continuous q-Hermite polynomials

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1: 18.28 Askey–Wilson Class

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18.28.16 $$H_n\left(\cos \theta |q\right)=\sum _{\mathrm{\ell }=0}^n\frac{{(q;q)}_n{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta }}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}={\mathrm{e}}^{\mathrm{i}n\theta }{}_2\varphi _0(\genfrac{}{}{0pt}{}{q^{-n},0}{-};q,q^n{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

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Defines:

$H_n\left(x|q\right)$: continuous $q$-Hermite polynomial

Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: 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.E16

Encodings:

pMML, png, TeX

See also:

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

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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

Encodings:

pMML, png, TeX

See also:

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

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

2: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►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).

3: 18.1 Notation

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

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4: 28.9 Zeros

… ►For real $q$ each of the functions ${\mathrm{ce}}_{2n}(z,q)$, ${\mathrm{se}}_{2n+1}(z,q)$, ${\mathrm{ce}}_{2n+1}(z,q)$, and ${\mathrm{se}}_{2n+2}(z,q)$ has exactly $n$ zeros in . 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 ${\mathit{He}}_{2n}\left(q^{1/4}(\pi -2z)\right)$, and the zeros of ${\mathrm{ce}}_{2n+1}(z,q)$ and ${\mathrm{se}}_{2n+2}(z,q)$ approach asymptotically the zeros of ${\mathit{He}}_{2n+1}\left(q^{1/4}(\pi -2z)\right)$. Here ${\mathit{He}}_n\left(z\right)$ denotes the Hermite polynomial of degree $n$ (§18.3). Furthermore, for $q>0$ ${\mathrm{ce}}_m(z,q)$ and ${\mathrm{se}}_m(z,q)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of $J_m\left(2\sqrt{q}\cos z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. …