Multiple orthogonal polynomials associated with Macdonald functions

We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bessel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Padé table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation. 1 Macdonald functions In this paper we will investigate certain polynomials satisfying orthogonality properties with respect to weight functions related to Macdonald functions Kν(z). These Macdonald functions are modified Bessel functions of the second kind and satisfy the differential equation zu + zu′ − (z + ν)u = 0, for which they are the solution that remains bounded as z tends to infinity on the real line. They can be given by the following integral representations Kν(z) = ( π 2z )1/2 e Γ(ν + 1/2) ∫ ∞ 0 et(1 + t 2z ) dt (1.1)