Abstract
We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These polynomials have lowering and raising operators, which lead to the Rodrigues formula, difference equation of order r+1, and explicit expressions for the coefficients of recurrence relation of order r+1. Some limit relations are obtained.
Highlights
Multiple orthogonal polynomials are polynomials that satisfy orthogonality conditions shared with respect to a set of measures [9,10,11,12,13,14,15,16,17]
They are related to the simultaneous rational approximation of a system of r analytic functions [18,19] and play an important role both in pure and applied mathematics
Α/q,~β−~ei (s) Mq,~n+~e i ( s ), which leads to the following r raising operators for the monic q-Meixner multiple orthogonal polynomials of the second kind α,β α,~β α/q,~β−~e
Summary
Hermite’s proof [1] of the transcendence of the number e uses the notion of simultaneous approximation, which was subsequently studied in approximation theory and number theory [2,3,4,5,6,7,8]. Multiple orthogonal polynomials are polynomials that satisfy orthogonality conditions shared with respect to a set of measures [9,10,11,12,13,14,15,16,17] They are related to the simultaneous rational approximation of a system of r analytic functions [18,19] and play an important role both in pure and applied mathematics (see for instance [20,21,22] as well as [23,24,25,26,27]).
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