continuous q-Hermite polynomials

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11: 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

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

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

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12: 18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(i) Recurrence Relations in $n$

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Continuous Hahn

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Continuous Hahn

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§18.22(iii) $x$-Differences

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Continuous Hahn

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13: 18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas

… ►For the Hahn polynomials $p_n(x)=Q_n(x;\alpha ,\beta ,N)$ and … ►

Continuous Hahn

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

… ►(For symmetry properties of $p_n(x;a,b,\overline{a},\overline{b})$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).) …

14: 32.16 Physical Applications

§32.16 Physical Applications

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Integrable Continuous Dynamical Systems

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15: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

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Hahn

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Continuous Hahn

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18.23.6 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a+\mathrm{i}x}{2\mathrm{\Re }a};-\mathrm{i}z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\overline{b}-\mathrm{i}x}{2\mathrm{\Re }b};\mathrm{i}z)=\sum _{n=0}^{\mathrm{\infty }}\frac{p_n(x;a,b,\overline{a},\overline{b})}{{\left(2\mathrm{\Re }a\right)}_n{\left(2\mathrm{\Re }b\right)}_n}{z}^n.$$

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Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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18.23.7 $${(1-{\mathrm{e}}^{\mathrm{i}\varphi }z)}^{-\lambda +\mathrm{i}x}{(1-{\mathrm{e}}^{-\mathrm{i}\varphi }z)}^{-\lambda -\mathrm{i}x}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(x;\varphi )z^n,$$ .

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

16: 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

… ►and completeness relation … ► … ►

§1.18(vii) Continuous Spectra: More General Cases

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17: 1.17 Integral and Series Representations of the Dirac Delta

… ►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that … ►for all functions $\varphi (x)$ that are continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►provided that $\varphi (x)$ is continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). … ►provided that $\varphi (x)$ is continuous and of period $2\pi$; see §1.8(ii). …

18: 28.9 Zeros

… ►They are continuous in $q$. For $q\to \mathrm{\infty }$ the zeros of ${\mathrm{ce}}_{2n}(z,q)$ and ${\mathrm{se}}_{2n+1}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n}\left(q^{1/4}(\pi -2z)\right)$, and the zeros of ${\mathrm{ce}}_{2n+1}(z,q)$ and ${\mathrm{se}}_{2n+2}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n+1}\left(q^{1/4}(\pi -2z)\right)$. Here ${\mathrm{𝐻𝑒}}_n\left(z\right)$ denotes the Hermite polynomial of degree $n$ (§18.3). …

19: 18.18 Sums

… ►Alternatively, assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(-1,1)$. … ►Assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(0,\mathrm{\infty })$. … ►Assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(-\mathrm{\infty },\mathrm{\infty })$. … ►

Ultraspherical

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Legendre

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20: 18.37 Classical OP’s in Two or More Variables

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§18.37(i) Disk Polynomials

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Definition in Terms of Jacobi Polynomials

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Definition in Terms of Jacobi Polynomials

… ►In one variable they are essentially ultraspherical, Jacobi, continuous $q$-ultraspherical, or Askey–Wilson polynomials. …For general $q$ they occur as Macdonald polynomials for root system $A_n$ , as Macdonald polynomials for general root systems, and as Macdonald–Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992).