Evaluations of some infinite series involving $ r $-Stirling numbers and harmonic numbers

Xudong Chen; Xudong Chen

doi:10.3934/math.2026491

AIMS Mathematics

2026, Volume 11, Issue 4: 11977-11992. doi: 10.3934/math.2026491

Previous Article Next Article

Research article Special Issues

Evaluations of some infinite series involving $ r $-Stirling numbers and harmonic numbers

Xudong Chen ^,

School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received: 05 January 2026 Revised: 21 April 2026 Accepted: 22 April 2026 Published: 28 April 2026
MSC : 05A19, 40A25, 11B73, 11M32

In this paper, we study infinite series involving $ r $-Stirling numbers of the first kind and harmonic numbers. As a result, we establish the integral expressions for three parametric $ r $-Stirling series, and show that these series are ultimately reducible to alternating multiple zeta values (AMZVs). As applications, the explicit AMZV expressions for some parametric series are determined, including some results presented by Wang and Chen, and some results similar to those of Hoffman.
- infinite series identities,
- $ r $-Stirling numbers of the first kind,
- harmonic numbers,
- alternating multiple zeta values
Citation: Xudong Chen. Evaluations of some infinite series involving $ r $-Stirling numbers and harmonic numbers[J]. AIMS Mathematics, 2026, 11(4): 11977-11992. doi: 10.3934/math.2026491

Related Papers:

Abstract

In this paper, we study infinite series involving $ r $-Stirling numbers of the first kind and harmonic numbers. As a result, we establish the integral expressions for three parametric $ r $-Stirling series, and show that these series are ultimately reducible to alternating multiple zeta values (AMZVs). As applications, the explicit AMZV expressions for some parametric series are determined, including some results presented by Wang and Chen, and some results similar to those of Hoffman.

References

[1]	M. E. Hoffman, Multiple harmonic series, Pacific J. Math., 152 (1992), 275–290. https://doi.org/10.2140/PJM.1992.152.275 doi: 10.2140/PJM.1992.152.275
[2]	M. E. Hoffman, The algebra of multiple harmonic series, J. Algebra, 194 (1997), 477–495. https://doi.org/10.1006/jabr.1997.7127 doi: 10.1006/jabr.1997.7127
[3]	D. Zagier, Values of zeta functions and their applications, In: A. Joseph, F. Murat, B. Prum, R. Rentscher, First European Congress of Mathematics Paris, July 6–10, 1992, Progress in Mathematics, Birkhäuser Basel, 120 (1994), 497–512. https://doi.org/10.1007/978-3-0348-9112-7_23
[4]	K. C. Au, Iterated integrals and multiple polylogarithm at algebraic arguments, arXiv: 2201.01676, 2022. https://doi.org/10.48550/arXiv.2201.01676
[5]	K. C. Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv: 2007.03957, 2024. https://doi.org/10.48550/arXiv.2007.03957
[6]	M. E. Hoffman, Harmonic-number summation identities, symmetric functions, and multiple zeta values, Ramanujan J., 42 (2017), 501–526. https://doi.org/10.1007/s11139-015-9750-4 doi: 10.1007/s11139-015-9750-4
[7]	M. Kuba, A. Panholzer, A note on harmonic number identities, Stirling series and multiple zeta values, Int. J. Number Theory, 15 (2019), 1323–1348. https://doi.org/10.1142/S179304211950074X doi: 10.1142/S179304211950074X
[8]	L. Comtet, Advanced combinatorics, D. Reidel Publishing Co., Dordrecht, 1974. https://doi.org/10.1007/978-94-010-2196-8
[9]	P. Sun, Computing the 5-order sums of $\zeta(k)$, Acta Math. Sinica (Chin. Ser.), 46 (2003), 297–302 (in Chinese). https://doi.org/10.12386/A2003sxxb0038 doi: 10.12386/A2003sxxb0038
[10]	W. Wang, Y. Chen, Explicit formulas of sums involving harmonic numbers and Stirling numbers, J. Difference Equ. Appl., 26 (2020), 1369–1397. https://doi.org/10.1080/10236198.2020.1842384 doi: 10.1080/10236198.2020.1842384
[11]	C. Xu, Evaluations of Euler-type sums of weight $\leq 5$, Bull. Malays. Math. Sci. Soc., 43 (2020), 847–877. https://doi.org/10.1007/s40840-018-00715-3 doi: 10.1007/s40840-018-00715-3
[12]	C. Xu, Y. Cai, On harmonic numbers and nonlinear Euler sums, J. Math. Anal. Appl., 466 (2018), 1009–1042. https://doi.org/10.1016/j.jmaa.2018.06.036 doi: 10.1016/j.jmaa.2018.06.036
[13]	M. Kuba, H. Prodinger, A note on Stirling series, Integers, 10 (2010), 393–406. https://doi.org/10.1515/integ.2010.034 doi: 10.1515/integ.2010.034
[14]	A. Z. Broder, The $r$-Stirling numbers, Discrete Math., 49 (1984), 241–259. https://doi.org/10.1016/0012-365X(84)90161-4 doi: 10.1016/0012-365X(84)90161-4
[15]	R. Merris, The $p$-Stirling numbers, Turkish J. Math., 24 (2000), 379–399. https://journals.tubitak.gov.tr/math/vol24/iss4/7
[16]	J. H. Conway, R. K. Guy, The book of numbers, Copernicus, New York, 1996. https://doi.org/10.1007/978-1-4612-4072-3
[17]	A. Dil, K. N. Boyadzhiev, Euler sums of hyperharmonic numbers, J. Number Theory, 147 (2015), 490–498. https://doi.org/10.1016/j.jnt.2014.07.018 doi: 10.1016/j.jnt.2014.07.018
[18]	I. Mező, A. Dil, Hyperharmonic series involving Hurwitz zeta function, J. Number Theory, 130 (2010), 360–369. https://doi.org/10.1016/j.jnt.2009.08.005 doi: 10.1016/j.jnt.2009.08.005
[19]	L. Kargın, M. Can, A. Dil, M. Cenkci, On evaluations of Euler-type sums of hyperharmonic numbers, Bull. Malays. Math. Sci. Soc., 45 (2022), 113–131. https://doi.org/10.1007/s40840-021-01179-8 doi: 10.1007/s40840-021-01179-8
[20]	R. Li, Euler sums of generalized hyperharmonic numbers, Funct. Approx. Comment. Math., 66 (2022), 179–189. https://doi.org/10.1007/s11139-023-00761-x doi: 10.1007/s11139-023-00761-x
[21]	C. Xu, X. Zhang, Y. Li, Euler sums of multiple hyperharmonic numbers, Lith. Math. J., 62 (2022), 412–419. https://doi.org/10.1007/s10986-022-09552-1 doi: 10.1007/s10986-022-09552-1
[22]	I. Mező, New properties of $r$-Stirling series, Acta Math. Hungar., 119 (2008), 341–358. https://doi.org/10.1007/s10474-007-7047-9 doi: 10.1007/s10474-007-7047-9
[23]	X. Chen, W. Wang, H. Zhang, Infinite series involving $r$-Stirling numbers and hyperharmonic numbers, 2025, in press.
[24]	Q. Ma, W. Wang, Infinite series identities on $r$-Stirling numbers, Rocky Mountain J. Math., 2024, in press.
[25]	Q. Ma, W. Wang, Riordan arrays and $r$-Stirling number identities, Discrete Math., 346 (2023), 113211. https://doi.org/10.1016/j.disc.2022.113211 doi: 10.1016/j.disc.2022.113211
[26]	I. Mező, Nonlinear Euler sums, Pacific J. Math., 272 (2014), 201–226. https://doi.org/10.2140/pjm.2014.272.201 doi: 10.2140/pjm.2014.272.201
[27]	C. Xu, Multiple zeta values and Euler sums, J. Number Theory, 177 (2017), 443–478. https://doi.org/10.1016/j.jnt.2017.01.018 doi: 10.1016/j.jnt.2017.01.018
[28]	W. Wang, C. Xu, Alternating multiple zeta values, and explicit formulas of some Euler-Apéry-type series, European J. Combin., 93 (2021), 103283. https://doi.org/10.1016/j.ejc.2020.103283 doi: 10.1016/j.ejc.2020.103283

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)