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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 = λ ν ( q ) :
28.15.2 a - ν 2 - q 2 a - ( ν + 2 ) 2 - q 2 a - ( ν + 4 ) 2 - = q 2 a - ( ν - 2 ) 2 - q 2 a - ( ν - 4 ) 2 - .
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