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 to regular functionals \(\textbf{u}\) and \(\textbf{v}\) , respectively. We assume that \(\begin{aligned} \sum _{j=1} ^{M} a_{j,n}\textrm{D}_x ^k P_{k+n-j} (z)=\sum _{j=1} ^{N} b_{j,n}\textrm{D}_x ^{m} Q_{m+n-j} (z)\;, \end{aligned}\) with \(k,m,M,N \in \mathbb {N}\) , \(a_{j,n}\) and \(b_{j,n}\) are sequences of complex numbers, \( 2\textrm{S}_xf(x(s))=(\triangle +2\,\textrm{I})f(z),~~ \textrm{D}_xf(x(s))=\frac{\triangle }{\triangle x(s-1/2)}f(z), \) \(z=x(s-1/2)\) , \(\triangle f(s)=f(s+1)-f(s)\) , \(\textrm{I}\) is the identity operator, and x defines a lattice. We show that under some natural conditions, the functionals \(\textbf{u}\) and \(\textbf{v}\) are connected by a rational factor whenever \(m=k\) , and for \(k>m\) , \(\textbf{u}\) and \(\textbf{S}_x ^{k-m} \textbf{v}\) are semiclassical functionals and in addition \(\textbf{S}_x\textbf{u}\) and \(\textbf{S}_x ^{k-m+1} \textbf{v}\) 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.