论文标题
整数的力量和第二类的史瓦特数字的总和
Sums of powers of integers and generalized Stirling numbers of the second kind
论文作者
论文摘要
通过将Newton-Gregory扩展应用于与整数$ s_k（n）= 1^k + 2^k + \ cdots + n^k $相关的多项式相关的多项式，我们得出了几个无限族的明确配方属的家庭，用于$ s_k（n）$。其中一个家庭涉及第二种的$ r $ stirling $ \ genfrac {\ {} {} {\}} {0pt} {} {} {} {k} {k} {j} { $ \ genfrac {\ {} {\}} {0pt} {} {} {k} {j} _ { - r} $，两个公式的家庭都由非阴性integer $ r $索引。作为副产品，我们获得了$ s_k（n）$的三个附加公式美元$ \ genfrac {\ {} {\}} {0pt} {} {} {k} {j} _ {k-j} $。此外，我们为$ \ genfrac {\ {} {\}} {0pt} {} {} {k} {k} {j} {j} _ {x} $和谐波数字提供了bernoulli polyenmials $ b_k（x-1）$的公式。
By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families involves the $r$-Stirling numbers of the second kind $\genfrac{\{}{\}}{0pt}{}{k}{j}_r$, $j=0,1,\ldots,k$, while the other involves their duals $\genfrac{\{}{\}}{0pt}{}{k}{j}_{-r}$, with both families of formulas being indexed by the non-negative integer $r$. As a by-product, we obtain three additional formulas for $S_k(n)$ involving the numbers $\genfrac{\{}{\}}{0pt}{}{k}{j}_{n+m}$, $\genfrac{\{}{\}}{0pt}{}{k}{j}_{n-m}$ (where $m$ is any given non-negative integer), and $\genfrac{\{}{\}}{0pt}{}{k}{j}_{k-j}$, respectively. Moreover, we provide a formula for the Bernoulli polynomials $B_k(x-1)$ in terms of $\genfrac{\{}{\}}{0pt}{}{k}{j}_{x}$ and the harmonic numbers.
