# §18.38 Mathematical Applications

## §18.38(i) Classical OP’s: Numerical Analysis

### Approximation Theory

The scaled Chebyshev polynomial $2^{1-n}\mathop{T_{n}\/}\nolimits\!\left(x\right)$, $n\geq 1$, enjoys the “minimax” property on the interval $[-1,1]$, that is, $|2^{1-n}\mathop{T_{n}\/}\nolimits\!\left(x\right)|$ 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).

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).

### Differential Equations

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).

## §18.38(ii) Classical OP’s: Other Applications

### Integrable Systems

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 $V_{n}(x)=\ifrac{2n\mathop{H_{n+1}\/}\nolimits\!\left(x\right)\mathop{H_{n-1}\/% }\nolimits\!\left(x\right)}{(\mathop{H_{n}\/}\nolimits\!\left(x\right))^{2}},$

with $\mathop{H_{n}\/}\nolimits\!\left(x\right)$ as in §18.3, satisfies the Toda equation

 18.38.2 $(\ifrac{{d}^{2}}{{dx}^{2}})\mathop{\ln\/}\nolimits V_{n}(x)=V_{n+1}(x)+V_{n-1}% (x)-2V_{n}(x),$ $n=1,2,\dots$.

### Complex Function Theory

 18.38.3 $\sum_{m=0}^{n}\mathop{P^{(\alpha,0)}_{m}\/}\nolimits\!\left(x\right)\geq 0,$ $-1\leq x\leq 1$, $\alpha>-1$, $n=0,1,\dots$,

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).

### Zonal Spherical Harmonics

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.

### Random Matrix Theory

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).

### Riemann–Hilbert Problems

See Deift (1998, Chapter 7) and Ismail (2005, Chapter 22).

For applications of Krawtchouk polynomials $\mathop{K_{n}\/}\nolimits\!\left(x;p,N\right)$ and $q$-Racah polynomials $\mathop{R_{n}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).