Christoph Muschielok | Searcharxiv

arXiv2022

Further Identities for $c_{mk}$- and $a_{mk}$-Weighted Sums and a Remark on a Representation of Pythagoras' Equation

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.

math.NT↗

arXiv2022

Another Approach on Power Sums

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 $ψ_m^{(a)}$, \[ S^{(a)}_m(n) = \sum_k c_{mk} ψ_k^{(a)}(n). \]

math.CO↗