# continuous q-ultraspherical 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
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, the Chebyshev polynomials $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, the Legendre polynomials $P_{n}\left(x\right)$, the Laguerre polynomials $L_{n}\left(x\right)$, and the Hermite polynomials $H_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … For another version of the discrete orthogonality property of the polynomials $T_{n}\left(x\right)$ see (3.11.9). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
##### 5: 18.30 Associated OP’s
###### §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. …
##### 6: 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: $h_{n}\left(x\,|\,q\right)$

• ##### 7: 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).
##### 9: 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. …
###### §18.28(vii) Continuous$q^{-1}$-Hermite Polynomials
For continuous $q^{-1}$-Hermite polynomials the orthogonality measure is not unique. …