continued-fraction equations

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1: 31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

2: 28.15 Expansions for Small $q$

… ►Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$: ►

28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

ⓘ

Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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3: 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. …

4: 28.34 Methods of Computation

… ►

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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5: 30.3 Eigenvalues

… ►

§30.3(iii) Transcendental Equation

►If $p$ is an even nonnegative integer, then the continued-fraction equation …

6: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►The eigenvalues $q_m$ satisfy the continued-fraction equation …

7: 28.6 Expansions for Small $q$

… ►Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: …

8: 29.3 Definitions and Basic Properties

… ►satisfies the continued-fraction equation …

9: 30.16 Methods of Computation

… ►Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93). …

10: 31.11 Expansions in Series of Hypergeometric Functions

… ►In this case the accessory parameter $q$ is a root of the continued-fraction equation …