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biconfluent Heun equation

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11: 31.17 Physical Applications
§31.17 Physical Applications
§31.17(i) Addition of Three Quantum Spins
§31.17(ii) Other Applications
For application of biconfluent Heun functions in a model of an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere see Boyd and Natarov (1998).
12: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
13: 31.4 Solutions Analytic at Two Singularities: Heun Functions
§31.4 Solutions Analytic at Two Singularities: Heun Functions
To emphasize this property this set of functions is denoted by … The eigenvalues q m satisfy the continued-fraction equationThe set q m depends on the choice of s 1 and s 2 . The solutions (31.4.3) are called the Heun functions. …
14: 31.6 Path-Multiplicative Solutions
§31.6 Path-Multiplicative Solutions
A further extension of the notation (31.4.1) and (31.4.3) is given by
15: Gerhard Wolf
Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. … Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. …
  • 16: 31.18 Methods of Computation
    §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.
    17: 31.3 Basic Solutions
    §31.3(i) Fuchs–Frobenius Solutions at z = 0
    H ( a , q ; α , β , γ , δ ; z ) denotes the solution of (31.2.1) that corresponds to the exponent 0 at z = 0 and assumes the value 1 there. …
    §31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
    §31.3(iii) Equivalent Expressions
    For example, H ( a , q ; α , β , γ , δ ; z ) is equal to …
    18: 31.16 Mathematical Applications
    §31.16 Mathematical Applications
    §31.16(i) Uniformization Problem for Heun’s Equation
    It describes the monodromy group of Heun’s equation for specific values of the accessory parameter.
    §31.16(ii) Heun Polynomial Products
    19: 31.7 Relations to Other Functions
    §31.7(i) Reductions to the Gauss Hypergeometric Function
    Other reductions of H to a F 1 2 , with at least one free parameter, exist iff the pair ( a , p ) takes one of a finite number of values, where q = α β p . …
    §31.7(ii) Relations to Lamé Functions
    equation (31.2.1) becomes Lamé’s equation with independent variable ζ ; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …
    20: 31.9 Orthogonality
    §31.9(i) Single Orthogonality
    For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).
    §31.9(ii) Double Orthogonality
    Heun polynomials w j = Hp n j , m j , j = 1 , 2 , satisfy …and the integration paths 1 , 2 are Pochhammer double-loop contours encircling distinct pairs of singularities { 0 , 1 } , { 0 , a } , { 1 , a } . …