Closed form summation of $C$-finite sequences

Abstract:

We consider sums of the form \[ \sum _{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)\dots F_k(a_kn+b_kj+c_k),\] in which each $\{F_i(n)\}$ is a sequence that satisfies a linear recurrence of degree $D(i)<\infty$, with constant coefficients. We assume further that the $a_i$’s and the $a_i+b_i$’s are all nonnegative integers. We prove that such a sum always has a closed form, in the sense that it evaluates to a linear combination of a finite set of monomials in the values of the sequences $\{F_i(n)\}$ with coefficients that are polynomials in $n$. We explicitly describe two different sets of monomials that will form such a linear combination, and give an algorithm for finding these closed forms, thereby completely automating the solution of this class of summation problems. We exhibit tools for determining when these explicit evaluations are unique of their type, and prove that in a number of interesting cases they are indeed unique. We also discuss some special features of the case of “indefinite summation", in which $a_1=a_2=\cdots = a_k = 0$.