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1: 18.19 Hahn Class: Definitions

§18.19 Hahn Class: Definitions

… ►These eight further families can be grouped in two classes of OP’s: ►

Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ 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). ►

Hahn, Krawtchouk, Meixner, and Charlier

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2: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

3: 22.22 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C13). …

4: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

5: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

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6: 18.25 Wilson Class: Definitions

§18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. …The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn). … ►

§18.25(ii) Weights and Standardizations: Continuous Cases

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Table 18.25.2: Wilson class OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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7: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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§18.21(i) Dualities

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§18.21(ii) Limit Relations and Special Cases

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Figure 18.21.1: Askey scheme. … Magnify

8: 18.1 Notation

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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Hahn Class OP’s

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Wilson Class OP’s

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$q$-Hahn Class OP’s

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Askey–Wilson Class OP’s

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9: 16.27 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C19). …

10: 23.24 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C15). …