orthogonal polynomials and other functions

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1: 18.37 Classical OP’s in Two or More Variables

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Definition in Terms of Jacobi Polynomials

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2: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

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3: 18.1 Notation

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Classical OP’s

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Hahn Class OP’s

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Wilson Class OP’s

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Disk: $R_{m,n}^{(\alpha )}\left(z\right)$.

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Triangle: $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$.

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4: 18.38 Mathematical Applications

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5: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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Hahn $\to$ Jacobi

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Meixner $\to$ Laguerre

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Charlier $\to$ Hermite

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Meixner–Pollaczek $\to$ Laguerre

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6: 18.3 Definitions

§18.3 Definitions

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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 $-1-\beta >\alpha >-1$ a finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ is orthogonal on $(1,\mathrm{\infty })$ with weight function $w(x)={(x-1)}^{\alpha }{(x+1)}^{\beta }$. …

7: 13.18 Relations to Other Functions

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§13.18(v) Orthogonal Polynomials

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8: Richard A. Askey

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. …

9: 13.6 Relations to Other Functions

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§13.6(v) Orthogonal Polynomials

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10: 18.41 Tables

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§18.41(i) Polynomials

►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_n\left(x\right)$ and $U_n\left(x\right)$, and $0.5,1,3,5,10$ for $L_n\left(x\right)$ and $H_n\left(x\right)$. … ►

§18.41(iii) Other Tables

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