§18.38 Mathematical Applications

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,

with $H_n\left(x\right)$ as in §18.3, satisfies the Toda equation

The Askey–Gasper inequality

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).

Ultraspherical polynomials are zonal spherical harmonics. As such they have many applications. See, for example, Andrews et al. (1999, Chapter 9). See also §14.30.

Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. See Fyodorov (2005) and Deift (1998, Chapter 5).

See Deift (1998, Chapter 7) and Ismail (2009, Chapter 22).

See Deans (1983, Chapters 4, 7).

For group-theoretic interpretations of OP’s see Vilenkin and Klimyk (1991, 1992, 1993).

For applications of Krawtchouk polynomials $K_n(x;p,N)$ and $q$-Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).

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

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

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

Then

where

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

commutes with $K_0,K_1,K_2$, that is $K_{j}Q=Q{K}_j$, and satisfies

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

has a solution

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.

Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and $q$-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7).

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.

EOP’s are the subject of recent work on rational solutions to the fourth Painlevé equation, see Clarkson (2003a) and Marquette and Quesne (2016),where use of Hermite EOP’s makes a connection to quantum mechanics.

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