Multiple binomial sums

TL;DR

This paper characterizes multiple binomial sums via their generating functions as diagonals of rational functions and introduces algorithms for their equality and recurrence relation determination, improving computational efficiency.

Contribution

It establishes a novel characterization of multiple binomial sums through rational function diagonals and develops practical algorithms for their analysis.

Findings

Sequences are characterized as diagonals of rational functions.

Algorithms effectively decide equality and compute recurrences.

Approach avoids spurious singularities and certificates.

Abstract

Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with algebraic generating function. We study the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is twofold. Firstly, we show that a univariate sequence is a multiple binomial sum if and only if its generating function is the diagonal of a rational function. Secondly, we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. In conjunction with geometric simplifications of the integral representations, this approach behaves well in practice. The process avoids the computation of certificates and the problem of the appearance of spurious singularities that afflicts discrete creative telescoping, both in theory and in practice.