Stieltjes fraction (S-fraction)
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1: 3.10 Continued Fractions
§3.10 Continued Fractions
… ►Stieltjes Fractions
… ►is called a Stieltjes fraction (-fraction). … ►Jacobi Fractions
… ►For the same function , the convergent of the Jacobi fraction (3.10.11) equals the convergent 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
… ►Histogram Approach
… ►Derivative Rule Approach
…3: 2.6 Distributional Methods
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§2.6(ii) Stieltjes Transform
… ►The Stieltjes transform of 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 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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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
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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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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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Partial fractions expansions and identities for products of Bessel functions.
J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
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6: 31.15 Stieltjes Polynomials
§31.15 Stieltjes Polynomials
… ►§31.15(ii) Zeros
… ►This is the Stieltjes electrostatic interpretation. … ►§31.15(iii) Products of Stieltjes Polynomials
…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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Numerical stability in evaluating continued fractions.
Math. Comp. 28 (127), pp. 795–810.
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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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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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8: 10.55 Continued Fractions
§10.55 Continued Fractions
►For continued fractions for and 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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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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Gevrey asymptotics and Stieltjes transforms of algebraically decaying functions.
Proc. Roy. Soc. London Ser. A 458, pp. 625–644.
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