Asymptotic form and infinite product representation of solution of a second order initial value problem with a complex parameter and a finite number of turning points

The paper studies the differential equation * $y'' + (\rho ^2 \varphi ^2 (x) - q(x))y = 0$ on the interval I = 0, 1], containing a finite number of zeros 0 < x ₁ < x ₂ < ... < x _m < 1 of ? ², i.e. so-called turning points. Using asymptotic estimates from 6] for appropriate fundamental systems of solutions of (*) as |ρ| → ∞, it is proved that, if there is an asymptotic solution of the initial value problem generated by (*) in the interval 0, x ₁), then the asymptotic solutions in the remaining intervals can be obtained recursively. Furthermore, an infinite product representation of solutions of (*) is studied. The representations are useful in the study of inverse spectral problems for such equations.