- Research
- Open access
- Published:
Some extensions for the several combinatorial identities
Advances in Difference Equations volume 2021, Article number: 38 (2021)
Abstract
In this paper, we give some extensions for Mortenson’s identities in series with the Bell polynomial using the partial fraction decomposition. As applications, we obtain some combinatorial identities involving the harmonic numbers.
1 Introduction
The higher-order harmonic numbers are defined by
When \(n = 1\), they reduce to the classical harmonic numbers \(H_{n}=H_{n}^{(1)}\).
We also define the generalized higher-order harmonic numbers \(H_{n}^{(r)}(z)\) as
When \(z = 0\), they reduce to the higher-order harmonic numbers \(H_{n}^{(r)}(0)= H_{n}^{(r)}\).
The standard Bell polynomials are presented in Comtet’s book [5]. The modified Bell polynomials \(L_{n}(x_{1},x_{2},\ldots )\) are defined by
This expansion gives
Mortenson [9, p. 990, Lemma 3.1] gave the following identities:
which are called Mortenson’s identities, where \((z)_{n}\) are the Pochhammer symbols defined by \((z)_{0} = 1\), \((z)_{n}=z(z+1)\cdots (z+n-1)\).
H.M. Srivastava, J. Choi, G. Dattoli, and A. Sofo et al. investigated some infinite combinatorial series identities involving the harmonic numbers and generalized harmonic numbers by applying the hypergeometric series, Vandermonde convolutions, and Riemann zeta and polygamma functions; for details, see [1, 6–8, 10–13]. W. Chu studied some finite combinatorial identities involving the harmonic numbers by applying the partial fraction decomposition [2–4].
In this paper, we give some extensions of Mortenson’s identities using the partial fraction decomposition. We obtain some new or old combinatorial identities involving the harmonic numbers and generalized harmonic numbers and propose two open problems.
2 Extensions of Mortenson’s identities
First, we give an extension of Mortenson’s identity (4).
Theorem 1
For \(n \in \mathbb{N}\), \(r \in \mathbb{N}_{0}\), and \(\ {x} > 0\), we have
where \(U_{k}=(-1)^{k-1 }H^{(k)}_{n}(-{x})+H^{(k)}_{n+1}({x}-1)\).
Proof
By means of the standard partial fraction decomposition we easily obtain
Multiplying both sides of (7) by z and then letting \(z \rightarrow \infty \), we obtain
This completes the proof. □
We next give an extension of Mortenson’s identity (5) by Theorem 1.
Theorem 2
For \(n, r, M \in \mathbb{N}\) and \(x \geq -j\), \(j=1,2,\ldots , M\), we have
where \({U_{k}}=(-1)^{k-1}H_{n}^{(k)}(-x-j-1)+H_{n+1}^{(k)}({x}+j)\).
Proof
Letting \(x \longmapsto x+1\) in (6) and then letting \(x \longmapsto x+j+1\), we obtain
The proof is complete. □
Taking \({x} = -1\) in (8), we easily obtain the following corollary.
Corollary 3
Let \(n, r, M \in \mathbb{N}\).
When \(1 \leq M \leq n\), we have
When \(M>n\), we have
where \({V_{k}}=(-1)^{k-1}H_{n}^{(k)}(-j)+H_{n+1}^{(k)}(j-1)\).
3 Some applications and two open problems
In this section, we deduce several combinatorial identities involving the harmonic numbers from Theorems 1 and 2. We also suggest two open problems on Mortenson’s identities.
Setting \(r=0,1\) in (6), we obtain the familiar formulas
respectively. Setting \(r=1\) in Corollary 3, we obtain the following combinatorial identities involving the harmonic numbers:
Finally, we propose the following two open problems.
Open Problem 1
For \(m>n\), how do we calculate the combinatorial sums
Open Problem 2
For \(n \in \mathbb{N}\), \(m,r,M \in \mathbb{N}_{0}\), \({x} > 0\), what are the combinatorial sums
Availability of data and materials
Not applicable.
References
Choi, J., Srivastava, H.M.: Some summation formulas involving harmonic numbers and generalized harmonic numbers. Math. Comput. Model. 54, 2220–2234 (2011)
Chu, W.: Partial fraction decompositions and harmonic number identities. J. Comb. Math. Comb. Comput. 60, 139–153 (2007)
Chu, W.: Summation formulae involving harmonic numbers. Filomat 26, 143–152 (2012)
Chu, W., Fu, A.M.: Dougall–Dixon formula and harmonic number identities. Ramanujan J. 18, 11–31 (2009)
Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht and Boston (1974)
Dattoli, G., Licciardi, S., Sabia, E., Srivastava, H.M.: Some properties and generating functions of generalized harmonic numbers. Mathematics 7, Article ID 577 (2019)
Dattoli, G., Srivastava, H.M.: A note on harmonic numbers, umbral calculus and generating functions. Appl. Math. Lett. 21, 686–693 (2008)
Gould, H.W., Srivastava, H.M.: Some combinatorial identities associated with the Vandermonde convolution. Appl. Math. Comput. 84, 97–102 (1997)
Mortenson, E.: Supercongruences between truncated \(_{2}F_{1}\) hypergeometric functions and their Gaussian analogs. Trans. Am. Math. Soc. 355, 987–1007 (2003)
Rassias, Th.M., Srivastava, H.M.: Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers. Appl. Math. Comput. 131, 593–605 (2002)
Sofo, A., Srivastava, H.M.: Identities for the harmonic numbers and binomial coefficients. Ramanujan J. 25, 93–113 (2011)
Sofo, A., Srivastava, H.M.: A family of shifted harmonic sums. Ramanujan J. 37, 89–108 (2015)
Srivastava, H.M., Raina, R.K.: Some combinatorial series identities. Math. Proc. Camb. Philos. Soc. 96, 9–13 (1984)
Acknowledgements
The authors greatly appreciate the reviewer’s comments and suggestions, which improved our paper.
Funding
The present investigation was supported by Natural Science Foundation General Project of Chongqing, China under Grant cstc2019jcyj-msxmX0143.
Author information
Authors and Affiliations
Contributions
There was an equal amount of contributions from two authors. The authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Dedicated to Honor Professor Hari Mohan Srivastava on his 80th Birth Anniversary
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Xi, GW., Luo, QM. Some extensions for the several combinatorial identities. Adv Differ Equ 2021, 38 (2021). https://doi.org/10.1186/s13662-020-03171-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-03171-1