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

TL;DR

This paper explores properties of expansion coefficients in dual bases, provides explicit formulas involving Stirling numbers, and connects these to classical number theory concepts like Pythagoras' theorem and Mersenne numbers.

Contribution

It introduces explicit formulas for the coefficients c_{mk} using Stirling numbers and extends the understanding of sums involving these coefficients.

Findings

Explicit expression for c_{mk} in terms of Stirling numbers

Evaluation of sums T_m^1, T_m^2, T_m^3 involving coefficients

Connections to Mersenne numbers and Pythagoras' equation

Abstract

We present some properties of the expansion coefficients $a_{mk}$ and $c_{mk}$ of a pair of dual bases, \[ n^m = \sum_{k=2}^m c_{mk} \psi_{k}(n), \] and \[ \psi_m(n) = n + (m-1)(n-1) B_{n-1,m-1}, \] 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: \[ \psi_m(n) - n = \psi_n(m) - m. \] With this equation, we evaluate sums of the form \[ T^\alpha_m = \sum_{k=2}^m c_{mk} k^\alpha.\] 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.