On the representability of sequences as constant terms

On the representability of sequences as constant terms
Alin Bostan, Armin Straub, Sergey YurkevichJournal of Number Theory — Volume 253, 2023, Pages 235-256


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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    author = {Alin Bostan and Armin Straub and Sergey Yurkevich},
    title = {On the representability of sequences as constant terms},
    journal = {Journal of Number Theory},
    year = {2023},
    volume = {253},
    pages = {235--256},
    doi = {10.1016/j.jnt.2023.06.015},