Zeros transfer for recursively defined polynomials

Research in Number Theory, vol. 9

Abstract

Abstract The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.

Authors 3

1. Bernhard E. Heim corresponding Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University · University of Cologne

   Affiliation as printed

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

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

2. Markus Neuhauser Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University · Kutaisi International University

   Affiliation as printed

   Kutaisi International University, 5/7, Youth Avenue, 4600, Kutaisi, Georgia

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

3. Robert Tröger Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University

   Affiliation as printed

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