Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 003, 14 pages      arXiv:2006.15534      https://doi.org/10.3842/SIGMA.2021.003

The Expansion of Wronskian Hermite Polynomials in the Hermite Basis

Codruţ Grosu a and Corina Grosu b
a) Google Zürich, Brandschenkestrasse 110, Zürich, Switzerland
b) Department of Applied Mathematics, Politehnica University of Bucharest, Splaiul Independentei 313, Bucharest, Romania

Received July 08, 2020, in final form January 04, 2021; Published online January 09, 2021

Abstract
We express Wronskian Hermite polynomials in the Hermite basis and obtain an explicit formula for the coefficients. From this we deduce an upper bound for the modulus of the roots in the case of partitions of length 2. We also derive a general upper bound for the modulus of the real and purely imaginary roots. These bounds are very useful in the study of irreducibility of Wronskian Hermite polynomials. Additionally, we generalize some of our results to a larger class of polynomials.

Key words: Wronskian; Hermite polynomials; Schrödinger operator.

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