Xiaoyu Liu
Department of Mathematics and Three Gorges Mathematics Research Center, China Three Gorges University, Yichang, Hubei, China.
*Corresponding author: Xiaoyu Liu
Abstract
This note primarily focuses on the expansion of the powers of arctan(x) series, exploring variable transformations, differentiation, integration, and subsequent evaluations for such series. Firstly, by substituting specific values for the variable x in the series with powers of arctan(x) and its derivative, some Euler sum identities involving harmonic numbers can be derived. Secondly, an alternative series representation for the powers of arctan(x) was identified, revealing a connection between these two forms of representation. Additionally, ongoing calculations were performed for the integration of powers of arctan(x), and Taylor series expansions of such integrals were carried out. This process led to the derivation of additional Euler sum identities containing harmonic numbers. Moreover, we observed that these derived Euler sum identities are closely linked to certain special constants, such as being expressible in terms of π, Catalan constant, and others. Some theorems and corollarys are provided, along with examples where specific values are assigned to the variables.
References
[1] Euler L. (2015). Correspondence of Leonhard Euler with Christian Goldbach. Birkhäuser.
[2] Sofo A, Nimbran A S. (2019). Euler sums and integral connections. Mathematics, 7(9): 833.
[3] SMezo I, CENKCİ M. (2017). Evaluation of Euler-like sums via Hurwitz zeta values. Turkish Journal of Mathematics, 41(6): 1640-1655.
[4] Kunzhen Zhang, Xinhua Xiong. (2023). Harmonic Number Identities from Log-integral Transformation. Journal of Applied Mathematics and Computation, 7(1), 83-89.
[5] Jung M H, Cho Y J, Choi J S. (2004). Euler sums evaluatable from integrals. Communications of the Korean Mathematical Society, 19(3): 545-555.
[6] Xu C, Yan Y, Shi Z. (2016). Euler sums and integrals of polylogarithm functions. Journal of Number Theory, 165: 84-108.
[7] Sofo A. (2022). Evaluating log-tangent integrals via Euler sums. Mathematical Modelling and Analysis, 27(1): 1-18.
[8] Sofo A, Batir N. (2022). Moments of log-tanh integrals. Integral Transforms and Special Functions, 33(6): 434-448.
[9] Freitas P. (2005). Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums. Mathematics of Computation, 74(251): 1425-1440.
[10] Bradley D M. (1999). A class of series acceleration formulae for Catalan's constant. The Ramanujan Journal, 3: 159-173.
[11] Milgram M. (2006). A new series expansion for integral powers of arctangent. Integral Transforms and Special Functions, 17(7): 531-538.
How to cite this paper
Generating a Series of Euler Sums with Harmonic Numbers via (arctan x)^p Expansions
How to cite this paper: Xiaoyu Liu. (2023) Generating a Series of Euler Sums with Harmonic Numbers via (arctan x)^p Expansions. Journal of Applied Mathematics and Computation, 7(4), 473-477.
DOI: http://dx.doi.org/10.26855/jamc.2023.12.006