classical orthogonal polynomials

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21: 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

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See also:

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

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22: Bibliography R

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The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.

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External Links:

MathReview (Wadim Zudilin), ZentralBlatt

Referenced by:

§18.30(viii).

Encodings:

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23: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

… ►See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).

24: 18.12 Generating Functions

§18.12 Generating Functions

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25: Bibliography I

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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:

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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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Notes:

With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.

External Links:

ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt

Referenced by:

§18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

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26: 3.5 Quadrature

… ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►The monic version $p_n(x)$ and orthonormal version $q_n(x)$ of a classical orthogonal polynomial are obtained by dividing the orthogonal polynomial by $k_n$ respectively $\sqrt{h_n}$, with $k_n$ and $h_n$ as in Table 18.3.1. Below we give for the classical orthogonal polynomials the recurrence coefficients ${\alpha }_n$ and ${\beta }_n$ in (3.5.30). … ►

Table 3.5.17_5: Recurrence coefficients in (3.5.30) and (3.5.30_5) for monic versions $p_n(x)$ and orthonormal versions $q_n(x)$ of the classical orthogonal polynomials.

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$p_n(x)$	$q_n(x)$	${\alpha }_n$	${\beta }_n$	$h_0$
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27: Bibliography G

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Positive sums of the classical orthogonal polynomials. SIAM J. Math. Anal. 8 (3), pp. 423–447.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (W. A. Al-Salam)

Referenced by:

(18.14.25), (18.14.26), (18.14.27).

Encodings:

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Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation, Oxford University Press, New York.

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Notes:

OPQ, a package of Matlab routines for generating classical and Sobolev orthogonal polynomials, accompanies this book.

External Links:

OPQ, ISBN 0-19-850672-4, MathReview Entry, ZentralBlatt

Referenced by:

(18.2.27), (18.2.28), (18.2.29), (18.2.30), §18.2(ix), §18.40(ii), §18.40(ii), §18.40(i), 3rd item, §3.5(v), Erratum (V1.0.28) for Table 18.3.1.

Encodings:

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28: Bibliography D

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Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162 (10), pp. 1793–1804.

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External Links:

ISSN 0021-9045, Document, FullText, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

(18.16.12), (18.16.13), §18.16(ii), §18.16(ii), §18.16(iv).

Encodings:

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Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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External Links:

ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

Encodings:

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29: Bibliography

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Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(18.27.12_5), (18.27.6).

Encodings:

BibTeX

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30: Bibliography K

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Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.

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External Links:

MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.37(ii).

Encodings:

BibTeX

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