Stieltjes fraction (S-fraction)
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11: 18.29 Asymptotic Approximations for -Hahn and Askey–Wilson Classes
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18.29.2
; fixed.
►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006).
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12: 1.4 Calculus of One Variable
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Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals
… ► … ► … ► … ► See Riesz and Sz.-Nagy (1990, Ch. 3). …13: Bibliography
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Zeros of Stieltjes and Van Vleck polynomials and applications.
J. Math. Anal. Appl. 110 (2), pp. 327–339.
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Zeros of Stieltjes and Van Vleck polynomials.
Trans. Amer. Math. Soc. 252, pp. 197–204.
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Recurrence relations, continued fractions, and orthogonal polynomials.
Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
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14: 1.14 Integral Transforms
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§1.14(vi) Stieltjes Transform
►The Stieltjes transform of a real-valued function is defined by … … ►Inversion
… ►Laplace Transform
…15: Bibliography H
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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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Definitions of Stieltjes Integrals of the Riemann Type.
Amer. Math. Monthly 45 (5), pp. 265–278.
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16: 6.9 Continued Fraction
§6.9 Continued Fraction
…17: 18.13 Continued Fractions
18: Bibliography M
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Explicit error terms for asymptotic expansions of Stieltjes transforms.
J. Inst. Math. Appl. 22 (2), pp. 129–145.
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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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An Introduction to the Fractional Calculus and Fractional Differential Equations.
A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.
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A new Stirling series as continued fraction.
Numer. Algorithms 56 (1), pp. 17–26.
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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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19: 18.2 General Orthogonal Polynomials
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►More generally than (18.2.1)–(18.2.3), may be replaced in (18.2.1) by , where the measure is the Lebesgue–Stieltjes measure corresponding to a bounded nondecreasing function on the closure of with an infinite number of points of increase, and such that for all .
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§18.2(x) Orthogonal Polynomials and Continued Fractions
… ►Using the terminology of §1.12(ii), the -th approximant of the continued fraction …20: 13.5 Continued Fractions
§13.5 Continued Fractions
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13.5.1
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►This continued fraction converges to the meromorphic function of on the left-hand side everywhere in .
For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).
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►This continued fraction converges to the meromorphic function of on the left-hand side throughout the sector .
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