Abstract and Applied Analysis

Volume 2014 (2014), Article ID 501906, 10 pages

http://dx.doi.org/10.1155/2014/501906

Research Article

Summation Formulas Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers

Junesang Choi

Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea

Received 19 May 2014; Accepted 30 June 2014; Published 21 July 2014

Academic Editor: S. D. Purohit

Copyright © 2014 Junesang Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for ₂F ₁(1).

1. Introduction and Preliminaries

The generalized harmonic numbers of order which are defined by (cf. [1]; see also [2, 3], [4, page 156], and [5, Section 3.5])

are the harmonic numbers. Here and denote the set of positive integers and the set of complex numbers, respectively, and we assume that

The generalized harmonic functions are defined by (see [2, 6]; see also [7, 8])

so that, obviously,

Equation (1) can be written in the following form:

by recalling the well-known (easily derivable) relationship between the Riemann zeta function and the Hurwitz (or generalized) zeta function (see [4, equation ]):

The polygamma functions are defined by

where is the familiar gamma function and the psi-function is defined by

A well-known (and potentially useful) relationship between the polygamma functions and the generalized zeta function is given by

It is also easy to have the following expression (cf. [4, equation 1.2(54)]):

which immediately gives another expression for as follows (cf. [9, equation (20)]):

The following identity was discovered by Euler in 1775 and has a long history (see, e.g., [10, page 252 et seq.]):

Identity (13) is a special case of the following more general sum due to Euler:

or, equivalently,

where (and in what follows) an empty sum is understood to be nil.

Many different techniques have been used, in the vast mathematical literature, in order to evaluate harmonic sums of the types (13) and (15). For example, D. Borwein and J. M. Borwein [11] established the following interesting sums by applying Parseval's identity to a Fourier series and contour integrals to a generating function:

where, in light of Euler sum (18), nonlinear harmonic sums (16) and (17) are substantially the same, since it is easily verified that

Euler started this line of investigation in the course of his correspondence with Goldbach beginning in 1742 and he was the first to consider the linear harmonic sums:

Euler, whose investigations were completed by Nielsen in 1906 (see Nielsen [12]), showed that the linear harmonic sums in (20) can be evaluated in terms of zeta values in the following special cases: , , odd, and even, but with the pair being restricted to a finite set of the so-called exceptional configurations (see Flajolet and Salvy [13]). Of these special cases, in the ones with , if is known, then can be found by means of the symmetry relation:

and vice versa (see also [1, page 140, Proposition 6]. Some typical examples are

Rather extensive numerical search for linear relations between linear Euler sums and polynomials in zeta values (see Bailey et al. [14]; see also Flajolet and Salvy [13]) strongly suggests that Euler found all the possible evaluations of linear harmonic sums.

The nonlinear harmonic sums involve products of at least two harmonic numbers. Let be a partition of an integer into summands, so that and . The Euler sum of index , is defined by

where the quantity is called the weight and the quantity is the degree. A few basic nonlinear sums were recently evaluated by de Doelder [15] by invoking their relations with the Eulerian beta integrals or with polylogarithm functions. A detailed numerical search was conducted by Bailey et al. [14] who showed the existence of many surprising evaluations like

Flajolet and Salvy [13] clearly and extensively analyzed most of the hitherto known evaluations for Euler sums and multiple zeta functions (see also Hoffman [16] and Zagier [17]). For a remarkably clear and insightful exposition of several important results and conjectures concerning multiple polylogarithms and the multiple zeta functions (including especially a broad survey of recent works on multiple zeta series and Euler sums of arbitrary degree), the interested reader should refer also to a survey-cum-expository paper by Bowman and Bradley [18], which contains a fairly comprehensive bibliography of as many as 83 further references on the subject.

Shen [19] investigated the connections between the Stirling numbers of the first kind and the Riemann zeta function by means of the Gauss summation formula (28) for the hypergeometric series. In the course of his analysis, Shen [19] proved some known identities like (13), (18), and (22). In fact, by employing the univariate series expansion of classical hypergeometric formulas, Shen [19] and Choi and Srivastava [20, 21] investigated the evaluation of infinite series related to generalized harmonic numbers. On the other hand, more summation formulas have been systematically derived by Chu [22] and Chu and de Donno [23] who developed fully this approach to the multivariate case. We chose to recall two identities:

Many formulas of finite series involving binomial coefficients, the Stirling numbers of the first and second kinds, harmonic numbers, and generalized harmonic numbers have also been investigated in diverse ways (see, e.g., [2, 23–32]).

Here we show how one can obtain further interesting and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for . For example, see the identities in Corollary 5:

Relevant connections between some of the identities presented here with those in earlier works are also pointed out.

2. Infinite Series Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers

We begin by recalling well-known Gauss’s summation formula for :

where denotes the Pochhammer symbol defined (for ) by

For convenient reference, without proof, we collect a set of easily derivable formulas necessary to provide further interesting identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers asserted as in the following lemma.

Lemma 1. Each of the following identities holds true:

Now we are ready to present certain general identities of infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers as in the following theorem.

Theorem 2. Each of the following summation formulas holds true:

Proof. Differentiating each side of (28) with respect to the variable and using some suitable identities in Section 1 and Lemma 1, we obtain (31). Differentiating each side of (31) with respect to the variable , we get (32). Similarly we prove (33) and (34). Differentiating each side of (31) with respect to the variable and using some suitable identities in Section 1 and Lemma 1, we obtain (35). Similarly we prove (36), (37), and (38).

Setting and in (31) to (34) and using some suitable identities in Section 1, we obtain some interesting identities involving harmonic numbers and generalized harmonic numbers given in the following corollary.

Corollary 3. Each of the following identities holds true:

Differentiating only the left-hand side of (34) with respect to the variable twice and setting and in each expression, in view of the identities (39), we may guess two identities asserted by the following conjecture.

Conjecture 4. Each of the following identities may hold true:

Setting and in (35) to (44) and using some suitable identities in Section 1 and Lemma 1, we obtain a set of very interesting identities in the following corollary.

Corollary 5. Each of the following identities holds true:

Differentiating only the left-hand side of (38) with respect to the variable twice and setting and in each expression, in view of the identities (41) to (44), we may guess two identities asserted by the following conjecture.

Conjecture 6. Each of the following identities may hold true:

Remark 7. It is interesting to observe that the number of terms in the numerator of each of the left-hand sides of (41) to (46) is equal to the number of partitions of   , respectively. For example, for (46), all the partitions of are as follows:

where denotes    and + is translated into a multiplication of its corresponding . The coefficient of each of the right-hand sides of (41) to (46) has the following rule:

The remaining thing is to find a rule that dominates the coefficients of each term of the numerator of the left-hand sides of (41) to (46).

Differentiating each side of (28) with respect to the variable successively and using some suitable identities in Section 1 and Lemma 1, we obtain a set of infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers different from those in Theorem 2 as in the following theorem.

Theorem 8. Each of the following summation formulas holds true:

Setting and in (49) and using some suitable identities in Section 1 and special values of -function (see, e.g., [4, Section 1.2] and [5, Section 1.3]), we obtain a set of interesting infinite series involving binomial coefficients and harmonic numbers given in the following corollary.

Corollary 9. Each of the following identities holds true:

3. Finite Series Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers

Setting in some chosen formulas in Theorems 2 and 8 and using some suitable identities in Section 1 and the following known and easily derivable formula:

we obtain a set of finite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers given in the following theorem.

Theorem 10. Each of the following finite summation formulas holds true:

where the empty sum is (as usual) understood to be nil throughout this paper,

Setting and in (52) to (53) and using some suitable identities in Section 1, we obtain a set of interesting identities involving binomial coefficients, harmonic numbers, and generalized harmonic numbers given in the following corollary.

Corollary 11. Each of the following identities holds true:

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by Dongguk University Research Fund of 2013. Also, it was supported, in part, by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2010-0011005).

References

V. S. Adamchik and H. M. Srivastava, “Some series of the zeta and related functions,” Analysis, vol. 18, no. 2, pp. 131–144, 1998.

J. Choi and H. M. Srivastava, “Some summation formulas involving harmonic numbers and generalized harmonic numbers,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2220–2234, 2011.

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1989.

H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic, London, UK, 2001.

H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science, Amsterdam, The Netherlands, 2012.

J. Choi, “Certain summation formulas involving harmonic numbers and generalized harmonic numbers,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 734–740, 2011.

T. M. Rassias and H. M. Srivastava, “Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 593–605, 2002.

A. Sofo and H. M. Srivastava, “Identities for the harmonic numbers and binomial coefficients,” Ramanujan Journal, vol. 25, no. 1, pp. 93–113, 2011.

M. W. Coffey, “On some series representations of the Hurwitz zeta function,” Journal of Computational and Applied Mathematics, vol. 216, no. 1, pp. 297–305, 2008.

B. C. Berndt, Ramanujan's Notebooks, Part I, Springer, New York, NY, USA, 1985.

D. Borwein and J. M. Borwein, “On an intriguing integral and some series related to ,” Proceedings of the American Mathematical Society, vol. 123, no. 4, pp. 1191–1198, 1995.

N. Nielsen, Die Gammafunktion, Chelsea, New York, NY, USA, 1965.

P. Flajolet and B. Salvy, “Euler sums and contour integral representations,” Experimental Mathematics, vol. 7, no. 1, pp. 15–35, 1998.

D. H. Bailey, J. M. Borwein, and R. Girgensohn, “Experimental evaluation of Euler sums,” Experimental Mathematics, vol. 3, no. 1, pp. 17–30, 1994.

P. J. de Doelder, “On some series containing and for certain values of and ,” Journal of Computational and Applied Mathematics, vol. 37, no. 1–3, pp. 125–141, 1991.

M. E. Hoffman, “Multiple harmonic series,” Pacific Journal of Mathematics, vol. 152, no. 2, pp. 275–290, 1992.

D. Zagier, “Values of Zeta functions and their applications,” in Proceedings of the 1st European Congress of Mathematics, Volume II (Paris, France, 1992), A. Joseph, F. Mignot, F. Murat, B. Prum, and R. Rentschler, Eds., Progress in Mathematics 120, pp. 497–512, Birkhäauser, Basel, Switzerland, 1994.

D. Bowman and D. M. Bradley, Multiple Polylogarithms : A brief survey in q-Series with Applications to Combinatorics , Number Theory, and Physics (Papers from the Conference held at the University of Illinois, Urbana, Ill, USA, October 2000) (B. C. Berndt and K. Ono, Editors), pp. 71–92, Contemporary Mathematics 291 , American Mathematical Society, Providence, RI, USA, 2001

L. Shen, “Remarks on some integrals and series involving the Stirling numbers and ,” Transactions of the American Mathematical Society, vol. 347, no. 4, pp. 1391–1399, 1995.

J. Choi and H. M. Srivastava, “Certain classes of infinite series,” Monatshefte für Mathematik, vol. 127, no. 1, pp. 15–25, 1999.

J. Choi and H. M. Srivastava, “Explicit evaluation of Euler and related sums,” Ramanujan Journal, vol. 10, no. 1, pp. 51–70, 2005.

W.-C. Chu, “Hypergeometric series and the Riemann zeta function,” Acta Arithmetica, vol. 82, no. 2, pp. 103–118, 1997.

W. Chu and L. de Donno, “Hypergeometric series and harmonic number identities,” Advances in Applied Mathematics, vol. 34, no. 1, pp. 123–137, 2005.

H. Alzer, D. Karayannakis, and H. M. Srivastava, “Series representations for some mathematical constants,” Journal of Mathematical Analysis and Applications, vol. 320, no. 1, pp. 145–162, 2006.

J. H. Conway and R. K. Guy, The Book of Numbers, Springer, Berlin, Germany, 1996.

D. Cvijović, “The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 4040–4043, 2010.

G. Dattoli and H. M. Srivastava, “A note on harmonic numbers, umbral calculus and generating functions,” Applied Mathematics Letters, vol. 21, no. 7, pp. 686–693, 2008.

D. H. Greene and D. E. Knuth, Birkhauser, Berlin, Germany, 3rd edition, 1990.

P. Paule and C. Schneider, “Computer proofs of a new family of harmonic number identities,” Advances in Applied Mathematics, vol. 31, no. 2, pp. 359–378, 2003.

C. Schneider, “Solving parameterized linear difference equations in -fields,” Technical Report 02-03, RISC-Linz, Johannes Kepler University, Linz, Austria, 2002, http://www.risc.uni-linz.ac.at/research/combinat/risc/publications.

M. Z. Spivey, “Combinatorial sums and finite differences,” Discrete Mathematics, vol. 307, no. 24, pp. 3130–3146, 2007.

WolframMathWorld, http://mathworld.wolfram.com/HarmonicNumber.html