  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.
