# continuous q-Hermite polynomials

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##### 1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
###### §31.5 Solutions Analytic at Three Singularities: Heun Polynomials
is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. …
##### 3: 24.1 Special Notation
###### Bernoulli Numbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
###### Euler Numbers and Polynomials
The notations $E_{n}$, $E_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
##### 4: 18.3 Definitions
###### §18.3 Definitions
For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …
###### Bessel polynomials
Bessel polynomials are often included among the classical OP’s. …
##### 5: 18.1 Notation
• Continuous Hahn: $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$.

• Continuous Dual Hahn: $S_{n}\left(x;a,b,c\right)$.

• Continuous $q$-Ultraspherical: $C_{n}\left(x;\beta\,|\,q\right)$.

• Continuous $q$-Hermite: $H_{n}\left(x\,|\,q\right)$.

• Continuous $q^{-1}$-Hermite: $h_{n}\left(x\,|\,q\right)$

• ##### 6: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
###### §18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
Ismail (1986) gives asymptotic expansions as $n\to\infty$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)$ the leading term is given by … 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).
##### 7: 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 $W_{n}\left(x;a,b,c,d\right)$, continuous dual Hahn polynomials $S_{n}\left(x;a,b,c\right)$, Racah polynomials $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$, and dual Hahn polynomials $R_{n}\left(x;\gamma,\delta,N\right)$. … Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is $(0,\infty)\cup S$, where $S$ is a specific finite set, e. …
###### 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. …
##### 9: 18.19 Hahn Class: Definitions
###### §18.19 Hahn Class: Definitions
• 1.

Hahn class (or linear lattice class). These are OP’s $p_{n}(x)$ where the role of $\frac{\mathrm{d}}{\mathrm{d}x}$ is played by $\Delta_{x}$ 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.

• The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). …
##### 10: 18.26 Wilson Class: Continued
###### 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. …