关于二项式定理与离散卷积之间的联系
On the link between binomial theorem and discrete convolution
AI总结：
本文引入多项式$\mathbf{P}^{m}_{b}(x)$并建立奇数次幂的多项式恒等式，借此揭示了二项式定理与奇数次幂离散卷积的联系，并将其推广至多项式情形，最后通过Mathematica程序验证。
摘要：
设$\mathbf{P}^{m}_{b}(x)$为关于$x$的$2m+1$次多项式且$b \in \mathbb{R}$，\\[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r \\]其中$\mathbf{A}_{m,r}$为实系数。本文引入多项式$\mathbf{P}^{m}_{b}(x)$并研究其性质，基于该多项式建立了奇数次幂的多项式恒等式。基于此奇数次幂恒等式，我们探讨了二项式定理与奇数次幂离散卷积之间的联系，并将该关系进一步推广至多项式情形。所有发现均通过Mathematica程序验证。
英文摘要：
Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree polynomial in $x$ and $b \in \mathbb{R}$ \[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r \] where $\mathbf{A}_{m,r}$ are real coefficients. In this manuscript, we introduce the polynomial $\mathbf{P}^{m}_{b}(x)$ and study its properties, establishing a polynomial identity for odd-powers in terms of this polynomial. Based on mentioned polynomial identity for odd-powers, we explore the connection between the Binomial theorem and discrete convolution of odd-powers, further extending this relation to the multinomial case. All findings are verified using Mathematica programs.
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。
