§18.38 Mathematical Applications

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

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

Keywords:: Chebyshev polynomials, classical orthogonal polynomials
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¶ Approximation Theory

Keywords:: Chebyshev polynomials

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).

¶ Quadrature

Keywords:: classical orthogonal polynomials, quadrature

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

Keywords:: Chebyshev polynomials, classical orthogonal polynomials, 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

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¶ Integrable Systems

Keywords:: Bäcklund transformations, Hermite polynomials, Lax pairs, Toda equation, classical orthogonal polynomials, soliton theory

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $n$ : nonnegative integer and $x$ : real variable
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with $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ as in §18.3, satisfies the Toda equation

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : nonnegative integer and $x$ : real variable
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¶ Complex Function Theory

Keywords:: Askey–Gasper inequality, Bieberbach conjecture, Jacobi polynomials, classical orthogonal polynomials

The Askey–Gasper inequality

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

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
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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

Keywords:: spherical harmonics, ultraspherical polynomials, 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

Keywords:: Hermite polynomials, classical orthogonal polynomials, 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

Keywords:: Riemann–Hilbert problems, classical orthogonal polynomials

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

¶ Radon Transform

Keywords:: Radon transform, classical orthogonal polynomials

See Deans (1983, Chapters 4, 7).

§18.38(iii) Other OP’s

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¶ Group Representations

Keywords:: group representations, orthogonal polynomials

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

¶ Coding Theory

Keywords:: Krawtchouk polynomials, coding theory, $q$ -Racah polynomials

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).