On the canonical solution of indefinite problem with m turning points of even order

We consider the differential equationequation(∗)y″+(ρ2ϕ2(x)−q(x))y=0y″+(ρ2ϕ2(x)−q(x))y=0 on a finite interval I  , say I=[0,1]I=[0,1], where I contains m turning points, that is here, zeros of ϕ, under the assumption that one of the turning points is of odd order while the others are of even order. Using of the asymptotic estimates provided in [W. Eberhard, G. Freiling, A. Schneider, Connection formulae for second-order differential equations with a complex parameter and having an arbitrary number of turning points, Math. Nachr. 165 (1994) 205–229] for a special fundamental system of solutions of (∗) in I, we study the infinite product representation of solutions of (∗). Such representations are relevant to the inverse spectral problem.