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On the generalized eta- and theta-transformation formulas as the Hecke modular relation

  • Ruiyang Li , Takako Kuzumaki EMAIL logo and Shigeru Kanemitsu
Published/Copyright: May 7, 2025
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Abstract

We prove that both the eta transformation formula and the transformation formula for Dieter’s Lambert series are consequences of the Hecke functional equation for the generating zeta-function. Then using the product expression for the theta-function, we prove the theta-transformation formula under a general modular substitution is a consequence of those two transformation formulas, and consequently, theta transformation formula itself is a consequence of the Hecke modular relation.

MSC 2010: 11F03; 01A55; 40A30; 42A16

Dedicated to Professor Dr. Masanobu Kaneko with friendship and great respect


  1. (Communicated by Marco Cantarini)

Acknowledgement

The authors are grateful to the referees for their comments and suggestions for the improvement of the paper.

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Received: 2024-05-25
Accepted: 2024-11-05
Published Online: 2025-05-07
Published in Print: 2025-04-28

© 2025 Mathematical Institute Slovak Academy of Sciences

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