orthogonal polynomials and other functions

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11: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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Legendre, Ultraspherical, and Jacobi

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Jacobi $\to$ Laguerre

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Laguerre $\to$ Hermite

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

12: 18.20 Hahn Class: Explicit Representations

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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13: Bibliography I

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On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:

ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

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An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.

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External Links:

ISSN 0030-8730, FullText, MathReview (T. Erber), ZentralBlatt

Referenced by:

§18.39(v).

Encodings:

BibTeX

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More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.

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External Links:

ISSN 0163-0563, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.39(v).

Encodings:

BibTeX

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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External Links:

ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt

Referenced by:

§1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

BibTeX

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14: 18.26 Wilson Class: Continued

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§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

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§18.26(iv) Generating Functions

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15: 18.36 Miscellaneous Polynomials

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§18.36(ii) Sobolev Orthogonal Polynomials

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§18.36(iii) Multiple Orthogonal Polynomials

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§18.36(iv) Orthogonal Matrix Polynomials

… ►implying that, for $n\ge k$, the orthogonality of the $L_n^{(-k)}\left(x\right)$ with respect to the Laguerre weight function $x^{-k}{\mathrm{e}}^{-x}$, $x\in [0,\mathrm{\infty })$. … ►

§18.36(vi) Exceptional Orthogonal Polynomials

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16: 18.33 Polynomials Orthogonal on the Unit Circle

§18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(i) Definition

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Szegő–Askey

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§18.33(v) Biorthogonal Polynomials on the Unit Circle

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Recurrence Relations

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17: 18.5 Explicit Representations

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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18: 15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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15.9.10 $$P_n^{(\lambda )}(x;\varphi )=\frac{{\left(2\lambda \right)}_n}{n!}{\mathrm{e}}^{n\mathrm{i}\varphi }F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}x}{2\lambda };1-{\mathrm{e}}^{-2\mathrm{i}\varphi }).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

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19: 18.30 Associated OP’s

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§18.30(i) Associated Jacobi Polynomials

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20: 18.2 General Orthogonal Polynomials

… ►If polynomials $p_n(x)$ are generated by recurrence relation (18.2.8) under assumption of inequality (18.2.9_5) (or similarly for the other three forms) then the $p_n(x)$ are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function $\alpha (x)$ on $(a,b)$ yielding the orthogonality realtion (18.2.4_5) is guaranteed. …