Bilateral series weighted by periodic functions

Jing Li; Wenchang Chu; Jing Li; Wenchang Chu

doi:10.3934/math.20261234

AIMS Mathematics

2026, Volume 11, Issue 9: 31250-31266. doi: 10.3934/math.20261234

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Research article

Bilateral series weighted by periodic functions

Jing Li ¹,
Wenchang Chu ^{2
,
,}

1.
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, China
2.
Independent Researcher, Via Dalmazio Birago 9/E, Lecce 73100, Italy

Received: 15 May 2026 Revised: 10 September 2026 Accepted: 16 September 2026 Published: 23 September 2026
MSC : Primary 30E20, Secondary 11M32

By incorporating the derivative operator $ \theta $ and hyperbolic cotangent function, a general summation formula is established for the "Dirichlet–like" bilateral $ \Omega $-series. Seven classes of infinite series identities are exemplified as consequences. They do not seem to have previously appeared in the literature.
- "Dirichlet–like" bilateral series,
- derivative operator,
- periodic function,
- binomial sum by telescoping,
- hyperbolic tangent (cotangent) function
Citation: Jing Li, Wenchang Chu. Bilateral series weighted by periodic functions[J]. AIMS Mathematics, 2026, 11(9): 31250-31266. doi: 10.3934/math.20261234

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Abstract

By incorporating the derivative operator $ \theta $ and hyperbolic cotangent function, a general summation formula is established for the "Dirichlet–like" bilateral $ \Omega $-series. Seven classes of infinite series identities are exemplified as consequences. They do not seem to have previously appeared in the literature.

References

[1]	K. N. Boyadzhiev, Derivative polynomials for tanh, tan, sech and sec in explicit form, Fibonacci Quart. 45 (2007), 291–303. https://doi.org/10.1080/00150517.2007.12428197 doi: 10.1080/00150517.2007.12428197
[2]	R. Chapman, Evaluating $\zeta(2)$, Available from: . https://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf
[3]	W. Chu, C. Y. Wang, Convolution formulae for Bernoulli numbers, Integral Transforms Spec. Funct. 21 (2010), 437–457. https://doi.org/10.1080/10652460903360861 doi: 10.1080/10652460903360861
[4]	L. Comtet, Advanced Combinatorics, Dordrecht–Holland, The Netherlands, 1974 (Chap. Ⅲ).
[5]	P. Flajolet, B. Salvy, Euler Sums and Contour Integral Representations, Exper. Math. 7 (1998), 15–35. https://doi.org/10.1080/10586458.1998.10504356 doi: 10.1080/10586458.1998.10504356
[6]	I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Academic Press, 2007, Elsevier Inc.
[7]	R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics, Addison-Wesley Publ. Company, Reading, massachusetts, 1989.
[8]	N. Karjanto, R. Emmanuelle, B. Sriraman, Functional Dirichlet series, integral representations of a generalized zeta function, and applications to sustainability, in Bharath Sriraman (ed.) "Handbook of Visual, Experimental and Computational Mathematics", pages 391–439, Springer Book, 2026. https://doi.org/10.1007/978-3-032-16368-4_91
[9]	C. Wright, T. J.Osler, Four derivations of an interesting bilateral series generalizing the series for zeta of 2, Int. J. Math. Educ. Sci., 44 (2013), 456–461. https://doi.org/10.1080/0020739X.2012.714495 doi: 10.1080/0020739X.2012.714495

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