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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 {\it D. Schmidt} and {\it 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.
34M40Stokes phenomena and connection problems (ODE in the complex domain)
34A30Linear ODE and systems, general
34E05Asymptotic expansions (ODE)
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