FORMULAS FOR COEFFICIENTS OF POLYNOMIALS ASSIGNED TO ARITHMETIC FUNCTIONS

Journal of Mathematical Sciences, vol. 290, pp. 612–622

Abstract

Abstract We associate recursively defined polynomials with normalized (non-vanishing) arithmetic functions g and h . Let $$P_0^{g,h}(x):=1$$ P 0 g , h ( x ) : = 1 . Then, $$ P_n^{g,h}(x):= \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). $$ P n g , h ( x ) : = x h ( n ) ∑ k = 1 n g ( k ) P n - k g , h ( x ) . For special functions g and h , we obtain the D’Arcais polynomials, which are associated with the coefficients of powers of the Dedekind $$\eta $$ η -function and related to a hook length formula by Nekrasov and Okounkov. More examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and Laguerre polynomials. We provide explicit formulas and identities for the coefficients of $$P_n^{g,h}(x)$$ P n g , h ( x ) which separate the impact of g and h . Finally, we mention several applications.

Authors 2

1. Bernhard E. Heim Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University · University of Cologne

   Affiliation as printed

   Department of Mathematics and Computer Science, Division of Mathematics, University of Cologne, Weyertal 86–90, 50931, Cologne, Germany

   Lehrstuhl A für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

2. Markus Neuhauser corresponding Aachen Lehrstuhl für Geometrie und Analysis

   RWTH Aachen University · Kutaisi International University

   Affiliation as printed

   Kutaisi International University (KIU), Youth Avenue, Turn 5/7, Kutaisi, 4600, Georgia

   Lehrstuhl für Geometrie und Analysis, RWTH Aachen University, 52056, Aachen, Germany