complex orthogonal polynomials

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31: 18.11 Relations to Other Functions

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Ultraspherical

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Laguerre

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§18.11(ii) Formulas of Mehler–Heine Type

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Jacobi

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Laguerre

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32: 18.35 Pollaczek Polynomials

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18.35.7 $${(1-z{\mathrm{e}}^{\mathrm{i}\theta })}^{-\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{(1-z{\mathrm{e}}^{-\mathrm{i}\theta })}^{-\lambda -\mathrm{i}{\tau }_{a,b}(\theta )}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(\cos \theta ;a,b)z^n,$$ , .

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Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.8))(proved)

Permalink:

http://dlmf.nist.gov/18.35.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

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33: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

… ►See Hahn (1980), where corresponding results are given when $x$ is replaced by a complex variable $z$ that is bounded away from the orthogonality interval $[-1,1]$. … ►For large $\beta$, fixed $\alpha$, and $0\le n/\beta \le c$, Dunster (1999) gives asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ that are uniform in unbounded complex $z$-domains containing $z=\pm 1$. … ►For an asymptotic expansion of $P_n^{(\alpha ,\beta )}\left(z\right)$ as $n\to \mathrm{\infty }$ that holds uniformly for complex $z$ bounded away from $[-1,1]$, see Elliott (1971). … ►See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).

34: Bibliography C

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Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:

Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).

External Links:

CAOP

Referenced by:

1st item.

Encodings:

BibTeX

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Asymptotics of the largest zeros of some orthogonal polynomials. J. Phys. A 31 (25), pp. 5525–5544.

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External Links:

ISSN 0305-4470, Document, MathReview (Tim H. Baker), ZentralBlatt

Referenced by:

§18.26(v), §18.29.

Encodings:

BibTeX

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An Introduction to Orthogonal Polynomials. Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York.

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External Links:

ISBN 0-677-04150-0, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§1.16(ix), (18.2.33), (18.2.36), (18.2.38), §18.2(v), §18.2(iii), §18.2(iv), §18.2(vii), §18.2(x), §18.20(i), (18.30.25), §18.30(ii), §18.30, §18.40(ii), Ch.18.

Encodings:

BibTeX

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Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

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External Links:

ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt

Referenced by:

§18.28(iv).

Encodings:

BibTeX

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Basic hypergeometric functions and orthogonal Laurent polynomials. Proc. Amer. Math. Soc. 140 (6), pp. 2075–2089.

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External Links:

ISSN 0002-9939, Link, MathReview (Roelof Koekoek), ZentralBlatt

Referenced by:

§18.33(iv).

Encodings:

BibTeX

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35: 18.34 Bessel Polynomials

§18.34 Bessel Polynomials

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§18.34(ii) Orthogonality

… ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : … ►In this limit the finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty })$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty })$ (see (18.34.5_5)). …

36: 28.31 Equations of Whittaker–Hill and Ince

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§28.31(ii) Equation of Ince; Ince Polynomials

… ►When $p$ is a nonnegative integer, the parameter $\eta$ can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. … … ►The normalization is given by … ►More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by …

37: 18.26 Wilson Class: Continued

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§18.26(ii) Limit Relations

… ►See also Figure 18.21.1. ►

§18.26(iii) Difference Relations

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§18.26(iv) Generating Functions

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§18.26(v) Asymptotic Approximations

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38: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

►Ismail (1986) gives asymptotic expansions as $n\to \mathrm{\infty }$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_n(\cos \theta ;a,b,c,d|q)$ the leading term is given by … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

39: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

Encodings:

BibTeX

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Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3446-0, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

(18.33.17), (18.33.18), (18.33.21), (18.33.22), (18.33.23), (18.33.24), (18.33.25), (18.33.26), (18.33.27), (18.33.28), (18.33.30), (18.33.31), (18.33.32), §18.33(vi), §18.33(i), §18.33(vi).

Encodings:

BibTeX

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Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3675-7, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

§18.33(i), §18.33(vi).

Encodings:

BibTeX

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On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, pp. 731–737.

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External Links:

ISSN 0002-9939, Document, Link, MathReview (A. C. Offord)

Referenced by:

§18.35(i).

Encodings:

BibTeX

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Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.

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Notes:

American Mathematical Society Colloquium Publications, Vol. 23

External Links:

MathReview (J. Burlak)

Referenced by:

Ch.14, §9.1, Notations A.

Encodings:

BibTeX

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40: Bibliography L

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Orthogonal polynomials, duality and association schemes. SIAM J. Math. Anal. 13 (4), pp. 656–663.

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External Links:

ISSN 0036-1410, Document, MathReview (C. L. Parihar), ZentralBlatt

Referenced by:

§18.28(viii), §18.28(xi), §18.38(iii).

Encodings:

BibTeX

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Orthogonal Polynomials for Exponential Weights. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 4, Springer-Verlag, New York.

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External Links:

ISBN 0-387-98941-2, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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Orthogonal polynomials for exponential weights $x^{2\rho }{e}^{-2Q(x)}$ on $[0,d)$ . J. Approx. Theory 134 (2), pp. 199–256.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Leonid B. Golinskiĭ)

Referenced by:

§18.32.

Encodings:

BibTeX

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Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

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External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§18.15(vi), §18.16(vi).

Encodings:

BibTeX

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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

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External Links:

ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.34(iii), §24.11, §24.16(i), §24.16(i).

Encodings:

BibTeX

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