonal polynomials associated with the modified Bessel function $K_\mu(z)$ or Macdonald function are considered. The corresponding weight function is $\lambda^k \rho_{k+\nu+1}(t)/ k!$, where $\ k \in \mathbb{N}_0, \ t \ge 0,\ \nu > -1,\ 0 < \lambda < 1,\ \rho_{\mu}(z) = 2 z^{\mu/2} K_\mu\left( 2\sqrt z\right)$. The limit case $t=0$ corresponds to the Meixner polynomials. Various properties, differential-difference recurrence relations are established. The modified sequence of polynomials with the weight $\lambda^k \rho_{k+\nu+1}(\lambda t)/ k! $ is investigated as well.