Möbius Covariance and Coefficient Duality: From Bernoulli Series to Enumerative Applications

Abstract

A coefficient duality first encountered for formal Bernoulli series is shown to be equivalent to a general Möbius 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 Möbius-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.

Disclosure

“directions for extending the framework, while the cubic multisection above suggests a systematic study of higher root-of-unity filters and their relation to known lacunary recurrences. Acknowledgements The author acknowledges the use of OpenAI’s ChatGPT as an assistive tool in the preparation of this manuscript, including literature discovery, exploratory derivations, symbolic and computational checks, organization of the exposition, and language and LaTeX editing. All mathematical statem”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T05:16:15.041008Z

Structural counts