On the Constant Terms of Certain Laurent Series

Barry Brent

doi:10.20944/preprints202306.1164.v4

Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 15 June 2023 / Approved: 16 June 2023 / Online: 16 June 2023 (03:45:35 CEST)
Version 2 : Received: 18 June 2023 / Approved: 19 June 2023 / Online: 19 June 2023 (05:14:08 CEST)
Version 3 : Received: 19 June 2023 / Approved: 20 June 2023 / Online: 20 June 2023 (02:58:25 CEST)
Version 4 : Received: 21 June 2023 / Approved: 21 June 2023 / Online: 21 June 2023 (02:46:26 CEST)
Version 5 : Received: 26 June 2023 / Approved: 26 June 2023 / Online: 26 June 2023 (05:23:28 CEST)
Version 6 : Received: 28 June 2023 / Approved: 28 June 2023 / Online: 29 June 2023 (02:07:30 CEST)

We study the divisibility properties of the constant terms of certain meromorphic modular forms for Hecke groups. We relate those properties to those of some sequences that have already appeared in the literature. For possible use in later drafts, we show how to invert the map taking a Laurent series $f(x) = 1/x + \sum_{n=0}^{\infty}a_{n+1} x^n$ to the sequence of constant terms of its positive powers. At the end of the article, we construct from elementary arithmetic functions some meromorphic but not necessarily modular functions and study their constant terms.

Laurent series; constant term; modular form; integer partition

Computer Science and Mathematics, Algebra and Number Theory

Comment 1

Received: 21 June 2023
Commenter: Barry Brent
Commenter's Conflict of Interests: Author

Comment: Added a corollary to Theorem 1 and made small revisions to abstract and introduction to account for the new material.

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On the Constant Terms of Certain Laurent Series

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