# q-Hahn class orthogonal polynomials

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##### 2: 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
• 3.

As given by a Rodrigues formula (18.5.5).

• Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). … 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). … For another version of the discrete orthogonality property of the polynomials $T_{n}\left(x\right)$ see (3.11.9). …
##### 7: 18.19 Hahn Class: Definitions
###### §18.19 HahnClass: Definitions
These eight further families can be grouped in two classes of OP’s: …
###### Hahn, Krawtchouk, Meixner, and Charlier
These polynomials are orthogonal on $(-\infty,\infty)$, and are defined as follows. …A special case of (18.19.8) is $w^{(1/2)}(x;\pi/2)=\frac{\pi}{\cosh\left(\pi x\right)}$.
##### 8: 18.24 Hahn Class: Asymptotic Approximations
###### §18.24 HahnClass: Asymptotic Approximations
For an asymptotic expansion of $P^{(\lambda)}_{n}\left(nx;\phi\right)$ as $n\to\infty$, with $\phi$ fixed, see Li and Wong (2001). …Corresponding approximations are included for the zeros of $P^{(\lambda)}_{n}\left(nx;\phi\right)$.
###### Approximations in Terms of Laguerre Polynomials
Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.
##### 9: 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).
##### 10: 18.42 Software
For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3).
• CAOP (website). Computer Algebra and Orthogonal Polynomials.