Discrete orthogonal polynomials associated with Macdonald function

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 $k\in {\mathbb{N}}_0,t\ge 0,\nu >-1,0<\lambda <1,{\rho }_{\mu }(z)=2z^{\mu /2}{K}_{\mu }\left(2\sqrt{z}\right)$. 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.