EOP (exceptional orthogonal polynomials)

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1: 18.1 Notation

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$x,y,t$	real variables.
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EOP’s	exceptional orthogonal polynomials.

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2: 18.36 Miscellaneous Polynomials

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§18.36(vi) Exceptional Orthogonal Polynomials

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3: 18.39 Applications in the Physical Sciences

… ►c) A Rational SUSY Potential argument … ►and eigenvalues $n+3$, with $n$ as above, with $w(x)$ the weight function of (18.36.10), and ${\widehat{H}}_{n+3}\left(x\right)$ a type III Hermite EOP defined by (18.36.8) and (18.36.9). … ►This seems odd at first glance as ${\widehat{H}}_{n+3}\left(x\right)$ is a polynomial of order $n+3$ for $n=0,1,2,\mathrm{\dots }$, seemingly suggesting that for $n=0$, this being the first excited state, i. …Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. … ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). …

4: 18.38 Mathematical Applications

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Quadrature

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Riemann–Hilbert Problems

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Radon Transform

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Group Representations

… ►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures. …