Duality and Difference Operators for Matrix Valued Discrete Polynomials on the Nonnegative Integers

Abstract

In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators acting on the matrix orthogonal polynomials. These operators belong to the so called Fourier algebras, which play a key role in the construction of the families. In order to illustrate duality, we describe a family of Charlier type matrix orthogonal polynomials with explicit shift operators which allow us to find explicit formulas for three term recurrences, difference operators and squared norms. These are the essential ingredients for the construction of different dual families.

Matrix exceptional Laguerre polynomials

“…Proof. We first note that ( 16) and ( 17) are equivalent using ( 13) and (14). So it suffices to prove ( 17), which we expand as…”

“…Observe that the term 𝜙 ′ 𝑊𝐹 * 2 (𝜙 ′ ) * on the left-hand side is equal to 𝜙 ′ 𝐹 2 𝑊(𝜙 ′ ) * and on the righthand side by (13). Next, we use (14) in the terms 𝜙(𝑊𝐹 * 2 ) ′ (𝜙 ′ ) * and 𝜙 ′ (𝐹 2 𝑊) ′ 𝜙 * and (15) in the terms 𝜙(𝐹 1 𝑊) ′ 𝜙 * to see that we have to show that…”

“…[12], but it can be done for more examples of this nature for either differential or difference operators. 11,13,14 Take 𝑇 0 , the second-order difference operator defined in Ref. = 𝟏 so that 𝜆 = 0.…”

“…[32, p. 127]), and using Ref. [55,Theorem 3,(14)], the indicial equation for (A.2) becomes det ( 𝜇(𝜇 − 1) + 𝜇𝐵 1 (0) + 𝐵 2 (0)…”