The factorial-basis method for finding definite-sum solutions of linear recurrences with polynomial coefficients

Abstract

The problem of finding a nonzero solution of a linear recurrence $Ly=0$ with polynomial coefficients where y has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of Chen and Kauers (2017, Sec. 8), has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a quasi-triangular, shift-compatible factorial basis $B={〈P_k\left(n\right)〉}_{k=0}^{\infty }$ of the polynomial space $K\left[n\right]$ over a field $K$ of characteristic zero, computes a recurrence satisfied by the coefficient sequence $c={〈c_k〉}_{k=0}^{\infty }$ of the solution $y_n={\sum }_{k=0}^{\infty }{c}_k{P}_k\left(n\right)$ (where, thanks to the quasi-triangularity of $B$, the sum on the right terminates for each $n\in N$). More generally, if $B$ is m-sieved for some $m\in N$, our algorithm computes a system of m recurrences satisfied by the m-sections of the coefficient sequence c. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of $Ly=0$.