Recurrence Relations for Wronskian Hermite Polynomials

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.

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• Exceptional Laguerre polynomials
• A new recurrence formula for generic exceptional orthogonal polynomials
• Enumeration of Standard Young Tableaux
• Complex exceptional orthogonal polynomials and quasi-invariance
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Cited by in corpus (4)

• Coefficients of Wronskian Hermite polynomials
• General solution of the exceptional Hermite differential equation and its minimal surface representation
• Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients
• The Expansion of Wronskian Hermite Polynomials in the Hermite Basis