Harmonic Numbers and Cubed Binomial Coefficients

Anthony Sofo Victoria University College, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia Correspondence should be addressed to Anthony Sofo, anthony.sofo@vu.edu.au Received 18 January 2011; Accepted 3 April 2011 Academic Editor: Toufik Mansour Copyright q 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. 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.


Introduction
The well-known Riemann zeta function is defined as 1.1 The generalized harmonic numbers of order α are given by and where ψ z denotes the Psi, or digamma function defined by and the Gamma function Γ z ∞ 0 u z−1 e −u du, for Ê z > 0.
Variant Euler sums of the form 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 , ∞ n 1 H 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.

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 a be a positive real number, z ≥ 0, n is a positive integer and let Q an, z an z z −1 be an analytic function of z.Then, for z 0 and a 1.

2.1
Theorem 2.2.Let a, b, c, d ≥ 0 be real positive numbers, |t| ≤ 1, p ≥ 0 and let j, k, l, m ≥ 0 be real positive numbers.Then where International Journal of Combinatorics is the classical Beta function.Differentiating with respect to the parameter j, and utilizing Lemma 2.1 implies the resulting equation is as follows: In the following three corollaries we encounter harmonic numbers at possible rational values of the argument, of the form H α r/b −1 where r 1, 2, 3, . . . ,k, α 1, 2, 3, . . . and k ∈ AE.
The polygamma function ψ α z is defined as To evaluate H α r/b −1 we have available a relation in terms of the polygamma function ψ α z , for rational arguments z, where ζ z is the Riemann zeta function.We also define The evaluation of the polygamma function ψ α r/b at rational values of the argument can be explicitly done via a formula as given by K ölbig 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 .

6 International Journal of Combinatorics
where

2.15
Now, by interchanging sums, we have n bn r 3 .

2.16
We can evaluate 2.17 here we have used the result from 18 n≥1 ψ n n bn r

2.18
Now using 2.7 and 2.8 , we may write

2.20
Substituting 2.19 , 2.20 into 2. 16 where XR k and Y R k are given by 2.12 and 2.13 , respectively, on simplifying the identity 2.11 is realized.
For k 1 and b 1 the following identity is valid:

2.22
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 a 1, d c b > 0, t 1, p 0, j 0, and let l m k ≥ 1 be a positive integer.Then where XR k is given by 2.12 and Y R k is given by 2.13 .
Proof.Following similar steps to Corollary 2.3, we may write

2.26
By substituting 2.26 into 2.25 and collecting zeta functions, the identity 2.24 is obtained.
For k 1 and b 1 the following identity is valid: × ln 1 − x • x a−1 y b−1 z c−1 w d dx dy dz dw.

2.28
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 a 1, d c b > 0, t 1, j 0, and let l m k ≥ 1 be a positive integer.Then

2.30
where XR k is given by 2.12 and Y R k is given by 2.13 .
Proof.We follow similar steps as the previous corollary so that n≥1 H

2.31
After much algebraic simplification, the following identity is obtained:

2.32
Now we can substitute 2.32 into 2.31 , collecting zeta functions and using 2.12 and 2.13 for XR k and Y R k , respectively, the identity 2.30 is obtained.
Some specific examples of Corollary 2.7 are as follows.
For k 1 and b 1 the following identity is valid, n≥1 H  , for q 1, 2, 3, . . ., 2.36 with its associated corollaries.This work will be investigated in a forthcoming paper.

Remark 2 . 9 .
Theoretically it should be possible to obtain an integral representation for the general sum 2 H 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.