# q-zAl-Salam--Chihara 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. …
##### 6: 18.1 Notation
###### Wilson Class OP’s
• Al-SalamChihara: $Q_{n}\left(x;a,b\,|\,q\right)$.

• Nor do we consider the shifted Jacobi polynomials: …
###### §18.28(iv) $q^{-1}$-Al-Salam–ChiharaPolynomials
For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006).
###### §18.28(v) Continuous $q$-Ultraspherical Polynomials
These polynomials are also called Rogers polynomials. …
##### 8: 18.30 Associated OP’s
###### §18.30(ii) Associated Legendre Polynomials
For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12). …
##### 9: 18.38 Mathematical Applications
###### Integrable Systems
Ultraspherical polynomials are zonal spherical harmonics. …
###### Group Representations
For applications of Krawtchouk polynomials $K_{n}\left(x;p,N\right)$ and $q$-Racah polynomials $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987). …
##### 10: 18.2 General Orthogonal Polynomials
###### Kernel Polynomials
See Chihara (1978, Ch. I, §8). … See Chihara (1978, pp. 86–89), and, in slightly different notation, Ismail (2009, §§2.3, 2.6, 2.10), where it is assumed that $\mu_{0}=1$. …