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Journal Articles Journal of Number Theory Year : 2023

On the representability of sequences as constant terms

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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hal-03907115 , version 1 (19-12-2022)
hal-03907115 , version 2 (22-12-2022)

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Alin Bostan, Armin Straub, Sergey Yurkevich. On the representability of sequences as constant terms. Journal of Number Theory, 2023, 253, pp.235-256. ⟨10.1016/j.jnt.2023.06.015⟩. ⟨hal-03907115v2⟩
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