Coherent pair of measures for orthogonal polynomials on lattices

We consider two sequences of orthogonal polynomials $(P_n{)}_{n\ge 0}$ and $(Q_n{)}_{n\ge 0}$ with respect regular functionals $\mathbf{u}$ and $\mathbf{v}$, 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, $$2{\mathrm{S}}_{x}f(x(s))=(\mathrm{△}+2\mathrm{I})f(z),{\mathrm{D}}_{x}f(x(s))=\frac{\mathrm{△}}{\mathrm{△}x(s-1\mathrm{/}2)}f(z),$$ $z=x(s-1\mathrm{/}2)$, $\mathrm{I}$ is the identity operator, $x$ defines a lattice, and $\mathrm{△}f(s)=f(s+1)-f(s)$. We show that under some natural conditions, the functionals $\mathbf{u}$ and $\mathbf{v}$ are connected by a rational factor whenever $m=k$, and for $k>m$, $\mathbf{u}$ and ${\mathbf{S}}_x^{k-m}\mathbf{v}$ are semiclassical functionals and in addition ${\mathbf{S}}_x\mathbf{u}$ and ${\mathbf{S}}_x^{k-m+1}\mathbf{v}$ 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.