# 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
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)). …
##### 6: 18.30 Associated OP’s
###### Associated Legendre Polynomials
For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12); on associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). …
###### §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: 18.1 Notation
###### Wilson Class OP’s
• Al-SalamChihara: $Q_{n}\left(x;a,b\,|\,q\right)$.

• Nor do we consider the shifted Jacobi polynomials: …
##### 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
###### §18.2(ii) $x$-Difference Operators
This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials18.20(i)). …