asymptotics of Jacobi polynomials

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1: 18.15 Asymptotic Approximations

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§18.15(i) Jacobi

… ►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 $z=\pm 1$. …This reference also supplies asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ for large $n$, fixed $\alpha$, and $0\le \beta /n\le c$. … ►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). The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7). …

2: 18.16 Zeros

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Asymptotic Behavior

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3: 15.12 Asymptotic Approximations

… ►See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1). …

4: Bibliography W

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Asymptotic monotonicity of the relative extrema of Jacobi polynomials. Canad. J. Math. 46 (6), pp. 1318–1337.

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External Links:

ISSN 0008-414X, MathReview (S. I. Drobnies), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

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External Links:

ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

BibTeX

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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.

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External Links:

ISSN 1364-5021, Document, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

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5: 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. …

6: Bibliography E

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Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

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7: Bibliography I

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Asymptotics of the Askey-Wilson and $q$-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

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External Links:

ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

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8: Bibliography F

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A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math. 37 (5), pp. 979–1007.

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External Links:

ISSN 0008-414X, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.15(i), §18.16(ii).

Encodings:

BibTeX

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9: Bibliography B

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The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

BibTeX

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10: Bibliography D

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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§15.12(iii), §18.15(i), §18.15(vi).

Encodings:

BibTeX

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