International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2008, Article IDΒ 794181, 11 pages

http://dx.doi.org/10.1155/2008/794181

## Sums of Reciprocals of Triple Binomial Coefficients

School of Computer Science and Mathematics, Victoria University, P.O. Box 14428, Melbourne, VIC 8001, Australia

Received 28 August 2007; Accepted 17 December 2007

Academic Editor: GeorgeΒ Andrews

Copyright Β© 2008 A. 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.

#### Abstract

We investigate the integral representation of infinite sums involving the reciprocals of triple binomial coefficients. We also recover some wellknown properties of (3) and extend the range of results given by other authors.

#### 1. Introduction

In this paper, we investigate the summation of the reciprocal of triple products of combinatorial coefficients. In particular, we develop integral representations forand their alternating series counterparts.

For the representation of sums of reciprocals of single and double binomial coefficients, one may refer to some results in the papers [1β3], see also [4].

For designated cases of the parameter values , various particular sums may be expressed in terms of and . For many interesting properties of the Zeta function, the reader is referred to [5].

The representation of sums in terms of integrals is extremely useful because it allows one to estimate bounds on the sums in cases they cannot be written in closed form. Convexity properties for sums may also be investigated.

ApΓ©ry's [6], see also Beukers [7], proof of the irrationality of uses an elementary and quite complicated construction of the approximants to this number based on a recurrence relation. The integral representationfor the sequence was proposed.

It is important to note that other integral representations of are available in terms of both single and double integrals. Guillera and Sondow [8] list a number of them including the classical resultsIn a recent paper, Muzaffar [9] also obtained some results of the combinatorial typeby utilising the power series expansion of , and are constants depending on In this paper, we complement and extend some of the results given by Muzaffar.

There are some identities in the literature involving reciprocals of triple products of combinatorial coefficients, one prominent identity is the Dougall identity, see [10] or [11],for

#### 2. The Main Results

In this section, we develop integral identities for reciprocals of triple products of binomial coefficients.

Theorem 2.1. *For , and positive real
numbers and then **and similarly **where *

*Proof. *Consider (2.1):where is the
classical Gamma function and is the Beta
function. It holds that
by an allowable change of
integral and sum, and hence we have
which is the result
(2.2).

To prove identity (2.3), consider (2.1) and expand as
follows:which is the result (2.3). The
results (2.6) and (2.7) may be obtained in a similar fashion and therefore will
not be pursued here.

The hypergeometric representation (2.4) and (2.8) can be obtained by the consideration of the ratio of successive terms (2.1) and (2.5), respectively.

We may also
note that from known properties of the hypergeometric function, we may write,
from (2.4) and
(2.8),*Examples**Example 3.1. *
It holds
that

Other integral representations of do exist, some
of which are as follows.

Finch [12] gave the
expression

Lord [13]
posed the problem to show thatthe last three expressions are
directly from
(2.2),
(2.3), and
(2.4), respectively.

Nan-Yue and Williams [14] also gavewhere = golden ratio
= .*Example 3.2. *
It holds
that
For , and , we have the
values *Example 3.3. * For the
alternating case,*Example 3.4. *
It holds
that
where is Catalan's
constant, is the golden
ratio, and = silver ratio
=

Now consider the following theorem, which is a generalisation of Theorem 2.1.

Theorem 3.5. *For , and positive real
numbers and with then **where is given by
(2.9) and **is Pochhammer's symbol.*

*Proof. *Consider (3.14):By an allowable change of
integral and sum, we havewhich is the result
(3.17).

To arrive at the result (3.16), considerby an allowable change of sum and
integral, hencewhich is the result (3.16).

The hypergeometric representations (3.13) and (3.18) can be obtained by the consideration of the ratio of successive terms (3.9) and (3.14), respectively.

In the case when Theorem 3.5 reduces to Theorem 2.1.

*Examples**Example 4.1. *
It holds
that
*Example 4.2. *
It holds
that

#### 3. Conclusion

We have provided triple integral identities for sums of the reciprocal of triple binomial coefficients. In doing so, we have recovered the standard representation for and have generalised and extended some results published previously by other authors.

In another forum, we will extend our results to consider binomial coefficients of the form

#### Acknowledgment

This paper was completed while the author was a Visiting Professor at the Dipartimento di Sistemi e Informatica, Universita di Firenze. I wish to express my sincere thanks to Professor Sprugnoli for his hospitality.

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