含有调和数平方的无穷级数恒等式
Infinite Series Identities Involving Quadratic Harmonic Numbers
摘要: 本文研究两类含有以下广义调和数平方的组合恒等式。首先通过组合分析中的Abel分部求和引理获得含有两个差分对{Ak,Bk}和{Ak′,Bk′}的无穷级数求和公式，即定理2。然后选取恰当的序列{Ak,Ak′}和{Bk,Bk′}，利用定理2，证明含有广义调和数hk2(a,b)和hk2(a,b)的无穷级数恒等式。最后对参数a和b取特殊值，进一步获得一些新的π，Catalan常数和ln2的无穷级数求和公式。
Abstract: In this paper, we study two combinatorial identities including the following quadratic generalized harmonic numbers First, the modified Abel lemma on summation by parts in combination analysis is employed to obtain summation formulae of infinite series involving two difference pairs {Ak,Bk} and {Ak′,Bk′}, that is Theorem 2. Then applying Theorem 2 through appropriate sequences {Ak,Ak′} and {Bk,Bk′}, we establish infinite series identities involving generalized harmonic numbers hk2(a,b) and hk2(a,b). Finally, by selecting special values for parameters a and b, several new infinite series are obtained for , Catalan constant and ln2 as consequences.
文章引用：王晓元, 刘筱蒙. 含有调和数平方的无穷级数恒等式[J]. 理论数学, 2023, 13(3): 516-525. https://doi.org/10.12677/PM.2023.133055

参考文献

[1] Paule, P. and Schneider, C. (2003) Computer Proofs of a New Family of Harmonic Number Identities. Advances in Applied Mathematics, 31, 359-378. [Google Scholar] [CrossRef] 
[2] Chu, W.C. and Fu, A.M. (2009) Dougall-Dixon Formula and Harmonic Number Identities. The Ramanujan Journal, 18, 11-31. [Google Scholar] [CrossRef] 
[3] Wang, W.P. and Jia, C.Z. (2014) Harmonic Number Identities via the Newton-Andrews Method. The Ramanujan Journal, 35, 263-285. [Google Scholar] [CrossRef] 
[4] Wei, C.A. and Gong, D.X. (2014) The Derivative Operator and Harmonic Number Identities. The Ramanujan Journal, 34, 361-371. [Google Scholar] [CrossRef] 
[5] Wei, C.A., Yan, Q.L. and Gong, D.X. (2015) A Family of Summation Formulae Involving Harmonic Numbers. Integral Transforms and Special Functions, 26, 667-677. [Google Scholar] [CrossRef] 
[6] Liu, H.M. and Wang, W.P. (2019) Gauss’s Theorem and Harmonic Number Summation Formulae with Certain Mathematical Constants. Journal of Difference Equations and Applications, 25, 313-330. [Google Scholar] [CrossRef] 
[7] 刘红梅. 含有中心二项式系数以及广义调和数的无穷级数恒等式[J]. 数学物理学报, 2020, 40(3): 631-640.
[8] Kargin, L., Cenkci, M., Dil, A. and Can, M. (2022) Gener-alized Harmonic Number via Poly-Bernoulli Polynomials. Publicationes Mathematicae-Debrecen, 100, 365-386.
[9] Wang, R. and Wuyungaowa (2022) Generalized Harmonic Numbers with Combinatorial Sequences. Journal of Applied Mathematics and Physics, 10, 1602-1618. [Google Scholar] [CrossRef] 
[10] Chu, W.C. (2006) Bailey’s Very Well-Poised -Series Identity. Journal of Combinatorial Theory, Series A, 113, 966-979. [Google Scholar] [CrossRef] 
[11] Chu, W.C. (2012) Infinite Series Identities on Harmonic Numbers. Results in Mathematics, 61, 209-221. [Google Scholar] [CrossRef] 
[12] Wang, X.Y. (2018) Infinite Series Containing Generalized Har-monic Numbers. Results in Mathematics, 73, Article No. 24. [Google Scholar] [CrossRef] 
[13] Wang, X.Y. and Chu, W.C. (2018) Infinite Series Identities In-volving Quadratic and Cubic Harmonic Numbers. Publicacions Matemàtiques, 62, 285-300. [Google Scholar] [CrossRef] 
[14] Wang, J. and Wei, C.A. (2018) Four Families of Summation Formulas Involving Generalized Harmonic Numbers. The Ramanujan Journal, 45, 73-94. [Google Scholar] [CrossRef] 
[15] 王晓元, 贾利琴. 含有调和数的无穷级数恒等式[J]. 大连交通大学学报, 2018, 39(5): 118-120.
[16] Chen, K.-W. and Chen, Y.-H. (2020) Infinite Series Containing Generalized Harmonic Functions. Notes on Number Theory and Discrete Mathematics, 26, 85-104. [Google Scholar] [CrossRef] 
[17] Wade, W.R. (2014) Introduction to Analysis. 4th Edition, Pearson Education Limited, London.

为你推荐