Definite Sums as Solutions of Linear Recurrences With Polynomial Coefficients

We present an algorithm which, given a linear recurrence operator $L$ with polynomial coefficients, $m\in \mathbb{N}\setminus \{0\}$, $a_1,a_2,\ldots,a_m \in $\mathbb{N}$\setminus{0}$and$b_1,b_2,\ldots,b_m \in $\mathbb{K}$$, returns a linear recurrence operator $L^{\mathrm{\prime }}$ with rational coefficients such that for every sequence $h$, $$L\left(\sum _{k=0}^{\mathrm{\infty }}\prod _{i=1}^m\left(\genfrac{}{}{0px}{}{a_{i}n+b_i}{k}\right)h_k\right)=0$$ if and only if $L^{\mathrm{\prime }}h=0$.