Multiple binomial sums

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.

References in corpus (4)

• A Fast Approach to Creative Telescoping
• The Ring of Malcev-Neumann Series and the Residue Theorem
• Counting With Irrational Tiles
• Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables

Cited by in corpus (10)

• Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers
• Discrete analogues of Macdonald-Mehta integrals
• Constructing minimal telescopers for rational functions in three discrete variables
• New representations for all sporadic Apéry-like sequences, with applications to congruences
• Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems
• Calabi--Yau Operators
• Plane bipolar orientations and quadrant walks
• Creative Telescoping on Multiple Sums
• Reduction-Based Creative Telescoping for Algebraic Functions
• Some Open Problems related to Creative Telescoping