On Sums of Binomial Coefficients

We investigate the integral representation of infinite sums involving the ratio of binomial coefficients. We also recover some well-known properties of ζ (3) and extend the range of results given by other authors.


Introduction
In this paper we investigate the summation of the ratio of products of combinatorial coefficients.In particular, we develop integral representations for ´.
For the representation of sums of reciprocals of single and double binomial coefficients, in integral form, one may refer to some results in the papers [5], [4] and [7], see also the book [6].
For designated cases of the parameter values (a, b, c, j, k, l, m, p, t), various particular sums may be expressed in terms of ζ(2) and ζ(3).For many interesting properties of the Zeta function the interested reader is refered to the internet site [9].
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, see [8].
Apéry's [1], see also Beukers [2], proof of the irrationality of ζ(3) uses an elementary and quite complicated construction of the approximants α n βn ∈ Q to this number based on a recurrence relation.The integral representation for the sequence {α n , β n } was proposed.
More recently Rhin and Viola [3] introduced the integral in their study of an irrationality measure for ζ(3).

The Main Results
In this section we develop integral identities for reciprocals of triple products of binomial coefficients.
The following lemma is given Lemma 1.For a, b and c positive real numbers and t ∈ R let The consecutive partial derivative operator of the continuous function Proof.The proof follows by noting that x ∂f ∂x = af and

By induction we see that
We may write and by the recurrence of Stirling numbers of the second kind, S (p + 1, p) = S (p, p − 1) + pS (p, p) we have that Now we investigate the following theorem Theorem 1.For a, b and c positive real numbers and j, k, l ≥ 0, t ∈ R, where where Γ (•) is the classical Gamma function and B (•, •) is the Beta function.
by an allowable change of integral and sum.By Lemma 1 which is the result (2.4). 2 The hypergeometric representation (2.5) can be obtained by the consideration of the ratio of successive terms (2.3).We may also note that from known properties of the hypergeometric function, we may write, from (2.5): ´2 dx dy dz.

For a
Now consider the following lemma and theorem, which is a generalisation of Theorem 1.
Lemma 2. For a, b, c and m positive real numbers and t ∈ R let where f is given by (2.1).
The consecutive partial derivative operator of the continuous function where S (p, r) are Stirling numbers of the second kind.
Proof.We note that x ∂f ∂x = af and

By induction we see that
We may write Now we investigate the following theorem Theorem 2. For a, b, c and m positive real numbers j, k, l ≥ 0 with j + k + l ≥ m + p, t ∈ R, and p ≥ 0 then is Pochhammer's symbol, and T 5 = mT 0 where T 0 is given by (2.6).