These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. For further information see Koornwinder (1984b) and Kwon et al. (2006).

Similar OP’s can also be constructed for the Laguerre polynomials; see Koornwinder (1984b, (4.8)).

Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. For an introductory survey to this subject, see Marcellán et al. (1993) and Marcellán and Xu (2015). Other relevant references include Iserles et al. (1991) and Koekoek et al. (1998).

These are polynomials in one variable that are orthogonal with respect to a number of different measures. They are related to Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers. For further information see Ismail (2009, Chapter 23).

These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. For further information see Durán and Grünbaum (2005).

Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, $A_n{A}_{n-1}{C}_n>0$ for $n\ge 1$ as per (18.2.9_5). For the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ this requires, omitting all strictly positive factors,

This inequality is violated for $n=1$ if , seemingly precluding such an extension of the Laguerre OP’s into that regime. This is correct unless $\alpha$ is a negative integer $-k$, but then the polynomials are only defined for $n\ge k$.

The possibility of generalization to $\alpha =-k$, for $k\in \mathbb{N}$, is implicit in the identity Szegő (1975, page 102),

implying that, for $n\ge k$, the orthogonality of the $L_n^{(-k)}\left(x\right)$ with respect to the Laguerre weight function $x^{-k}{\mathrm{e}}^{-x}$, $x\in [0,\mathrm{\infty })$. This infinite set of polynomials of order $n\ge k$, the smallest power of $x$ being $x^k$ in each polynomial, is a complete orthogonal set with respect to this measure. These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness.

This lays the foundation for consideration of exceptional OP’s wherein a finite number of (possibly non-sequential) polynomial orders are missing, from what is a complete set even in their absence.

Exceptional type I $X_m$-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order $m$, or, said another way, the first $m$ polynomial orders, $0,1,\mathrm{\dots },m-1$ are missing. The exceptional type III $X_m$-EOP’s are missing orders $1,\mathrm{\dots },m$. See Liaw et al. (2016, Eqns. 1.1 and 1.2), for the origen of this type characterization. EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. See Gómez-Ullate et al. (2009) for an elementary introduction.

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. A broad overview appears in Milson (2017).