On the representability of sequences as constant terms

TL;DR

This paper classifies which sequences, defined as constant terms of polynomial powers, can be represented as diagonals of rational functions, focusing on linear recurrence and hypergeometric sequences.

Contribution

It provides a classification of constant term sequences as diagonals for linear recurrences and certain hypergeometric sequences, addressing a question posed by Don Zagier.

Findings

Classified constant term sequences satisfying linear recurrences.

Identified hypergeometric sequences that are constant terms.

Connected constant term sequences to diagonals of rational functions.

Abstract

A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the generating functions of such sequences are invariably diagonals of multivariate rational functions, and hence special period functions, it is a famous open question, raised by Don Zagier, to classify diagonals that are constant terms. In this paper, we provide such a classification in the case of sequences satisfying linear recurrences with constant coefficients. We also consider the case of hypergeometric sequences and, for a simple illustrative family of hypergeometric sequences, classify those that are constant terms.