On the solutions of double confluent Heun equations (Q1587843)

Here, the 4-parametric family of the double confluent Heun equations \[ x^{2}y''(x)+(\alpha x^{2}+x+\alpha)y'(x)+ \biggl(\beta_{1}+ \frac{1}{2} \biggr)\alpha x+ \biggl( \frac{\alpha^{2}}{2}-\gamma\biggr)+ \biggl(\beta_{-1}-\frac{1}{2} \biggr)\frac{\alpha}{x}y(x)= 0 \] is investigated. The equations of the family have two irregular singular points, 0 and \(\infty\). The author obtains new integral formulas for the bases of solutions near the singularities. The paper contains many direct computations, explicit formulas and relations for the Stokes and connection coefficients. Also it contains a detailed description of existing functional ties(symmetries) among the solutions to the equations of the family; their group structure is investigated. The results obtained complement those of \textit{D. Schmidt} and \textit{G. Wolf} [In: Alavi, Yousev (ed.) et al., Trends and developments in ordinary differential equations. Proceedings of the international symposium, Kalamazoo, MI, USA, May 20-22, 1993. Singapore: World Scientific, 293-303 (1994; Zbl 0902.34002)].