Jacobi fraction

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11: 18.30 Associated OP’s

… ►Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. … ►

§18.30(i) Associated Jacobi Polynomials

… ►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). For other results for associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). … ►See Ismail (2009, p. 46 ), where the $k$th corecursive polynomial is also related to an appropriate continued fraction, given here as its $n$th convergent, …

12: Bibliography J

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Fundamenta Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumptibus fratrum Bornträger.

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Notes:

Reprinted in Werke, Vol. I, pp. 49–239, Herausgegeben von C. W. Borchardt, Berlin, Verlag von G. Reimer, 1881.

External Links:

Werke TOC

Referenced by:

§27.13(iv).

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Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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External Links:

MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§8.25(iv).

Encodings:

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Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

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External Links:

FullText, MathReview (N. N. Abdelmalek), ZentralBlatt

Referenced by:

§3.10(iii).

Encodings:

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Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

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External Links:

ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

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13: 19.5 Maclaurin and Related Expansions

… ► ►For Jacobi’s nome $q$: ►

19.5.5 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right)=r+8r^2+84r^3+992r^4+\mathrm{\cdots },$$ $r=\frac{1}{16}{k}^2$, $0\le k\le 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function, $q$: nome and $k$: real or complex modulus

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

… ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). …

14: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

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

Referenced by:

§18.18(v).

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

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15: 31.11 Expansions in Series of Hypergeometric Functions

… ►In this case the accessory parameter $q$ is a root of the continued-fraction equation … ►In each case $P_j^6$ can be expressed in terms of a Jacobi polynomial (§18.3). …

16: 15.9 Relations to Other Functions

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Jacobi

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§15.9(ii) Jacobi Function

… ►The Jacobi transform is defined as … … ►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …

17: Bibliography L

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Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.

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Referenced by:

§3.10(iii), §3.10(iii), Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).

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Co-recursive associated Jacobi polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 203–213.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

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Continued Fractions with Applications. Studies in Computational Mathematics, North-Holland Publishing Co., Amsterdam.

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External Links:

ISBN 0-444-89265-6, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iv), §1.12(v), §15.7, §18.13, §3.10(ii), §3.10(iii), §3.11(iv), §4.25, §4.39, §4.9(i), §4.9(ii), §5.10, §6.9, §7.18(v), §7.9.

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Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.

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External Links:

Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

Encodings:

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Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

2nd item.

Encodings:

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