论文标题
整数幂的超级智能公式的决定性公式
A determinantal formula for the hyper-sums of powers of integers
论文作者
论文摘要
对于非阴性整数$ r $和$ m $，让$ s_m^{（r）}（n）$表示$ r $ -fold sumpation（或hyper-sum）在第一个$ n $阳性整数上，$ n $ pastic Integers to $ m $ th powers，带有初始条件$ s_m^{（0）}（0）}（n）= n^m $。在本文中，我们为$ s_m^{（r）}（n）$提供了一个新的确定公式。具体来说，我们表明，对于所有整数，$ r \ geq 0 $和$ m \ geq 1 $，$ s_m^{（r）}（n）$与$ s_1^{（r）}（r）}（n）$乘以低hessenberg Matrix $ m-1 $ m-1 $ nviars bernoulli and + norder norker y + n norker norker norker nork y nork bernoulli和bernouli norker norker and y + norder norkeri和bernouli in + n） \ frac {r} {2} $。此外，每当$ r \ geq 1 $时，评估此决定因素会给我们$ s_m^{（r）}（n）$作为$ s_1^{（r）}（r）}（r）}（n）$ timple $ n_r $ a $ n_r $ m-1 $ m-m-1 $的偶尔或奇数，取决于$ m $ nd $ m $ ogg ogg ogg ogg ogg ogg ogg。
For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold summation (or hyper-sum) over the first $n$ positive integers to the $m$th powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we derive a new determinantal formula for $S_m^{(r)}(n)$. Specifically, we show that, for all integers $r\geq 0$ and $m \geq 1$, $S_m^{(r)}(n)$ is proportional to $S_1^{(r)}(n)$ times the determinant of a lower Hessenberg matrix of order $m-1$ involving the Bernoulli numbers and the variable $N_r = n + \frac{r}{2}$. Furthermore, whenever $r\geq 1$, evaluating this determinant gives us $S_m^{(r)}(n)$ as $S_1^{(r)}(n)$ times an even or odd polynomial in $N_r$ of degree $m-1$, depending on whether $m$ is odd or even.
