Some formulas related to Euler’s product expansion for cosine function

Taekyun Kim and Dae San Kim
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 166–180
DOI: 10.7546/nntdm.2025.31.1.166-180
Full paper (PDF, 263 Kb)

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Authors and affiliations

Taekyun Kim
Department of Mathematics, Kwangwoon University
Seoul 139-701, Republic of Korea

Dae San Kim
Department of Mathematics, Sogang University
Seoul 121-742, Republic of Korea

Abstract

In this paper, we derive by using elementary methods some continued fractions, certain identities involving derivatives of \tan x, several expressions for \log \cosh x and an identity for \pi^{2}, from a series expansion of \tan x, which gives the product expansion of the cosine function.

Keywords

  • Continued fraction
  • Product expansion
  • Bernoulli numbers
  • Euler numbers

2020 Mathematics Subject Classification

  • 11A55
  • 11B68

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Manuscript history

  • Received: 23 January 2025
  • Revised: 27 April 2025
  • Accepted: 28 April 2025
  • Online First: 28 April 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Kim, T., & Kim, D. S. (2025). Some formulas related to Euler’s product expansion for cosine function. Notes on Number Theory and Discrete Mathematics, 31(1), 166-180, DOI: 10.7546/nntdm.2025.31.1.166-180.

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