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# On the representability of sequences as constant terms

On the representability of sequences as constant terms
Alin Bostan, Armin Straub, Sergey YurkevichPreprint — 2022

## 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 those diagonals which are constant terms. In this paper, we provide such a classification in the case of sequences satisfying linear recurrences with constant coefficients. We further consider the case of hypergeometric sequences and, for a simple illustrative family of hypergeometric sequences, classify those that are constant terms.

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