论文标题
$ c_ {mk} $ - 和$ a_ {mk} $ - 加权总和的进一步身份，以及关于pythagoras等方程的表示
Further Identities for $c_{mk}$- and $a_{mk}$-Weighted Sums and a Remark on a Representation of Pythagoras' Equation
论文作者
论文摘要
我们介绍了扩展系数的一些属性$ a_ {mk} $和$ c_ {mk} $的一对双基， \ [n^m = \ sum_ {k = 2}^m c_ {mk}ψ_{k}（n），\]和 \ [ψ_m（n）= n +（m-1）（n-1）b_ {n-1，m-1}，\]，我们在arxiv：2207.01935v1中介绍。在这里，$ b_ {a，b} =（a+b）！/（a！\，b！）$是二项式系数。我们通过根据第二类的stirling数字为他们提供明确的表达方式来扩展有关$ c_ {mk} $系数的知识。从二项式系数指数的互换性互换，遵循我们在这里使用的中心身份： \ [ψ_M（n） - n =ψ_n（m） - m。 \]在此方程式中，我们评估形式的总和 \ [t^α_m= \ sum_ {k = 2}^m c_ {mK} k^α。此外，我们指出$ t_m^2 $和$ t_m^3 $的连接与Mersenne号码（一般整数指数）和OEIS条目A024023。我们以对$ a_ {mk} $系数表示如何表示毕达哥拉斯的方程式的小评论。
We present some properties of the expansion coefficients $a_{mk}$ and $c_{mk}$ of a pair of dual bases, \[ n^m = \sum_{k=2}^m c_{mk} ψ_{k}(n), \] and \[ ψ_m(n) = n + (m-1)(n-1) B_{n-1,m-1}, \] we introduced earlier in arXiv:2207.01935v1. Here, $B_{a,b} = (a+b)!/(a!\,b!)$ is a binomial coefficient. We extend the knowledge on the $c_{mk}$ coefficients by giving an explicit expression for them in terms of the Stirling numbers of the second kind. From the interchangeability of the indices of the binomial coefficient, follows the central identity we use here: \[ ψ_m(n) - n = ψ_n(m) - m. \] With this equation, we evaluate sums of the form \[ T^α_m = \sum_{k=2}^m c_{mk} k^α.\] Explicitly, the case $T_m^1$ is handled. Furthermore, we indicate connections of $T_m^2$ and $T_m^3$ to the Mersenne numbers (general integer exponent) and the OEIS entry A024023. We conclude with a small remark on how we can represent Pythagoras' equation in terms of the $a_{mk}$ coefficients.
