Hermite EOP’s

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1: 18.36 Miscellaneous Polynomials

… ►Two representative examples, type I

X_{1}

-Laguerre, Gómez-Ullate et al. (2010), and type III

X_{2}

-Hermite, Gómez-Ullate and Milson (2014) EOP’s, are illustrated here. … ►

Type III $X_{2}$ -Hermite EOP’s

►Hermite EOP’s are defined in terms of classical Hermite OP’s. The type III

X_{2}

-Hermite EOP’s, missing polynomial orders

1

and

2

, are the complete set of polynomials, with real coefficients and defined explicitly as …

2: 18.39 Applications in the Physical Sciences

… ►c) A Rational SUSY Potential argument … ►Thus the two missing quantum numbers corresponding to EOP’s of order

1

and

2

of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. …

3: 18.38 Mathematical Applications

… ►Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. … ►EOP’s are the subject of recent work on rational solutions to the fourth Painlevé equation, see Clarkson (2003a) and Marquette and Quesne (2016),where use of Hermite EOP’s makes a connection to quantum mechanics. …

4: 18.1 Notation

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Classical OP’s

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Hermite: $H_{n}\left(x\right)$ , $\mathit{He}_{n}\left(x\right)$ .

… ►

Discrete $q$ -Hermite I: $h_{n}\left(x;q\right)$ .

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Discrete $q$ -Hermite II: $\tilde{h}_{n}\left(x;q\right)$ .

… ►

Continuous $q$ -Hermite: $H_{n}\left(x\,|\,q\right)$ .

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