# Closed form summation of C-finite sequences

Research paper by **Curtis Greene, Herbert S. Wilf**

Indexed on: **11 Feb '05**Published on: **11 Feb '05**Published in: **Mathematics - Combinatorics**

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#### Abstract

We consider sums of the form
\[\sum_{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)... F_k(a_kn+b_kj+c_k),\]
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 $a_1=a_2=... = a_k = 0$.