论文标题
关于另一种功率总和的笔记
A note on another approach on power sums
论文作者
论文摘要
在本说明中，我们首先回顾了Muschielok最近提出的新颖的权力总和：2207.01935V1，可以通过公式$ s_m^{（a）}（a）}（n）= \ sum_ {k} c_ {k} c_ {mk} c_ {mk} c_ {mk} c_k^ψ系数和基础功能的情况下$ψ_m^{（a）}（n）$满足递归属性$ψ_m^{（a+1）}（n）= \ sum_ {i = 1}^nψ_m^{（a）}（a）}（i）$。然后，我们指出了许多关于上述方法的补充事实，这些事实并未在Muschielok的论文中明确考虑。特别是，我们表明，对于任何给定的$ M $，可以通过仅涉及二项式系数的矩阵来获得$ c_ {mk} $的值。这可以与Muschielok的原始方法进行比较，其中$ c_ {mk} $的值可以通过反转涉及第一类的stirl数字的下三角矩阵来获得。另外，我们对系数的功能形式做出了猜想，$ c_ {m \，m-k} $。
In this note, we first review the novel approach to power sums put forward recently by Muschielok in arXiv:2207.01935v1, which can be summarized by the formula $S_m^{(a)}(n) = \sum_{k} c_{mk} ψ_k^{(a)}(n)$, where the $c_{mk}$'s are the expansion coefficients and where the basis functions $ψ_m^{(a)}(n)$ fulfil the recursive property $ψ_m^{(a+1)}(n)= \sum_{i=1}^n ψ_m^{(a)}(i)$. Then, we point out a number of supplementary facts concerning the said approach not contemplated explicitly in Muschielok's paper. In particular, we show that, for any given $m$, the values of the $c_{mk}$'s can be obtained by inverting a matrix involving only binomial coefficients. This may be compared with the original approach of Muschielok, where the values of the $c_{mk}$'s can be obtained by inverting a lower triangular matrix involving the Stirling numbers of the first kind. Also, we make a conjecture about the functional form of the coefficients $c_{m\, m-k}$.
