Further Identities for c_mk- and a_mk-Weighted Sums and a Remark on a Representation of Pythagoras' Equation

2022/08/23 by Christoph Muschielok, Muschielok, Christoph
Mathematics · #11B83 #Algebraic and Geometric Analysis #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2208.11032

openalex publication_date 2022/08/23 · openalex created_date 2022/08/26 · openalex updated_date 2026/07/28

Abstract

We present some properties of the expansion coefficients a_mk and c_mk of a pair of dual bases, n^m = ∑_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 = ∑_k=2^m c_mk k^α. Explicitly, the case T_m¹ is handled. Furthermore, we indicate connections of T_m² and T_m³ 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.

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