relation to classical theta functions

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11: 18.37 Classical OP’s in Two or More Variables

§18.37 Classical OP’s in Two or More Variables

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18.37.1 $$R_{m,n}^{(\alpha )}\left(r{\mathrm{e}}^{\mathrm{i}\theta }\right)={\mathrm{e}}^{\mathrm{i}(m-n)\theta }{r}^{|m-n|}\frac{P_{\min (m,n)}^{(\alpha ,|m-n|)}\left(2r^2-1\right)}{P_{\min (m,n)}^{(\alpha ,|m-n|)}\left(1\right)},$$ $r\ge 0$, $\theta \in ℝ$, $\alpha >-1$.

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Defines:

$R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $ℝ$: real line, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.37(ii)

Permalink:

http://dlmf.nist.gov/18.37.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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Definition in Terms of Jacobi Polynomials

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12: Bibliography I

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Theta Functions. Springer-Verlag, New York.

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Notes:

Die Grundlehren der mathematischen Wissenschaften, Band 194

External Links:

ISBN 3540056998, MathReview (H. Klingen), ZentralBlatt

Referenced by:

§21.5(ii), §21.8, Ch.21.

Encodings:

BibTeX

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A Classical Introduction to Modern Number Theory. 2nd edition, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97329-X, MathReview (Glenn Stevens), ZentralBlatt

Referenced by:

§24.10(i), §24.10(ii), §24.10(iii), §24.17(iii).

Encodings:

BibTeX

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Special Functions, $q$-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

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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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:

ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

Encodings:

BibTeX

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