Definite Sums as Solutions of Linear Recurrences With Polynomial Coefficients

Abstract

We present an algorithm which, given a linear recurrence operator $L$ with polynomial coefficients, $m\in \mathbb{N}\setminus \{0\}$, $a_1,a_2,\dots ,a_m\in \mathbb{N}\setminus \{0\}$ and $b_1,b_2,\dots ,b_m\in \mathbb{K}$, returns a linear recurrence operator $L^{\prime }$ with rational coefficients such that for every sequence $h$, \[ L\left(\sum_{k=0}^\infty \prod_{i=1}^m \binom{a_i n + b_i}{k} h_k\right) = 0 \] if and only if $L^{\prime }h=0$.

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• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals