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 [7–13] 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 Kölbig [14] (see also [15]) or Choi and Cvijović [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.