# §18.36 Miscellaneous Polynomials

## §18.36(i) Jacobi-Type 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)).

## §18.36(ii) Sobolev Orthogonal Polynomials

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).

## §18.36(iii) Multiple Orthogonal Polynomials

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).

## §18.36(iv) Orthogonal Matrix Polynomials

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).

## §18.36(v) Non-Classical Laguerre Polynomials $L^{(-k)}_{n}\left(x\right)$, $k=1,2\dots$

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 $n\geq 1$ as per (18.2.9_5). For the Laguerre polynomials $L^{(\alpha)}_{n}\left(x\right)$ this requires, omitting all strictly positive factors,

 18.36.1 $n+\alpha>0,$ $n\geq 1$. ⓘ Symbols: $n$: nonnegative integer Referenced by: Erratum (V1.2.0) §18.36 Permalink: http://dlmf.nist.gov/18.36.E1 Encodings: TeX, pMML, png See also: Annotations for §18.36(v), §18.36 and Ch.18

This inequality is violated for $n=1$ if $\alpha<-1$, 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 $n\geq k$.

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

 18.36.2 $L^{(-k)}_{n}\left(x\right)=(-1)^{k}\frac{(n-k)!}{n!}x^{k}L^{(k)}_{n-k}\left(x% \right),$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.36.E2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.36(v), §18.36 and Ch.18

implying that, for $n\geq k$, the orthogonality of the $L^{(-k)}_{n}\left(x\right)$ with respect to the Laguerre weight function $x^{-k}{\mathrm{e}}^{-x}$, $x\in[0,\infty)$. This infinite set of polynomials of order $n\geq k$, 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.

## §18.36(vi) Exceptional Orthogonal Polynomials

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, $0,1,\ldots,m-1$ are missing. The exceptional type III $X_{m}$-EOP’s are missing orders $1,\ldots,m$. 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).

### Type I $X_{1}$-Laguerre EOP’s

Consider the weight function

 18.36.3 $\hat{W}_{k}(x)=\frac{x^{k}{\mathrm{e}}^{-x}}{(x+k)^{2}},$ $k>0,x\in[0,\infty)$. ⓘ Symbols: $[\NVar{a},\NVar{b})$: half-closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.36.E3 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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

 18.36.4 $n\left(\left(x+k\right)^{2}(n+k-1)-k\right)\hat{L}^{(k)}_{n+1}\left(x\right)+(% n+k-1)\left(\left(x+k\right)^{2}(x-2n-k+1)+2k\right)\hat{L}^{(k)}_{n}\left(x% \right)+(n+k-2)\left(\left(x+k\right)^{2}(n+k)-k\right)\hat{L}^{(k)}_{n-1}% \left(x\right)=0,$ $k>0$, $n\geq 2$, ⓘ Defines: $\hat{L}^{(\NVar{k})}_{\NVar{n}}\left(\NVar{x}\right)$: exceptional Laguerre polynomial Symbols: $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.36.E4 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

initialized via :

 18.36.5 $\hat{L}^{(k)}_{n}\left(x\right)=-(x+k+1)L^{(k)}_{n-1}\left(x\right)+L^{(k)}_{n% -2}\left(x\right),$ $n\geq 1$, ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $\hat{L}^{(\NVar{k})}_{\NVar{n}}\left(\NVar{x}\right)$: exceptional Laguerre polynomial, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.36.E5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

resulting in orthogonality;

 18.36.6 $\int_{0}^{\infty}\hat{L}^{(k)}_{n}\left(x\right)\hat{L}^{(k)}_{m}\left(x\right% )\hat{W}_{k}(x)\,\mathrm{d}x=\frac{(n+k)\Gamma\left(n+k-1\right)}{(n-1)!}% \delta_{n,m}.$

The $y(x)=\hat{L}^{(k)}_{n}\left(x\right)$ 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 $T_{k}(y)\equiv-xy^{\prime\prime}+\frac{x-k}{x+k}((x+k+1)y^{\prime}-y)=(n-1)y.$ ⓘ Symbols: $\equiv$: equals by definition, $y$: real variable, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.36(vi) Permalink: http://dlmf.nist.gov/18.36.E7 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

The restriction to $n\geq 1$ 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).

### 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

 18.36.8 $\hat{H}_{0}\left(x\right)=2^{\ifrac{3}{2}}{\pi}^{-\ifrac{1}{4}},$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\hat{H}_{\NVar{n}}\left(\NVar{x}\right)$: exceptional Hermite polynomial and $x$: real variable Referenced by: §18.39(i) Permalink: http://dlmf.nist.gov/18.36.E8 Encodings: TeX, pMML, png See also: Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18
 18.36.9 $\hat{H}_{n+3}\left(x\right)=\frac{(4x^{2}+2)H_{n+1}\left(x\right)+8xH_{n}\left% (x\right)}{{\pi}^{1/4}\sqrt{2^{n+1}(n+3)n!}}=\frac{\mathscr{W}\left\{H_{1}% \left(x\right),H_{2}\left(x\right),H_{n+3}\left(x\right)\right\}}{{\pi}^{1/4}% \sqrt{2^{n+7}(n+1)(n+2)(n+3)!}},$ $n=0,1,\dots$, ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\mathscr{W}$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\hat{H}_{\NVar{n}}\left(\NVar{x}\right)$: exceptional Hermite polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Referenced by: §18.39(i) Permalink: http://dlmf.nist.gov/18.36.E9 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

and orthonormal with respect to the weight function

 18.36.10 $w(x)=\frac{{\mathrm{e}}^{-x^{2}}}{\left(4x^{2}+2\right)^{2}},$ $x\in(-\infty,\infty)$. ⓘ Symbols: $\in$: element of, $\mathrm{e}$: base of natural logarithm, $(\NVar{a},\NVar{b})$: open interval, $w(x)$: weight function and $x$: real variable Referenced by: §18.39(i), Erratum (V1.2.0) §18.36 Permalink: http://dlmf.nist.gov/18.36.E10 Encodings: TeX, pMML, png See also: Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

In §18.39(i) it is seen that the functions, $\sqrt{w(x)}\hat{H}_{n+3}\left(x\right)$, 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).