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Multiple q-Kravchuk polynomials

J. Arvesú, A. M. Ramírez-Aberasturis

Abstract

We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribution. The lowering and raising operators as well as the Rodrigues formula for these polynomials are obtained. The difference equation of order r+1 is studied. The connection via limit relation between four types of Kravchuk polynomials is discussed.

Multiple q-Kravchuk polynomials

Abstract

We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribution. The lowering and raising operators as well as the Rodrigues formula for these polynomials are obtained. The difference equation of order r+1 is studied. The connection via limit relation between four types of Kravchuk polynomials is discussed.
Paper Structure (6 sections, 6 theorems, 67 equations)

This paper contains 6 sections, 6 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. Multiple Kravchuk polynomials
  4. Multiple $q$-Kravchuk polynomials
  5. Difference equation for multiple $q$-Kravchuk polynomials
  6. Connection with multiple Kravchuk polynomials

Key Result

Lemma 3.2

For monic $q$-Kravchuk multiple orthogonal polynomials we have $r$ raising operators where $\vec{\beta}_{i,q^{2}}=(\beta_{1},\ldots,q^{2}\beta_{i},\ldots,\beta_{r})$ and

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 6 more