论文标题
权力总和的另一种方法
Another Approach on Power Sums
论文作者
论文摘要
我们表明，对于某些多项式〜$ψ^{（a）} _ m（n）$，带有属性\ [ψ^{（a+1）} _ m（n）= \ sum_ {ν= 1}^nψ_m^{（a+sum_m^{（a）}（a）}（a）}（a）}（n月）， $ a，m，n \ in \ mathbb {n} _0 $）。我们将这些多项式用作表达单一〜$ n^m $的基础。一旦确定了扩展系数，我们就可以表达$ m $ - th power总和〜$ s^{（a）} _ m（n）$的任何顺序$ a $ a $，\ [s^{（a）} _ m（a）_ m（n）= \ sum_a = \ sum_a = 1} \sum_{ν_1=1}^{ν_2} ν_1^m, \] in a very convenient way by exploiting the summation property of the $ψ_m^{(a)}$, \[ S^{(a)}_m(n) = \sum_k c_{mk} ψ_k^{(a)}(n). \]
We show that explicit forms for certain polynomials~$ψ^{(a)}_m(n)$ with the property \[ ψ^{(a+1)}_m(n) = \sum_{ν=1}^n ψ_m^{(a)}(ν) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express the monomials~$n^m$. Once the expansion coefficients are determined, we can express the $m$-th power sums~$S^{(a)}_m(n)$ of any order $a$, \[ S^{(a)}_m(n) = \sum_{ν_a = 1}^n \cdots \sum_{ν_2 = 1}^{ν_3} \sum_{ν_1=1}^{ν_2} ν_1^m, \] in a very convenient way by exploiting the summation property of the $ψ_m^{(a)}$, \[ S^{(a)}_m(n) = \sum_k c_{mk} ψ_k^{(a)}(n). \]
