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The double confluent Heun equation: Characteristic exponent and connection formulae. (English) Zbl 0849.34004
Summary: The connection relations between the solutions of different type are obtained for the double confluent Heun equation, a second-order linear differential equation with two irregular singular points of unit rank. For the relevant quantity which determines the characteristic exponent and also enters the connection coefficients, two entirely different representations are given. One is essentially a finite determinant (of size 4 by 4 or 3 by 3 or 2 by 2, depending on details of the derivation) the elements of which are Taylor series at half the convergence radius with recursively available coefficients. The other one is an asymptotic expansion in terms of the recursively known coefficients of the formal power series solutions of the differential equation at one of the irregular singular points. In terms of the same coefficients, a series representation converging like a power series at half the convergence radius is obtained for the other relevant quantity which enters the connection coefficients. Of the same type is a numerically stable explicit representation of the coefficients of the Floquet solutions.
MSC:
34M99Differential equations in the complex domain
34A30Linear ODE and systems, general
34E05Asymptotic expansions (ODE)