continued fraction

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31: 5.15 Polygamma Functions

… ►For $B_{2k}$ see §24.2(i). ►For continued fractions for ${\psi }^{\prime }\left(z\right)$ and ${\psi }^{\prime \prime }\left(z\right)$ see Cuyt et al. (2008, pp. 231–238).

32: 29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …

33: 10.74 Methods of Computation

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§10.74(v) Continued Fractions

►For applications of the continued-fraction expansions (10.10.1), (10.10.2), (10.33.1), and (10.33.2) to the computation of Bessel functions and modified Bessel functions see Gargantini and Henrici (1967), Amos (1974), Gautschi and Slavik (1978), Tretter and Walster (1980), Thompson and Barnett (1986), and Cuyt et al. (2008). …

34: 13.29 Methods of Computation

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§13.29(v) Continued Fractions

►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of $M(n,b,x)$, when $b$ and $x$ are real and $n$ is a positive integer. …

35: 8.17 Incomplete Beta Functions

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§8.17(v) Continued Fraction

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36: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(v) Continued Fraction

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37: 29.3 Definitions and Basic Properties

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§29.3(iii) Continued Fractions

… ►satisfies the continued-fraction equation … ►The continued fraction following the second negative sign on the left-hand side of (29.3.10) is finite: it equals 0 if $p=0$, and if $p>0$, then the last denominator is ${\beta }_0-H$. If $\nu$ is a nonnegative integer and $2p\le \nu$, then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with $2m\le \nu$. …

38: 18.40 Methods of Computation

… ►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. … ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …

39: 30.3 Eigenvalues

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§30.3(iii) Transcendental Equation

►If $p$ is an even nonnegative integer, then the continued-fraction equation …If $p=0$ or $p=1$, the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if $p>1$ its last denominator is ${\beta }_0-\lambda$ or ${\beta }_1-\lambda$. …

40: 28.34 Methods of Computation

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Solution of the continued-fraction equations (28.6.16)–(28.6.19) and (28.15.2) by successive approximation. See Blanch (1966), Shirts (1993a), and Meixner and Schäfke (1954, §2.87).

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