# 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 2830.
##### 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 $a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.$
##### 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
• (e)

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).

• ##### 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