18 Orthogonal PolynomialsApplications18.37 Classical OP’s in Two or More Variables18.39 Applications in the Physical Sciences

- §18.38(i) Classical OP’s: Numerical Analysis
- §18.38(ii) Classical OP’s: Mathematical Developments and Applications
- §18.38(iii) Other OP’s

The monic Chebyshev polynomial ${2}^{1-n}{T}_{n}\left(x\right)$, $n\ge 1$, enjoys the ‘minimax’ property on the interval $[-1,1]$, that is, $|{2}^{1-n}{T}_{n}\left(x\right)|$ has the least maximum value among all monic polynomials of degree $n$. In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. For these results and applications in approximation theory see §3.11(ii) and Mason and Handscomb (2003, Chapter 3), Cheney (1982, p. 108), and Rivlin (1969, p. 31).

Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). This process has been generalized to spectral methods for solving partial differential equations. For further information see Mason and Handscomb (2003, Chapters 10 and 11), Gottlieb and Orszag (1977, pp. 7–19), and Guo (1998, pp. 120–151). See also the paragraph on DVRs, below.

Classical OP’s play a fundamental role in Gaussian quadrature. If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the $n$th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding $2n-1$. See §3.5(v).

The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature
abscissas and weights, have led to *discrete variable representations*, or DVRs, of Sturm–Liouville and other differential operators.
The terminology DVR arises as an otherwise continuous variable, such as the co-ordinate $x$, is replaced by its values at a finite set of zeros of appropriate
OP’s resulting in expansions using functions localized at these points. Light and Carrington Jr. (2000) review and extend the one-dimensional analysis to solution
of multi-dimensional many-particle systems, where the sparse nature of the resulting matrices is highly advantageous. Schneider et al. (2016)
discuss DVR/Finite Element solutions of the time-dependent Schrödinger equation. Each of these typically require a particular non-classical weight
functions and analysis of the corresponding OP’s. These are listed in §18.39(iii). These methods have become known as
*pseudo-spectral*, and are overviewed in Cerjan (1993), and Shizgal (2015).

The Toda equation provides an important model of a completely integrable system. It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. For instance,

18.38.1 | $${V}_{n}(x)=\frac{2n{H}_{n+1}\left(x\right){H}_{n-1}\left(x\right)}{{({H}_{n}\left(x\right))}^{2}},$$ | ||

with ${H}_{n}\left(x\right)$ as in §18.3, satisfies the Toda equation

18.38.2 | $$\frac{{d}^{2}}{{dx}^{2}}\mathrm{ln}{V}_{n}(x)={V}_{n+1}(x)+{V}_{n-1}(x)-2{V}_{n}(x),$$ | ||

$n=1,2,\mathrm{\dots}$. | |||

The *Askey–Gasper inequality*

18.38.3 | $$\sum _{m=0}^{n}{P}_{m}^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_{n}}{n!}{}_{3}{}^{}F_{2}^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ | ||

$x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots}$, | |||

also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. See de Branges (1985). For the generalized hypergeometric function ${}_{3}{}^{}F_{2}^{}$ see (16.2.1).

See Deans (1983, Chapters 4, 7).

The $3j$ symbol (34.2.6), with an alternative expression as a terminating ${}_{3}{}^{}F_{2}^{}$ of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. The orthogonality relations in §34.3(iv) for the $3j$ symbols can be rewritten in terms of orthogonality relations for Hahn or dual Hahn polynomials as given by §§18.2(i), 18.2(iii) and Table 18.19.1 or by §18.25(iii), respectively. See Koornwinder (1981, §4) for details.

The $6j$ symbol (34.4.3), with an alternative expression as a terminating balanced ${}_{4}{}^{}F_{3}^{}$ of unit argument, can be expressend in terms of Racah polynomials (18.26.3). The orthogonality relations (34.5.14) for the $6j$ symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). See Wilson (1980, §5) for details.

A symmetric Laurent polynomial is a function of the form

$$f(z)={c}_{0}+\sum _{k=1}^{n}{c}_{k}({z}^{k}+{z}^{-k}).$$ |

Define operators ${K}_{0}$ and ${K}_{1}$ acting on symmetric Laurent polynomials by ${K}_{0}=L$ ($L$ given by (18.28.6_2)) and $({K}_{1}f)(z)=(z+{z}^{-1})f(z)$. Define a further operator ${K}_{2}$ by

18.38.4 | $${K}_{2}={[{K}_{0},{K}_{1}]}_{q},$$ | ||

where the $q$-commutator of operators $X$ and $Y$ is defined by

18.38.5 | $${[X,Y]}_{q}={q}^{\frac{1}{2}}XY-{q}^{-\frac{1}{2}}YX.$$ | ||

Then

18.38.6 | ${[{K}_{1},{K}_{2}]}_{q}$ | $=B{K}_{1}+{C}_{0}{K}_{0}+{D}_{0},$ | ||

${[{K}_{2},{K}_{0}]}_{q}$ | $=B{K}_{0}+{C}_{1}{K}_{1}+{D}_{1},$ | |||

where

18.38.7 | $B$ | $={\left(1-{q}^{-1}\right)}^{2}({e}_{3}+q{e}_{1}),$ | ||

${C}_{0}$ | $={\left(q-{q}^{-1}\right)}^{2},$ | |||

${C}_{1}$ | $={q}^{-1}{\left(q-{q}^{-1}\right)}^{2}{e}_{4},$ | |||

${D}_{0}$ | $=-{q}^{-3}{\left(1-q\right)}^{2}\left(1+q\right)\left({e}_{4}+q{e}_{2}+{q}^{2}\right),$ | |||

${D}_{1}$ | $=-{q}^{-3}{\left(1-q\right)}^{2}\left(1+q\right)\left({e}_{1}{e}_{4}+q{e}_{3}\right),$ | |||

and ${e}_{1},{e}_{2},{e}_{3},{e}_{4}$ are the *elementary symmetric polynomials*
in $a,b,c,d$ given by $a+b+c+d$, $ab+ac+\mathrm{\cdots}+cd$, $abc+abd+acd+bcd$,
$abcd$, respectively. A further operator, the so-called
*Casimir operator*

18.38.8 | $$Q=\begin{array}{l}\left({q}^{-\frac{1}{2}}-{q}^{\frac{3}{2}}\right){K}_{0}{K}_{1}{K}_{2}+q{K}_{2}^{2}+B\left({K}_{0}{K}_{1}+{K}_{1}{K}_{0}\right)\\ \phantom{\rule{2em}{0ex}}+q{C}_{0}{K}_{0}^{2}+{q}^{-1}{C}_{1}{K}_{1}^{2}+\left(1+q\right){D}_{0}{K}_{0}+\left(1+{q}^{-1}\right){D}_{1}{K}_{1}\end{array}$$ | ||

commutes with ${K}_{0},{K}_{1},{K}_{2}$, that is ${K}_{j}Q=Q{K}_{j}$, and satisfies

18.38.9 | $$Q={Q}_{0},$$ | ||

where ${Q}_{0}$ is a constant with explicit expression in terms of ${e}_{1},{e}_{2},{e}_{3},{e}_{4}$ and $q$ given in Koornwinder (2007a, (2.8)).

The abstract associative algebra with generators ${K}_{0},{K}_{1},{K}_{2}$ and relations (18.38.4), (18.38.6) and with the constants $B,{C}_{0},{C}_{1},{D}_{0},{D}_{1}$ in (18.38.6) not yet specified, is called the Zhedanov algebra or Askey–Wilson algebra AW(3). If we consider this abstract algebra with additional relation (18.38.9) and with dependence on $a,b,c,d$ according to (18.38.7) then it is isomorphic with the algebra generated by ${K}_{0}=L$ given by (18.28.6_2), $({K}_{1}f)(z)=(z+{z}^{-1})f(z)$ and ${K}_{2}$ given by (18.38.4), and ${K}_{0},{K}_{1},{K}_{2}$ act on the linear span of the Askey–Wilson polynomials (18.28.1). See Zhedanov (1991), Granovskiĭ et al. (1992, §3), Koornwinder (2007a, §2) and Terwilliger (2011). Similar algebras can be associated with all families of OP’s in the $q$-Askey scheme and the Askey scheme.

The *Dunkl operator*, introduced by Dunkl (1989),
is an operator associated with reflection groups or root systems
which has terms involving first order partial derivatives and reflection terms.
Analogues of the original Dunkl operator (the rational case)
were introduced by Heckman and Cherednik for the trigonometric
case, and by Cherednik for the $q$-case. Algebraic structures were
built of which special representations involve Dunkl type operators.
In the $q$-case this algebraic structure is called the
*double affine Hecke algebra* (DAHA), introduced by Cherednik.
Eigenvalue equations involving Dunkl type operators have
as eigenfunctions nonsymmetric analogues of multivariable special
functions associated with root systems. This gives also new structures
and results in the one-variable case, but the obtained nonsymmetric
special functions can now usually be written as a linear combination
of two known special functions.

In the one-variable case the Dunkl operator eigenvalue equation

18.38.10 | $${f}^{\prime}(x)+\left(\alpha +\frac{1}{2}\right)\frac{f(x)-f(-x)}{x}=\mathrm{i}\lambda f(x)$$ | ||

has a solution

18.38.11 | $$f(x)=\mathrm{\Gamma}\left(\alpha +1\right){\left(\frac{2}{\lambda x}\right)}^{\alpha}\left({J}_{\alpha}\left(\lambda x\right)+\mathrm{i}{J}_{\alpha +1}\left(\lambda x\right)\right),$$ | ||

where the Bessel function ${J}_{\nu}\left(z\right)$ is defined in (10.2.2).

For the one-variable $q$-case see Noumi and Stokman (2004), Koornwinder (2007a, §§3,4), Koornwinder and Bouzeffour (2011, §§4,5) and Terwilliger (2013). The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial ${R}_{n}(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial ${z}^{-1}(1-az)(1-bz){R}_{n-1}(z;qa,qb,c,d|q)$, where ${R}_{n}(z)$ is given in (18.28.1_5). See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas.

The solved Schrödinger equations of §18.39(i) involve *shape invariant* potentials,
and thus are in the family of *supersymmetric* or *SUSY* potentials. SUSY leads to algebraic simplifications in generating excited states,
and *partner potentials* with closely related energy spectra, from knowledge of a single ground state wave function.
These generalize the *ladder operators*, as reviewed and extended by Infeld and Hull (1951), and also called *creation and annilhilation* operators.
Overviews appear in Dutt et al. (1988), Cooper et al. (1995), and Quesne (2011).

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. Detailed examples appear in the Section on Miscellaneous OP’s §§18.36(v) and 18.36(vi). A review is Milson (2017). Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. Much of the exploration of the EOP’s is based on the operator algebra as developed in SUSY, above.

See §§18.32, 18.39(iii), and 32.15, for recent developments and applications.