On another extension of coherent pairs of measures

Abstract

Let $M$ and $N$ be fixed non-negative integer numbers and let ${\pi }_N$ be a polynomial of degree $N$. Suppose that $(P_n{)}_{n\ge 0}$ and $(Q_n{)}_{n\ge 0}$ are two orthogonal polynomial sequences such that %their derivatives of orders $k$ and $m$ (respectively) satisfy the structure relation $${\pi }_N(x)P_{n+m}^{(m)}(x)=\sum _{j=n-M}^{n+N}{r}_{n,j}{Q}_{j+k}^{(k)}(x)(n=0,1,\dots ),$$ where $r_{n,j}$ are complex number independent of $x$. It is shown that under natural constraints, $(P_n{)}_{n\ge 0}$ and $(Q_n{)}_{n\ge 0}$ 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 $(m,k)$.