Alternating series in terms of Riemann zeta function and Dirichlet beta function

Zhiling Fan; Wenchang Chu; Zhiling Fan; Wenchang Chu

doi:10.3934/era.2024058

Electronic Research Archive

2024, Volume 32, Issue 2: 1227-1238. doi: 10.3934/era.2024058

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Alternating series in terms of Riemann zeta function and Dirichlet beta function

Zhiling Fan ¹,
Wenchang Chu ^{2
,
,}

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

Received: 19 September 2023 Revised: 15 January 2024 Accepted: 25 January 2024 Published: 30 January 2024

By making use of the multisection series method, four classes of alternating infinite series are evaluated, in closed form, by the Riemann zeta function and the Dirichlet beta function.
- infinite summation,
- higher derivative,
- integer function,
- Riemann zeta function,
- Dirichlet beta function
Citation: Zhiling Fan, Wenchang Chu. Alternating series in terms of Riemann zeta function and Dirichlet beta function[J]. Electronic Research Archive, 2024, 32(2): 1227-1238. doi: 10.3934/era.2024058

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Abstract

By making use of the multisection series method, four classes of alternating infinite series are evaluated, in closed form, by the Riemann zeta function and the Dirichlet beta function.

References

[1]	T. M. Apostol, Zeta and related functions, in NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, (2010), 601–616.
[2]	G. Bretti, C. Cesarano, P. E. Ricci, Laguerre-type exponentials and generalized Appell polynomials, Comput. Math. Appl., 48 (2004), 833–839. https://doi.org/10.1016/j.camwa.2003.09.031 doi: 10.1016/j.camwa.2003.09.031
[3]	E. Guariglia, Fractional calculus, zeta functions and Shannon entropy, Open Math. 19 (2021), 87–100. https://doi.org/10.1515/math-2021-0010 doi: 10.1515/math-2021-0010
[4]	S. D. Lin, H. M. Srivastava, Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput., 154 (2004), 725–733 https://doi.org/10.1016/S0096-3003(03)00746-X doi: 10.1016/S0096-3003(03)00746-X
[5]	, W. Ramírez, D. Bedoya, A. Urieles, C. cesarano, M. Ortega, New $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomials and their matrices, Carpathian Math. Publ., 15 (2023), 449–467 https://doi.org/10.15330/cmp.15.2.449-467 doi: 10.15330/cmp.15.2.449-467
[6]	R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics, Addison-Wesley Professional, Massachusetts, 1989.
[7]	K. N. Boyadzhiev, Derivative polynomials for tanh, tan, sech and sec in explicit form, Fibonacci Quart., 45 (2007), 291–303.
[8]	W. Chu, C. Wang, Convolution formulae for Bernoulli numbers, Integr. Transf. Spec. Funct., 21 (2010), 437–457. https://doi.org/10.1080/10652460903360861 doi: 10.1080/10652460903360861
[9]	K. Stromberg, An Introduction to Classical Real Analysis, American Mathematical Society, California, 1981.
[10]	J. R. Higgins, Two basic formulae of Euler and their equivalence to Tschakalov's sampling theorem, Sampl. Theory Signal Image Process., 2 (2003), 259–270. https://doi.org/10.1007/BF03549398 doi: 10.1007/BF03549398

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