Hahn class orthogonal polynomials

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1: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

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

2: 18.19 Hahn Class: Definitions

§18.19 Hahn Class: Definitions

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Hahn, Krawtchouk, Meixner, and Charlier

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Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

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	$p_n(x)$	$X$	$w_x$	$h_n$
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Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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… ►A special case of (18.19.8) is $w^{(1/2)}(x;\pi /2)=\frac{\pi }{\cosh \left(\pi x\right)}$.

3: 18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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4: 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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5: 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

6: 18.27 $q$-Hahn Class

§18.27 $q$-Hahn Class

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§18.27(ii) $q$-Hahn Polynomials

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§18.27(iii) Big $q$-Jacobi Polynomials

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§18.27(iv) Little $q$-Jacobi Polynomials

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Limit Relations

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7: 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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8: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

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Hahn

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9: 18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(i) Recurrence Relations in $n$

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§18.22(ii) Difference Equations in $x$

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§18.22(iii) $x$-Differences

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10: 18.38 Mathematical Applications

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