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1: 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).
2: 31.12 Confluent Forms of Heun’s Equation
§31.12 Confluent Forms of Heun’s Equation
Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … This has regular singularities at z = 0 and 1 , and an irregular singularity of rank 1 at z = . …
3: 18.2 General Orthogonal Polynomials
Confluent Form
4: 35.7 Gaussian Hypergeometric Function of Matrix Argument
Confluent Form
5: 31.10 Integral Equations and Representations
6: 35.6 Confluent Hypergeometric Functions of Matrix Argument
Laguerre Form
7: Bibliography D
  • A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux (1978a) Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
  • 8: 13.15 Recurrence Relations and Derivatives
    13.15.9 W κ + 1 2 , μ - 1 2 ( z ) - z W κ , μ ( z ) + ( κ + μ - 1 2 ) W κ - 1 2 , μ - 1 2 ( z ) = 0 ,
    9: 13.6 Relations to Other Functions
    13.6.6 U ( a , a , z ) = z 1 - a U ( 1 , 2 - a , z ) = z 1 - a e z E a ( z ) = e z Γ ( 1 - a , z ) .
    10: 13.27 Mathematical Applications
    §13.27 Mathematical Applications
    Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. The elements of this group are of the form … …