Research Article | Open Access

Anthony Sofo, "Harmonic Numbers and Cubed Binomial Coefficients", *International Journal of Combinatorics*, vol. 2011, Article ID 208260, 14 pages, 2011. https://doi.org/10.1155/2011/208260

# Harmonic Numbers and Cubed Binomial Coefficients

**Academic Editor:**Toufik Mansour

#### Abstract

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

#### 1. Introduction

The well-known Riemann zeta function is defined as The generalized harmonic numbers of order are given by and for , where denotes the Euler-Mascheroni constant defined by and where denotes the Psi, or digamma function defined by and the Gamma function , for .

Variant Euler sums of the form have been considered by Chen [1], and recently Boyadzhiev [2] evaluated various binomial identities involving power sums with harmonic numbers . Other remarkable harmonic number identities known to Euler are there is also a recurrence formula which shows that in particular, for and more generally that is a rational multiple of . Another elegant recursion known to Euler [3] is Further work in the summation of harmonic numbers and binomial coefficients has also been done by Flajolet and Salvy [4] and Basu [5]. In this paper it is intended to add, in a small way, some results related to (1.7) and to extend the result of Cloitre, as reported in [6], for . Specifically, we investigate integral representations and closed form representations for sums of harmonic numbers and cubed binomial coefficients. The works of [7β13] also investigate various representations of binomial sums and zeta functions in simpler form by the use of the Beta function and other techniques. Some of the material in this paper was inspired by the work of Mansour, [8], where he used, in part, the Beta function to obtain very general results for finite binomial sums.

#### 2. Integral Representations and Identities

The following Lemma, given by Sofo [11], is stated without proof and deals with the derivative of a reciprocal binomial coefficient.

Lemma 2.1. *Let be a positive real number, , is a positive integer and let be an analytic function of . Then,
*

Theorem 2.2. *Let be real positive numbers, , and let be real positive numbers. Then
*

*Proof. *Expand
where
is the classical Beta function. Differentiating with respect to the parameter , and utilizing Lemma 2.1 implies the resulting equation is as follows:
for .

In the following three corollaries we encounter harmonic numbers at possible rational values of the argument, of the form where , and . The polygamma function is defined as To evaluate we have available a relation in terms of the polygamma function , for rational arguments , where is the Riemann zeta function. We also define The evaluation of the polygamma function at rational values of the argument can be explicitly done via a formula as given by KΓΆlbig [14] (see also [15]) or Choi and CvijoviΔ [16] in terms of the polylogarithmic or other special functions. Some specific values are given as and can be confirmed on a mathematical computer package, such as Mathematica [17].

Corollary 2.3. *Let and let be a positive integer. Then
**
where
*

*Proof. *Let
where
Now, by interchanging sums, we have
We can evaluate
here we have used the result from [18]

Now using (2.7) and (2.8), we may write
Similarly
Substituting (2.19), (2.20) into (2.16) where and are given by (2.12) and (2.13), respectively, on simplifying the identity (2.11) is realized.

For and the following identity is valid:

Theorem 2.4.

*Proof. *The proof of this theorem is very similar to that of Theorem 2.2 and will not be given here.

Corollary 2.5. *Let , and let be a positive integer. Then
**
where is given by (2.12) and is given by (2.13).*

*Proof. *Following similar steps to Corollary 2.3, we may write
and evaluate
By substituting (2.26) into (2.25) and collecting zeta functions, the identity (2.24) is obtained.

For and the following identity is valid:

Theorem 2.6.

*Proof. *
The proof of this theorem is very similar to that of Theorem 2.2 and will not be given here.

Corollary 2.7. *Let , and let be a positive integer. Then
**
where is given by (2.12) and is given by (2.13).*

*Proof. *We follow similar steps as the previous corollary so that
After much algebraic simplification, the following identity is obtained:
Now we can substitute (2.32) into (2.31), collecting zeta functions and using (2.12) and (2.13) for and , respectively, the identity (2.30) is obtained.

Some specific examples of Corollary 2.7 are as follows.

For and the following identity is valid, where is Catalan's constant, defined by and is the complete elliptic integral of the first kind. The degenerate case , gives the well-known result

*Remark 2.8. *Corollaries 2.3, 2.5, and 2.7 are important and can be evaluated as demonstrated independently of their integral representations. Similarly the proofs of Corollaries 2.3, 2.5, and 2.7 are not obvious therefore their explicit representations is desired.

*Remark 2.9. *Theoretically it should be possible to obtain an integral representation for the general sum
with its associated corollaries. This work will be investigated in a forthcoming paper.

#### References

- H. Chen, βEvaluations of some Variant Euler Sums,β
*Journal of Integer Sequences*, vol. 9, no. 2, article 06.2.3, p. 9, 2006. View at: Google Scholar | Zentralblatt MATH - K. N. Boyadzhiev, βHarmonic number identities via Euler's transform,β
*Journal of Integer Sequences*, vol. 12, no. 6, article 09.6.1, p. 8, 2009. View at: Google Scholar - L. Euler,
*Opera Omnia, Series 1*, vol. 15, Teubner, Berlin, Germany, 1917. - P. Flajolet and B. Salvy, βEuler sums and contour integral representations,β
*Experimental Mathematics*, vol. 7, no. 1, pp. 15β35, 1998. View at: Google Scholar | Zentralblatt MATH - A. Basu, βA new method in the study of Euler sums,β
*Ramanujan Journal*, vol. 16, no. 1, pp. 7β24, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Sondow and E. W. Weisstein, Harmonic number. From MathWorld-A Wolfram Web Rescources, http://mathworld.wolfram.com/HarmonicNumber.html.
- H. Alzer, D. Karayannakis, and H. M. Srivastava, βSeries representations for some mathematical constants,β
*Journal of Mathematical Analysis and Applications*, vol. 320, no. 1, pp. 145β162, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Mansour, βCombinatorial identities and inverse binomial coefficients,β
*Advances in Applied Mathematics*, vol. 28, no. 2, pp. 196β202, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Sofo, βIntegral forms of sums associated with harmonic numbers,β
*Applied Mathematics and Computation*, vol. 207, no. 2, pp. 365β372, 2009. View at: Publisher Site | Google Scholar - A. Sofo,
*Computational techniques for the summation of series*, Kluwer Academic Publishers/Plenum Publishers, New York, NY, USA, 2003. - A. Sofo, βSums of derivatives of binomial coefficients,β
*Advances in Applied Mathematics*, vol. 42, no. 1, pp. 123β134, 2009. View at: Publisher Site | Google Scholar - A. Sofo, βHarmonic numbers and double binomial coefficients,β
*Integral Transforms and Special Functions*, vol. 20, no. 11-12, pp. 847β857, 2009. View at: Publisher Site | Google Scholar - A. Sofo, βHarmonic sums and integral representations,β
*Journal of Applied Analysis*, vol. 16, no. 2, pp. 265β277, 2010. View at: Publisher Site | Google Scholar - K. S. Kölbig, βThe polygamma function and the derivatives of the cotangent function for rational arguments,β CERN-IT-Reports CERN-CN 96-005, 1996. View at: Google Scholar
- K. S. Kölbig, βThe polygamma function $\psi (x)$ for $x=1/4$ and $x=3/4$,β
*Journal of Computational and Applied Mathematics*, vol. 75, no. 1, pp. 43β46, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Choi and D. Cvijović, βValues of the polygamma functions at rational arguments,β
*Journal of Physics. A*, vol. 40, no. 50, pp. 15019β15028, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Wolfram Research Inc.,
*Mathematica*, Wolfram Research Inc., Champaign, Ill, USA. - Y. A. Brychkov,
*Handbook of Special Functions*, CRC Press, Boca Raton, Fla, USA, 2008.

#### Copyright

Copyright © 2011 Anthony Sofo. 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.