Coherent pair of measures for orthogonal polynomials on lattices

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.

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