Stieltjes fraction (S-fraction)

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1: 3.10 Continued Fractions

§3.10 Continued Fractions

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Stieltjes Fractions

… ►is called a Stieltjes fraction ($S$-fraction). … ►

Jacobi Fractions

… ►For the same function $f(z)$, the convergent $C_n$ of the Jacobi fraction (3.10.11) equals the convergent $C_{2n}$ of the Stieltjes fraction (3.10.6). …

2: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis. … ►

Stieltjes Inversion via (approximate) Analytic Continuation

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Histogram Approach

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Derivative Rule Approach

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3: 2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of $f(t)$ is defined by … ►For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …

4: 12.6 Continued Fraction

§12.6 Continued Fraction

►For a continued-fraction expansion of the ratio $U(a,x)/U(a-1,x)$ see Cuyt et al. (2008, pp. 340–341).

5: 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.

External Links:

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

Referenced by:

6th item.

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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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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6: 31.15 Stieltjes Polynomials

§31.15 Stieltjes Polynomials

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§31.15(ii) Zeros

… ►This is the Stieltjes electrostatic interpretation. … ►

§31.15(iii) Products of Stieltjes Polynomials

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7: Bibliography J

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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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8: 10.55 Continued Fractions

§10.55 Continued Fractions

►For continued fractions for ${𝗃}_{n+1}\left(z\right)/{𝗃}_n\left(z\right)$ and ${𝗂}_{n+1}^{(1)}\left(z\right)/{𝗂}_n^{(1)}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).

9: Bibliography W

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Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.

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Notes:

Reprinted by Chelsea Publishing, New York, 1973 and AMS Chelsea Publishing, Providence, RI, 2000.

External Links:

ISBN 0-8218-2106-7, MathReview (R. C. Buck), ZentralBlatt

Referenced by:

§3.10(iii), §4.25, §4.9(i), §4.9(ii), Ch.4, §5.10.

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

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Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions. Proc. Roy. Soc. London Ser. A 458, pp. 625–644.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

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10: 25.2 Definition and Expansions

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25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

►where the Stieltjes constants ${\gamma }_n$ are defined via ►

25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

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Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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