A P-recursive sequence is a sequence which satisfies a linear recurrence relation with polynomial coefficients. For a P-recursive sequence $\{a_n{\}}_{n\ge 0}$ and its power $\{a_n^m{\}}_{n\ge 0}$, we give a method to construct a polynomial $X(n)$ so that the sum of $\sum _{k=0}^{n-1}X(k)a_k^m$ equals a finite sum of the products of a polynomial and $a_{n-n_1}{a}_{n-n_2}\cdots a_{n-n_k}$, where $n_1,\dots ,n_k$ are integers. When these summands have a common factor $f(n)$, we will derive that $f(n)$ is a factor of the sum $\sum _{k=0}^{n-1}X(k)a_k^m$. This method can be used to derive some congruences. We give some examples to exhibit the applications of the method.

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P-递归序列的同余和伸缩

甲P -recursive序列是这样的序列，其满足与多项式的系数的线性递归关系。对于P递归序列$\{{\mathrm{一种}}_n{\}}_{n\ge 0}$ 和它的力量 $\{{\mathrm{一种}}_n^{米}{\}}_{n\ge 0}$，我们给出一种构造多项式的方法 $X(n)$ 所以总和 $\sum _{克=0}^{n-1}X(克){\mathrm{一种}}_{克}^{米}$ 等于多项式和 ${\mathrm{一种}}_{n-n_1}{\mathrm{一种}}_{n-n_2}\cdots {\mathrm{一种}}_{n-n_{克}}$， 在哪里 $n_1,\dots ,n_{克}$是整数。当这些被加数有一个公因数时$F(n)$，我们将得出 $F(n)$ 是总和的一个因数 $\sum _{克=0}^{n-1}X(克){\mathrm{一种}}_{克}^{米}$. 这个方法可以用来推导出一些同余。我们给出了一些例子来展示该方法的应用。