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# On the sum of inverse binomial

In this post we are discussing the sum

In Staver was the first to study the sum . He observed that . He was , then , able to extract the recursive relation

(1)

He then proved a great result which is well known in literature

(2)

Later, in Rocket combining the identity

(3)

along with induction he was able to give another proof of . In Surin provided another proof using the well known integral representation of the binomial coefficient,

(4)

Since then the cases and have been studied extensively. However, Mansour generalising the idea of Sury provided a theorem which states the following:

Theorem [Mansour]: Let be non negative integers and be given by

where are two functions defined on . Let , be two sequences and , be their corresponding generating functions. Then,

Proof: The proof is a standard generating type one and is left to the reader.

Using the above theorem along with equation we can generate wonderful stuff. For example:

Example 1: Pick and . Then

which , after a bit of transformations gives the general result

Fabulous, isn’t it? If we set we get equation . Of course there are other applications of the above theorem. We can establish a similar equality for the sum . Another relation that can be established by the above theorem is the following:

(5)

We are not gonna go into a deep analysis but the following equalities also hold:

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and

(7)