Coherent pair of measures for orthogonal polynomials on lattices

TL;DR

This paper introduces a new concept of coherent pairs of measures for orthogonal polynomials on lattices, establishing conditions under which their associated functionals are connected by rational factors, extending classical theory.

Contribution

It generalizes the theory of orthogonal polynomials on lattices by defining and analyzing $(M,N)$-coherent pairs of measures of order $(m,k)$ with new functional relations.

Findings

Functionals are connected by rational factors when $m=k$.

For $k>m$, functionals are semiclassical and connected via rational factors after applying shift operators.

Introduces the concept of $(M,N)$-coherent pairs of measures for orthogonal polynomials on lattices.

Abstract

We consider two sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ with respect regular functionals ${\bf u}$ and ${\bf v}$, respectively. We assume that $$\sum_{j=1} ^{M} a_{j,n}\mathrm{D}_x ^k P_{k+n-j} (z)=\sum_{j=1} ^{N} b_{j,n}\mathrm{D}_x ^{m} Q_{m+n-j} (z)\;,$$ with $k,m,M,N \in \mathbb{N}$, $a_{j,n}$ and $b_{j,n}$ are sequences of complex numbers, $$2\mathrm{S}_xf(x(s))=(\triangle +2\,\mathrm{I})f(z),~~ \mathrm{D}_xf(x(s))=\frac{\triangle}{\triangle x(s-1/2)}f(z),$$ $z=x(s-1/2)$, $\mathrm{I}$ is the identity operator, $x$ defines a lattice, and $\triangle f(s)=f(s+1)-f(s)$. We show that under some natural conditions, the functionals ${\bf u}$ and ${\bf v}$ are connected by a rational factor whenever $m=k$, and for $k>m$, ${\bf u}$ and ${\bf S}_x ^{k-m}{\bf v}$ are semiclassical functionals and in addition ${\bf S}_x{\bf u}$ and ${\bf S}_x ^{k-m+1}{\bf v}$ are connected by a rational factor. This leads to the notion of $(M,N)$-coherent pair of measures of order $(m,k)$ extended to orthogonal polynomials on lattices.