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arXiv · 2608.14931

M\"obius Covariance and Coefficient Duality: From Bernoulli Series to Enumerative Applications

Max A. Alekseyev
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Abstract

A coefficient duality first encountered for formal Bernoulli series is shown to be equivalent to a general M\"obius covariance law for formal power series. We obtain a structural characterization, an eigenspace interpretation, and a weighted form of this duality. The Catalan convolution and Chebyshev identities from the motivating Bernoulli setting extend to arbitrary M\"obius-covariant families and yield a general Ramanujan-type summation formula encompassing consecutive half-integer powers. The framework also recovers classical Bernoulli and Euler recurrences and produces recurrence families for colored matchings and generalized central trinomial coefficients, with further realizations from reflection-symmetric Appell sequences and Gorenstein Hilbert series.

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BibTeXRIS

Max A. Alekseyev. 2026-08-14. M\"obius Covariance and Coefficient Duality: From Bernoulli Series to Enumerative Applications. https://arxiv.org/abs/2608.14931

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