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On the central connection problem for the double confluent Heun equation. (English) Zbl 0919.34007

The author solves the central connection problem for the double confluent Heun equation \[ D^2y+\alpha \left(z+ {1\over z}\right) Dy+ \left(\left(\beta_1+ 1/2\right) \alpha z+\left({\alpha^2 \over 2}-\gamma\right) +(\beta_{-1}-1/2) {\alpha\over z}\right)y=0 \] where \(D\) denotes the differential operator \(z{d\over dz}\). By Laplace transform, the equation is reduced to a special confluent Heun equation. For the last equation the problem was solved by D. Schmidt and G. Wolf [In: A. Ronveux (ed.). Heun’s differential equations, Oxford (1995; Zbl 0847.34006) and In: Alavi, Yousev (ed.) et al., Trends and developments in ordinary differential equations. Proceedings of the international symposium, Kalamazoo, 293-303 (1994; Zbl 0902.34002)]. The coefficients of the central connection matrix are computed by limit formulae obtained by the author.

MSC:

34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34A30 Linear ordinary differential equations and systems
34E05 Asymptotic expansions of solutions to ordinary differential equations
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References:

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