Preprint Article Version 6 Preserved in Portico This version is not peer-reviewed

On the Constant Terms of Certain Laurent Series

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)

How to cite: Brent, B. On the Constant Terms of Certain Laurent Series. Preprints 2023, 2023061164. https://doi.org/10.20944/preprints202306.1164.v6 Brent, B. On the Constant Terms of Certain Laurent Series. Preprints 2023, 2023061164. https://doi.org/10.20944/preprints202306.1164.v6

Abstract

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.

Keywords

Laurent series; constant term; modular form; integer partition

Subject

Computer Science and Mathematics, Algebra and Number Theory

Comments (1)

Comment 1
Received: 29 June 2023
Commenter: Barry Brent
Commenter's Conflict of Interests: Author
Comment: Demoted Theorem 1 to equation (3).
Renamed the corollary to the (former) Theorem 1 as Theorem 1 itself.
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