semi-classical orthogonal polynomials

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11: 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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Hahn $\to$ Jacobi

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Meixner $\to$ Laguerre

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12: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

►For orthogonal polynomials see Chapter 18. …

13: Roelof Koekoek

… ►Koekoek is mainly a teacher of mathematics and has published a few papers on orthogonal polynomials. He is also author of the book Hypergeometric Orthogonal Polynomials and Their $q$-Analogues (with P. … ►

18 Orthogonal Polynomials

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14: 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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15: 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 $H_n\left(x\right)$. … ►For $P_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ see §3.5(v). …

16: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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§18.7(i) Linear Transformations

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Legendre, Ultraspherical, and Jacobi

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§18.7(ii) Quadratic Transformations

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§18.7(iii) Limit Relations

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17: 18.36 Miscellaneous Polynomials

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§18.36(ii) Sobolev Orthogonal Polynomials

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§18.36(iii) Multiple Orthogonal Polynomials

►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►

§18.36(iv) Orthogonal Matrix Polynomials

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§18.36(vi) Exceptional Orthogonal Polynomials

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18: 32.15 Orthogonal Polynomials

§32.15 Orthogonal Polynomials

►Let $p_n(\xi )$, $n=0,1,\mathrm{\dots }$, be the orthonormal set of polynomials defined by ►

32.15.1 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{4}{\xi }^4-z{\xi }^2\right)p_m(\xi )p_n(\xi )d\xi ={\delta }_{m,n},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $m$: integer, $n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

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32.15.2 $$a_{n+1}(z)p_{n+1}(\xi )=\xi p_n(\xi )-a_n(z)p_{n-1}(\xi ),$$

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Symbols:

$n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

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19: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

20: 18.38 Mathematical Applications

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Quadrature

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Riemann–Hilbert Problems

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Radon Transform

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Group Representations

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Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

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