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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

… ► … ► … ►For an example, see Figure 1.4.1 … ►If ${\varphi }^{\prime }(x)$ is continuous or piecewise continuous, then …

3: 18.30 Associated OP’s

… ►In the recurrence relation (18.2.8) assume that the coefficients $A_n$, $B_n$, and $C_{n+1}$ are defined when $n$ is a continuous nonnegative real variable, and let $c$ be an arbitrary positive constant. … ►

18.30.1 $$A_n{A}_{n+1}{C}_{n+1}>0,$$ $n\ge 0$.

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Symbols:

$n$: continuous nonnegative real, $A_n$: coefficient and $C_n$: coefficient

Permalink:

http://dlmf.nist.gov/18.30.E1

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Annotations for §18.30 and Ch.18

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18.30.3 $$p_{n+1}(x;c)=(A_{n+c}x+B_{n+c})p_n(x;c)-C_{n+c}{p}_{n-1}(x;c),$$ $n=0,1,\mathrm{\dots }$.

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $x$: real variable, $n$: continuous nonnegative real, $A_n$: coefficient, $B_n$: coefficient and $C_n$: coefficient

Referenced by:

§18.30, §18.30

Permalink:

http://dlmf.nist.gov/18.30.E3

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Annotations for §18.30 and Ch.18

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18.30.4 $$P_n^{(\alpha ,\beta )}(x;c)=p_n(x;c),$$ $n=0,1,\mathrm{\dots }$,

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Defines:

$P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial

Symbols:

$p_n(x)$: polynomial of degree $n$ , $x$: real variable and $n$: continuous nonnegative real

Permalink:

http://dlmf.nist.gov/18.30.E4

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Annotations for §18.30, §18.30 and Ch.18

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18.30.7 $$P_n(x;c)=\sum _{\mathrm{\ell }=0}^n\frac{c}{\mathrm{\ell }+c}{P}_{\mathrm{\ell }}\left(x\right)P_{n-\mathrm{\ell }}\left(x\right).$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $P_n(x;c)$: associated Legendre polynomial, $\mathrm{\ell }$: nonnegative integer, $x$: real variable and $n$: continuous nonnegative real

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E7

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Annotations for §18.30, §18.30 and Ch.18

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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(vii) Continuous $q^{-1}$-Hermite Polynomials

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18.28.18 $$h_n\left(\sinh t|q\right)=\sum _{\mathrm{\ell }=0}^n{q}^{\frac{1}{2}\mathrm{\ell }(\mathrm{\ell }+1)}\frac{{(q^{-n};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}{\mathrm{e}}^{(n-2\mathrm{\ell })t}={\mathrm{e}}^{nt}{}_1\varphi _1(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q{\mathrm{e}}^{-2t})={\mathrm{i}}^{-n}{H}_n\left(\mathrm{i}\sinh t|q^{-1}\right).$$

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Defines:

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

Symbols:

$H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}\varphi _s(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}\varphi _s(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E18

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Annotations for §18.28(vii), §18.28 and Ch.18

►For continuous $q^{-1}$-Hermite polynomials the orthogonality measure is not unique. …

9: 18.25 Wilson Class: Definitions

… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►

§18.25(ii) Weights and Normalizations: Continuous Cases

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Continuous Dual Hahn

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18.25.6 $$p_n(x)=S_n(x;a_1,a_2,a_3),$$

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Symbols:

$S_n(x;a,b,c)$: continuous dual Hahn polynomial, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable

Referenced by:

§18.25(ii)

Permalink:

http://dlmf.nist.gov/18.25.E6

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Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

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

10: 18.21 Hahn Class: Interrelations

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Continuous Hahn $\to$ Meixner–Pollaczek

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18.21.10 $$\underset{t\to \mathrm{\infty }}{lim}{t}^{-n}{p}_n(x-t;\lambda +\mathrm{i}t,-t\tan \varphi ,\lambda -\mathrm{i}t,-t\tan \varphi )=\frac{{(-1)}^n}{{(\cos \varphi )}^n}{P}_n^{(\lambda )}(x;\varphi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.22(ii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.21.E10

Encodings:

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See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.11 $$p_n(x;a,a+\frac{1}{2},a,a+\frac{1}{2})=2^{-2n}{\left(4a+n\right)}_n{P}_n^{(2a)}(2x;\frac{1}{2}\pi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E11

Encodings:

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See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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Figure 18.21.1: Askey scheme. …(This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify