18 Orthogonal PolynomialsOther Orthogonal Polynomials18.35 Pollaczek Polynomials18.37 Classical OP’s in Two or More Variables

- §18.36(i) Jacobi-Type Polynomials
- §18.36(ii) Sobolev Orthogonal Polynomials
- §18.36(iii) Multiple Orthogonal Polynomials
- §18.36(iv) Orthogonal Matrix Polynomials
- §18.36(v) Non-Classical Laguerre Polynomials ${L}_{n}^{(-k)}\left(x\right)$, $k=1,2\mathrm{\dots}$
- §18.36(vi) Exceptional Orthogonal Polynomials

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)).

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,

18.36.1 | $$n+\alpha >0,$$ | ||

$n\ge 1$. | |||

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),

18.36.2 | $${L}_{n}^{(-k)}\left(x\right)={(-1)}^{k}\frac{(n-k)!}{n!}{x}^{k}{L}_{n-k}^{(k)}\left(x\right),$$ | ||

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 origin 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).

Consider the weight function

18.36.3 | $${\widehat{W}}_{k}(x)=\frac{{x}^{k}{\mathrm{e}}^{-x}}{{(x+k)}^{2}},$$ | ||

$k>0,x\in [0,\mathrm{\infty})$. | |||

The resulting EOP’s, ${\widehat{L}}_{n}^{(k)}\left(x\right)$, $n=1,2,\mathrm{\dots}$ satisfy

18.36.4 | $$\begin{array}{l}\begin{array}{l}n\left({\left(x+k\right)}^{2}(n+k-1)-k\right){\widehat{L}}_{n+1}^{(k)}\left(x\right)\\ \phantom{\rule{2em}{0ex}}+(n+k-1)\left({\left(x+k\right)}^{2}(x-2n-k+1)+2k\right){\widehat{L}}_{n}^{(k)}\left(x\right)\\ \phantom{\rule{2em}{0ex}}+(n+k-2)\left({\left(x+k\right)}^{2}(n+k)-k\right){\widehat{L}}_{n-1}^{(k)}\left(x\right)\end{array}\\ \phantom{\rule{2em}{0ex}}=0,\end{array}$$ | ||

$k>0$, $n\ge 2$, | |||

initialized via :

18.36.5 | $${\widehat{L}}_{n}^{(k)}\left(x\right)=-(x+k+1){L}_{n-1}^{(k)}\left(x\right)+{L}_{n-2}^{(k)}\left(x\right),$$ | ||

$n\ge 1$, | |||

resulting in orthogonality;

18.36.6 | $${\int}_{0}^{\mathrm{\infty}}{\widehat{L}}_{n}^{(k)}\left(x\right){\widehat{L}}_{m}^{(k)}\left(x\right){\widehat{W}}_{k}(x)dx=\frac{(n+k)\mathrm{\Gamma}\left(n+k-1\right)}{(n-1)!}{\delta}_{n,m}.$$ | ||

The $y(x)={\widehat{L}}_{n}^{(k)}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients:

18.36.7 | $${T}_{k}(y)\equiv -x{y}^{\prime \prime}+\frac{x-k}{x+k}((x+k+1){y}^{\prime}-y)=(n-1)y.$$ | ||

The restriction to $n\ge 1$ is now apparent: (18.36.7) does not posses a solution if $y(x)$ is a constant. Completeness follows from the self-adjointness of ${T}_{k}$, Everitt (2008).

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

18.36.8 | $${\widehat{H}}_{0}\left(x\right)={2}^{3/2}{\pi}^{-1/4},$$ | ||

18.36.9 | $${\widehat{H}}_{n+3}\left(x\right)=\frac{(4{x}^{2}+2){H}_{n+1}\left(x\right)+8x{H}_{n}\left(x\right)}{{\pi}^{1/4}\sqrt{{2}^{n+1}(n+3)n!}}=\frac{\mathcal{W}\left\{{H}_{1}\left(x\right),{H}_{2}\left(x\right),{H}_{n+3}\left(x\right)\right\}}{{\pi}^{1/4}\sqrt{{2}^{n+7}(n+1)(n+2)(n+3)!}},$$ | ||

$n=0,1,\mathrm{\dots}$, | |||

and orthonormal with respect to the weight function

18.36.10 | $$w(x)=\frac{{\mathrm{e}}^{-{x}^{2}}}{{\left(4{x}^{2}+2\right)}^{2}},$$ | ||

$x\in (-\mathrm{\infty},\mathrm{\infty})$. | |||

In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with
a *rational* potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).