Formulas for coefficients of polynomials assigned to arithmetic functions

2020/10/15 by Bernhard Heim, Heim, Bernhard, Markus Neuhauser +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.2010.07890

Updated Version

openalex publication_date 2020/10/15 · openalex created_date 2020/10/22 · arxiv created 2020/11/19 · arxiv updated 2020/11/23 · openalex updated_date 2026/08/01

Abstract

We attach to normalized (non-vanishing) arithmetic functions g and h recursively defined polynomials. Let P₀^g,h(x):=1. Then P_n^g,h(x) := (x)/(h(n)) ∑_k=1ⁿ g(k) P_n-k^g,h(x). For special g and h, we obtain the D'Arcais polynomials, which are equal to the coefficients of the -zth powers of the Dedekind η-function and are also given by Nekrasov and Okounkov as a hook length formula. Examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and associated Laguerre polynomials. We present explicit formulas and identities for the coefficients of P_n^g,h(x) which separate the impact of g and h. Finally, we provide several applications.

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