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31: Bibliography M

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Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.

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External Links:

ISSN 0020-2932, Document, MathReview, ZentralBlatt

Referenced by:

§2.6(iii).

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

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A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

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ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

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A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

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ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt

Referenced by:

§5.10.

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32: Bibliography L

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Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

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External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

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

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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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Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

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The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

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MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

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33: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.

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MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§31.12.

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Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.

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ISSN 0037-959X, MathReview, ZentralBlatt

Referenced by:

§31.12.

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The KP equation with quasiperiodic initial data. Phys. D 123 (1-4), pp. 123–152.

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Notes:

Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997)

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ISSN 0167-2789, Document, MathReview (Pavel Krejčí), ZentralBlatt

Referenced by:

§21.9.

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The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.

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ISSN 0031-8914, Document, Link

Referenced by:

§18.40(ii).

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34: 18.30 Associated OP’s

… ►Assuming equation (18.2.8) with its initialization defines a set of OP’s, $p_n(x)$, the corresponding associated orthogonal polynomials of order $c$ are the $p_n(x;c)$ as defined by shifting the index $n$ in the recurrence coefficients by adding a constant $c$, functions of $n$, say $f(n)$, being replaced by $f(n+c)$. … ►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. … ►The $p_n^{(0)}(x)$ are also referred to as the numerator polynomials, the $p_n(x)$ then being the denominator polynomials, in that the $n$-th approximant of the continued fraction, $z\in ℂ$, … ►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, …

35: Bibliography R

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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.

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

Referenced by:

6th item.

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Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

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External Links:

ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§12.11(ii), §6.13.

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

Referenced by:

§10.23(ii).

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36: 8.19 Generalized Exponential Integral

… ►with $|\mathrm{ph}z|\le \pi$ in both equations. … ►again with $|\mathrm{ph}z|\le \pi$ in both equations. … ►

§8.19(vii) Continued Fraction

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37: Bibliography H

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Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

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Notes:

A reprinting with corrections appeared in 2000.

External Links:

ISBN 3-540-56670-8, MathReview, ZentralBlatt

Referenced by:

§3.7(v), §3.7(i).

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Applied and Computational Complex Analysis. Vol. 2: Special Functions—Integral Transforms—Asymptotics—Continued Fractions. Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1991.

External Links:

ISBN 0-471-01525-3, MathReview (A. E. Heins), ZentralBlatt

Referenced by:

§1.11(v), Ch.3.

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Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.

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MathReview (Henrik Pedersen), ZentralBlatt

Referenced by:

§32.9(iii).

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Estimates of the stability intervals for Hill’s equation. Proc. Amer. Math. Soc. 14 (6), pp. 930–932.

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FullText, MathReview (P. Ungar), ZentralBlatt

Referenced by:

§28.29(iii).

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Differential Equations: A Modern Approach. Holt, Rinehart and Winston, New York.

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Notes:

Reprinted by Dover in 1975.

External Links:

MathReview (Z. Opial), ZentralBlatt

Referenced by:

§29.3(vi).

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38: 3.3 Interpolation

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3.3.3_1 $${\tilde{\omega }}_k=\frac{1}{{\omega }_{n+1}^{\prime }(z_k)}=\underset{j=0\phantom{\prime }}{\overset{n\phantom{\prime }}{{\prod }^{\prime }}}\frac{1}{z_k-z_j}.$$

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Symbols:

$z_k$: nodes of function and ${\omega }_{n+1}(z)$: nodal polynomial

Referenced by:

§3.3(i), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/3.3.E3_1

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pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §3.3(i), §3.3 and Ch.3

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3.3.3_2 $$1=\sum _{k=0}^n{\mathrm{\ell }}_k(z)={\omega }_{n+1}(z)\sum _{k=0}^n\frac{{\tilde{\omega }}_k}{z-z_k},$$

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Symbols:

$z_k$: nodes of function, ${\mathrm{\ell }}_k(z)$: Lagrange interpolation polynomial and ${\omega }_{n+1}(z)$: nodal polynomial

Referenced by:

§3.3(i), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/3.3.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §3.3(i), §3.3 and Ch.3

… ►If $f$ and the $z_k$ ($=x_k$) are real, and $f$ is $n$ times continuously differentiable on a closed interval containing the $x_k$, then … ►It can be used for solving a nonlinear scalar equation $f(z)=0$ approximately. … ►For Hermite interpolation, trigonometric interpolation, spline interpolation, rational interpolation (by using continued fractions), interpolation based on Chebyshev points, and bivariate interpolation, see Bulirsch and Rutishauser (1968), Davis (1975, pp. 27–31), and Mason and Handscomb (2003, Chapter 6). …

39: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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§17.6(iv) Differential Equations

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$q$-Differential Equation

… ►(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions $a\to q^a$, $b\to q^b$, $c\to q^c$, followed by ${lim}_{q\to 1-}$. … ►

§17.6(vi) Continued Fractions

►For continued-fraction representations of the ${}_2{}^{}\varphi _1^{}$ function, see Cuyt et al. (2008, pp. 395–399).