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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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BibTeX

@article{constantterms-2022,
    author = {Alin Bostan and Armin Straub and Sergey Yurkevich},
    title = {On the representability of sequences as constant terms},
    journal = {Preprint},
    year = {2022},
}