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 $(m,k)$.

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关于相干测度对的另一种扩展

令$M$和$N$为固定的非负整数，令$\pi_N$为$N$次多项式。假设 $(P_n)_{n\geq0}$ 和 $(Q_n)_{n\geq0}$ 是两个正交多项式序列，使得它们的阶数 $k$ 和 $m$（分别）的导数满足结构关系 $$ \pi_N(x)\,P_{n+m}^{(m)}(x)= \sum_{j=nM}^{n+N}r_{n,j}Q_{j+k }^{(k)}(x)\quad (n=0,1,\ldots)\,, $$ 其中 $r_{n,j}$ 是独立于 $x$ 的复数。结果表明，在自然约束下，$(P_n)_{n\geq0}$和$(Q_n)_{n\geq0}$是半经典正交多项式序列。此外，它们对应的矩线性泛函在分布意义上通过有理修正相关。这导致了 $\pi_N-$coherent pair 的概念，索引为 $M$，顺序为 $(m,k)$。