arXiv 2301.095872023-01-23math.CO

关于几个二项式系数恒等式及其应用

On some Binomial Coefficient Identities with Applications

Necdet Batir, Sezer Sorgunand Sevda Atpinar

备注：: Submitted

摘要：

我们提出了一个不同的证明以下身份由于Munarini，推广了一个好奇的二项式身份西蒙斯。其中\begin{align*} \sum_{k=0}^{n}\binomα{n-k}\binom{β+k}{k}x^k &=\sum_{k=0}^{n}(-1)^{n+k}\binom{β-α+n}{n-k}\binom{β+k}{k}(x+1)^k, \end{align*}是非负整数，$α$和$β$是复数，它们不是负整数。我们的方法是基于一个特别有趣的组合的泰勒定理和Wilf-Zeilberger算法。我们还推广了Alzer和Kouba的一个组合恒等式，并给出了一个新的二项式和恒等式。作为应用，我们给出了许多调和数和恒等式。作为例子，我们证明了\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*}和\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*}

英文摘要：

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}\binomα{n-k}\binom{β+k}{k}x^k &=\sum_{k=0}^{n}(-1)^{n+k}\binom{β-α+n}{n-k}\binom{β+k}{k}(x+1)^k, \end{align*} where $n$ is a non-negative integer and $α$ and $β$ 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*}

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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