- Cosmin Pohoata, Generalizing a curious combinatorial identity, submitted. (Link to the PDF file)
This (very short) note is rather a consequence of founding Theorem 1 from my post from http://www.mathlinks.ro/viewtopic.php?p=1101215#1101215.)
In [5] Simons proved a binomial coefficient identity using repeated differentiation which can be equivalently written as
\sum_{k=0}^{n}{\binom{n}{k}\binom{n+k}{k}(-1)^{n-k}(1+x)^{k}}==\sum_{k=0}^{n}{\binom{n}{k}}{\binom{n+k}{k}x^{k}}.
Proofs of this identity have been given by Chapman [1] using generating functions and by Prodinger [3] using Cauchy’s integral formula. Using Prodinger’s approach Munarini [2] established some generalizations of Simons’ identity and other neat identities, some of them proved by Shattuck recently in [4], using elegant combinatorial arguments. In the following we present an alternate generalization, different from the ones given by Munarini. The proof is very elementary, based on a simple algebraic manipulation.
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