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1: 32.16 Physical Applications

§32.16 Physical Applications

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Integrable Continuous Dynamical Systems

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2: 1.4 Calculus of One Variable

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§1.4(ii) Continuity

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Absolutely Continuous Stieltjes Measure

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3: 18.25 Wilson Class: Definitions

… ►The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn). … ►Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is $(0,\mathrm{\infty })\cup S$, where $S$ is a specific finite set, e. … ►

§18.25(ii) Weights and Standardizations: Continuous Cases

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Continuous Dual Hahn

… ►Table 18.25.2 provides the leading coefficients $k_n$ (§18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …

4: About Color Map

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Continuous Phase Mapping

►For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. … ►

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Figure 3: Continuous phase mapping

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5: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►Ismail (1986) gives asymptotic expansions as $n\to \mathrm{\infty }$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

6: 18.1 Notation

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Continuous Hahn: $p_n(x;a,b,\overline{a},\overline{b})$.

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Continuous Dual Hahn: $S_n(x;a,b,c)$.

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Continuous $q$-Ultraspherical: $C_n(x;\beta |q)$.

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Continuous $q$-Hermite: $H_n\left(x|q\right)$.

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Continuous $q^{-1}$-Hermite: $h_n\left(x|q\right)$

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7: 18.26 Wilson Class: Continued

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Wilson $\to$ Continuous Dual Hahn

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Wilson $\to$ Continuous Hahn

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Continuous Dual Hahn $\to$ Meixner–Pollaczek

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Continuous Dual Hahn

… ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …

8: 18.28 Askey–Wilson Class

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§18.28(v) Continuous $q$-Ultraspherical Polynomials

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§18.28(vi) Continuous $q$-Hermite Polynomials

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§18.28(ix) Continuous $q$-Jacobi Polynomials

… ►Specialization to continuous $q$-ultraspherical: … ►

From Continuous $q$-Ultraspherical to Continuous $q$-Hermite

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9: 18.19 Hahn Class: Definitions

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Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

… ►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). … ►

Continuous Hahn

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10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►When $\alpha$ is absolutely continuous, i. … ►

§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

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§1.18(vii) Continuous Spectra: More General Cases

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