  Let $M$ and $N$ be fixed non-negative integer numbers and let $\pi_N$ be a
polynomial of degree $N$. Suppose that $(P_n)_{n\geq0}$ and $(Q_n)_{n\geq0}$
are two orthogonal polynomial sequences such that %their derivatives of orders
$k$ and $m$ (respectively) satisfy the structure relation $$
\pi_N(x)\,P_{n+m}^{(m)}(x)= \sum_{j=n-M}^{n+N}r_{n,j}Q_{j+k}^{(k)}(x)\quad
(n=0,1,\ldots)\,, $$ where $r_{n,j}$ are complex number independent of $x$. It
is shown that under natural constraints, $(P_n)_{n\geq0}$ and $(Q_n)_{n\geq0}$
are semiclassical orthogonal polynomial sequences. Moreover, their
corresponding moment linear functionals are related by a rational modification
in the distributional sense. This leads to the concept of $\pi_N-$coherent pair
with index $M$ and order $(m,k)$.

Classical Analysis and ODEs