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- W4317951269 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 $n$ is a non-negative integer and $alpha$ and $beta$ 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*} sum_{k=0}^{n}binom{n}{k}^2H_kH_{n-k}=binom{2n}{n} left((H_{2n}-2H_n)^2+H_{n}^{(2)}-H_{2n}^{(2)}right). end{align*}" @default.
- W4317951269 created "2023-01-25" @default.
- W4317951269 creator A5008046089 @default.
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- W4317951269 date "2023-01-23" @default.
- W4317951269 modified "2023-09-28" @default.
- W4317951269 title "On some Binomial Coefficient Identities with Applications" @default.
- W4317951269 doi "https://doi.org/10.48550/arxiv.2301.09587" @default.
- W4317951269 hasPublicationYear "2023" @default.
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