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

SIGMA 14 (2018), 048, 29 pages      arXiv:1801.07980      https://doi.org/10.3842/SIGMA.2018.048
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

Recurrence Relations for Wronskian Hermite Polynomials

Niels Bonneux and Marco Stevens
Department of Mathematics, University of Leuven, Celestijnenlaan 200B box 2400, 3001 Leuven, Belgium

Received January 25, 2018, in final form May 09, 2018; Published online May 16, 2018

Abstract
We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term recurrence relation for Hermite polynomials. The polynomials are defined using partitions of natural numbers, and the coefficients in the recurrence relation can be expressed in terms of the number of standard Young tableaux of these partitions. Using the recurrence relation, we provide another recurrence relation and show that the average of the considered polynomials with respect to the Plancherel measure is very simple. Furthermore, we show that some existing results in the literature are easy corollaries of the recurrence relation.

Key words: Wronskian; Hermite polynomials; partitions; recurrence relation.

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