论文标题
数字和整数的力量总和
Figurate numbers and sums of powers of integers
论文作者
论文摘要
最近，Marko和Litvinov（ML）指出，对于所有积极整数$ n $和$ p $ $F_{n}^{p-\ell}$ is the $n$-th hyper-tetrahedron number of dimension $p-\ell$ and $c_{p,\ell}$ denotes the number of $(p -\ell)$-dimensional facets formed by cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \ leq n-1 $。在本文中，我们表明，对于每个自然数量$ p $，ML猜想都是正确的。我们的证明依赖于ML猜想的有效性一定意味着$ C_ {p，\ ell} =（p- \ ell）！ s（p，p- \ ell）$，其中$ s（p，p，p- \ ell）$是第二类的stirling号码。此外，我们提供了表达功率之和$ \ sum_ {i = 1}^{n} i^p $的许多等效公式，作为数字的线性组合。
Recently, Marko and Litvinov (ML) conjectured that, for all positive integers $n$ and $p$, the $p$-th power of $n$ admits the representation $n^p = \sum_{\ell =0}^{p-1} (-1)^{l} c_{p,\ell} F_{n}^{p-\ell}$, where $F_{n}^{p-\ell}$ is the $n$-th hyper-tetrahedron number of dimension $p-\ell$ and $c_{p,\ell}$ denotes the number of $(p -\ell)$-dimensional facets formed by cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \leq n-1$. In this paper we show that the ML conjecture is true for every natural number $p$. Our proof relies on the fact that the validity of the ML conjecture necessarily implies that $c_{p,\ell} = (p-\ell)! S(p, p-\ell)$, where $S(p,p-\ell)$ are the Stirling numbers of the second kind. Furthermore, we provide a number of equivalent formulas expressing the sum of powers $\sum_{i=1}^{n} i^p$ as a linear combination of figurate numbers.
