On some Binomial Coefficient Identities with Applications
Necdet Batir, Sezer Sorgunand Sevda Atpinar

TL;DR
This paper provides new proofs and generalizations of binomial coefficient identities, introduces novel binomial sum identities, and applies these to derive harmonic number sum identities using advanced combinatorial techniques.
Contribution
It offers a new proof of Munarini's binomial identity, generalizes identities of Alzer and Kouba, and introduces new binomial sum identities with applications to harmonic number sums.
Findings
New proof of Munarini's binomial identity
Generalizations of identities by Alzer and Kouba
New binomial sum identities and harmonic number sum formulas
Abstract
We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k &=\sum_{k=0}^{n}(-1)^{n+k}\binom{\beta-\alpha+n}{n-k}\binom{\beta+k}{k}(x+1)^k, \end{align*} where is a non-negative integer and and are complex numbers, which are not negative integers. Our approach is based on a particularly interesting combination of the Taylor theorem and the Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to Alzer and Kouba, and offer a new binomial sum identity. Furthermore, as applications, we give many harmonic number sum identities. As examples, we prove that \begin{equation*} H_n=\frac{1}{2}\sum_{k=1}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{k}H_k \end{equation*} and \begin{align*}…
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Combinatorial Mathematics
