# Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers

- Junesang Choi
^{1}Email author

**2013**:49

https://doi.org/10.1186/1029-242X-2013-49

© Choi; licensee Springer 2013

**Received: **16 November 2012

**Accepted: **25 January 2013

**Published: **14 February 2013

## Abstract

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we show how one can obtain further interesting identities about certain finite series involving binomial coefficients, harmonic numbers and generalized harmonic numbers by applying the usual differential operator to a known identity.

**MSC:**11M06, 33B15, 33E20, 11M35, 11M41, 40C15.

## Keywords

## 1 Introduction and preliminaries

*s*are defined by (

*cf*. [1]; see also [2, 3], [[4], p.156] and [[5], Section 3.5])

*ψ*is defined by

*cf*. [[4], Eq. 1.2(54)]):

*cf*. [[9], Eq. (20)]):

*Sigma*package in [12], together with the following identity [[10], Identity 20]:

*et al.*[[14], Eq. (3.62)] proved, by using the principle of mathematical induction, that

Dattolli and Srivastava [16] proposed several generating functions involving harmonic numbers by making use of an interesting approach based on the umbral calculus. Subsequently, Cvijović [17] showed the truth of the conjectured relations in [16] by using simple analytical arguments.

For a concise and beautiful description of these numbers, we refer also to WolframMathWorld’s website [18].

As we have seen in the above brief eclectic review, harmonic and generalized harmonic numbers are involved in a variety of useful identities. Of course, certain interesting properties of harmonic and generalized harmonic numbers have been studied (see, *e.g.*, [19]). Here we aim at presenting further interesting identities about certain interesting finite series associated with binomial coefficients, harmonic numbers and generalized harmonic numbers.

## 2 Finite-series involving binomial coefficients, harmonic numbers and generalized harmonic numbers

*cf*. [[20], p.362, Entry (55.4.8)]; see also [[2], Eq. (2.6)]):

*a*and

*b*, respectively, using (1.8) and considering the following easily derivable identities:

we obtain the following formulas in Theorem 1.

Setting $a=b-1=0$ in (2.1), (2.5) and (2.6) and using (1.3) and (1.8), we get certain interesting finite-sum identities involving binomial coefficients and harmonic numbers, respectively, asserted by Corollary 1.

*and*

**Remark 1** In the course of presenting a closed-form evaluation of some useful series involving the generalized zeta function $\zeta (s,a)$, Choi *et al.* [21] made use of the identity (2.7) without its proof. Choi and Srivastava [2] proved Eq. (2.7) as a special case of (2.1) here and presented another illustrative proof.

Here we give the answers for $k=2$ and $k=3$ in (2.10) asserted by the following lemma.

**Lemma 1**

*Each of the following identities holds true*:

*and*

*Proof*We will prove only (2.11) by using the same method as in [[2], pp.2224-2225]. A similar argument will establish (2.12). We first recall two basic relations for binomial coefficients:

Applying (1.17) to (2.21), we get the desired identity (2.11). □

Applying (2.7) and (2.11) to (2.9) and considering (2.8), we obtain two interesting identities asserted by the following theorem.

**Theorem 2**

*Each of the following identities holds true*:

*and*

*a*and observing the following identity:

we obtain further interesting identities involving binomial coefficients and generalized harmonic functions asserted by the following theorem.

Setting $a=b-1=0$ in (2.25) and (2.26), we find certain interesting identities and using (2.8), respectively, assert the following corollary.

**Remark 2** As in getting the results in Theorem 3, it is seen that a variety of interesting identities involving the generalized harmonic numbers can be obtained by applying the differential operator to the parameters of known formulas.

## 3 Inverse relations and a question

Applying this inverse relation to the identities in Section 2, we obtain many formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers asserted by the following corollary.

By using the first one of (2.13), we find an identity in the following lemma.

**Lemma 2**

*If Eq*. (3.9)

*holds true*,

*then we obtain the following identity*:

Applying Eq. (3.10) to Eqs. (2.7), (2.8) and (2.27), we get some interesting identities asserted by the following corollary.

**Question** We conclude this paper by posing a natural question: *Under what conditions does* Eq. (3.9) *hold true*?

## Declarations

### Acknowledgements

Dedicated to Professor Hari M. Srivastava.

This work was supported by the *Basic Science Research Program through the National Research Foundation of the Republic of Korea* funded by the Ministry of Education, Science and Technology (2012-0002957).

## Authors’ Affiliations

## References

- Adamchik VS, Srivastava HM: Some series of the zeta and related functions.
*Analysis*1998, 18: 131–144.MathSciNetView ArticleGoogle Scholar - Choi J, Srivastava HM: Some summation formulas involving harmonic numbers and generalized harmonic numbers.
*Math. Computer Modelling*2011, 54: 2220–2234. 10.1016/j.mcm.2011.05.032MathSciNetView ArticleGoogle Scholar - Graham RL, Knuth DE, Patashnik O:
*Concrete Mathematics*. Addison-Wesley, Reading; 1989.Google Scholar - Srivastava HM, Choi J:
*Series Associated with the Zeta and Related Functions*. Kluwer Academic, Dordrecht; 2001.View ArticleGoogle Scholar - Srivastava HM, Choi J:
*Zeta and q-Zeta Functions and Associated Series and Integrals*. Elsevier, Amsterdam; 2012.Google Scholar - Choi J: Certain summation formulas involving harmonic numbers and generalized harmonic numbers.
*Appl. Math. Comput.*2011, 218: 734–740. doi:10.1016/j.amc.2011.01.062 10.1016/j.amc.2011.01.062MathSciNetView ArticleGoogle Scholar - Rassias TM, Srivastava HM: Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers.
*Appl. Math. Comput.*2002, 131: 593–605. 10.1016/S0096-3003(01)00172-2MathSciNetView ArticleGoogle Scholar - Sofo A, Srivastava HM: Identities for the harmonic numbers and binomial coefficients.
*Ramanujan J.*2011, 25: 93–113. 10.1007/s11139-010-9228-3MathSciNetView ArticleGoogle Scholar - Coffey MW: On some series representations of the Hurwitz zeta function.
*J. Comput. Appl. Math.*2008, 216: 297–305. 10.1016/j.cam.2007.05.009MathSciNetView ArticleGoogle Scholar - Spivey MZ: Combinatorial sums and finite differences.
*Discrete Math.*2007, 307: 3130–3146. 10.1016/j.disc.2007.03.052MathSciNetView ArticleGoogle Scholar - Paule P, Schneider C: Computer proofs of a new family of harmonic number identities.
*Adv. Appl. Math.*2003, 31: 359–378. 10.1016/S0196-8858(03)00016-2MathSciNetView ArticleGoogle Scholar - Schneider, C: Solving parameterized linear difference equations in ∏∑-fields. Technical Report 02–03, RISC-Linz, J. Kepler University, July 2002. Available at: http://www.risc.uni-linz.ac.at/research/combinat/risc/publicationsGoogle Scholar
- Greene DH, Knuth DE:
*Mathematics for the Analysis of Algorithms*. 3rd edition. Birkhäuser, Basel; 1990.View ArticleGoogle Scholar - Alzer H, Karayannakis D, Srivastava HM: Series representations for some mathematical constants.
*J. Math. Anal. Appl.*2006, 320: 145–162. 10.1016/j.jmaa.2005.06.059MathSciNetView ArticleGoogle Scholar - Chu W, De Donno L: Hypergeometric series and harmonic number identities.
*Adv. Appl. Math.*2005, 34: 123–137. 10.1016/j.aam.2004.05.003MathSciNetView ArticleGoogle Scholar - Dattoli G, Srivastava HM: A note on harmonic numbers, umbral calculus and generating functions.
*Appl. Math. Lett.*2008, 21: 686–693. 10.1016/j.aml.2007.07.021MathSciNetView ArticleGoogle Scholar - Cvijović D: The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers.
*Appl. Math. Comput.*2010, 215: 4040–4043. 10.1016/j.amc.2009.12.011MathSciNetView ArticleGoogle Scholar - WolframMathWorld. http://mathworld.wolfram.com/HarmonicNumber.htmlGoogle Scholar
- Conway JH, Guy RK:
*The Book of Numbers*. Springer, New York; 1996.View ArticleGoogle Scholar - Hansen ER:
*A Table of Series and Products*. Prentice-Hall, New Jersey; 1975.Google Scholar - Choi J, Cho YJ, Srivastava HM: Series involving the zeta function and multiple gamma functions.
*Appl. Math. Comput.*2004, 159: 509–537. 10.1016/j.amc.2003.08.134MathSciNetView ArticleGoogle Scholar - Riordan J:
*Combinatorial Identities*. Wiley, New York; 1968.Google Scholar

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