一类含多重调和数的无穷级数的算法
An Algorithm for a Class of Infinite Series Involving Multiple Harmonic Numbers
DOI: 10.12677/pm.2024.146258, PDF,    国家自然科学基金支持
作者: 陈 昕：浙江理工大学理学院，浙江 杭州
关键词: 多重调和数；多重zeta值；着色多重zeta值；无穷级数；Multiple Harmonic Numbers； Multiple Zeta Values； Colored Multiple Zeta Values； Infinite Series
摘要: 本文利用部分分式展开及和式变换的方法，为一类含多重调和数的无穷级数建立了递推关系，并证明了此类无穷级数可以用四级的着色多重zeta值表示，并给出了相应的算法及一些特例。
Abstract: In this paper, we use the methods of partial fraction decomposition and transformation of summations to establish a recurrence for a class of infinite series involving multiple harmonic numbers, and prove that such infinite series are expressible in terms of colored multiple zeta values of level four. The corresponding algorithm and some special cases are presented.
文章引用：陈昕. 一类含多重调和数的无穷级数的算法[J]. 理论数学, 2024, 14(6): 387-398. https://doi.org/10.12677/pm.2024.146258

参考文献

[1] Hoffman, M.E. (2019) An Odd Variant of Multiple Zeta Values. Communications in Number Theory and Physics, 13, 529-567. [Google Scholar] [CrossRef] 
[2] Hoffman, M. (1992) Multiple Harmonic Series. Pacific Journal of Mathematics, 152, 275-290. [Google Scholar] [CrossRef] 
[3] Zagier, D. (1992) Values of Zeta Functions and Their Applications. In: Joseph, A., et al., Eds., First European Congress of Mathematics, Birkhäuser, Paris, 497-512.
[4] Hoffman, M.E. (1997) The Algebra of Multiple Harmonic Series. Journal of Algebra, 194, 477-495. [Google Scholar] [CrossRef] 
[5] Au, K.C. (2020) Evaluation of One-Dimensional Polylogarithmic Integral, with Applications to Infinite Series.
[6] Feng, L. and Wang, W. (2015) An Algorithm for Computing Mixed Sums of Products of Bernoulli Polynomials and Euler Polynomials. Journal of Symbolic Computation, 66, 84-97. [Google Scholar] [CrossRef] 
[7] Wang, W. and Xu, C. (2021) Alternating Multiple Zeta Values, and Explicit Formulas of Some Euler-Apéry-Type Series. European Journal of Combinatorics, 93, Article ID: 103283. [Google Scholar] [CrossRef] 
[8] Xu, C. and Zhao, J. (2024) Apéry-Type Series and Colored Multiple Zeta Values. Advances in Applied Mathematics, 153, Article ID: 102610. [Google Scholar] [CrossRef] 
[9] Sun, Z.W. (2015) New Series for Some Special Values of L-Functions. Journal of Nanjing University, 32, 189-218.
[10] Kuba, M. and Panholzer, A. (2019) A Note on Harmonic Number Identities, Stirling Series and Multiple Zeta Values. International Journal of Number Theory, 15, 1323-1348. [Google Scholar] [CrossRef] 
[11] Comtet, L. (1974) Advanced Combinatorics. D. Reidel Publishing Co., Dordrecht.

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