The Canonical Product of the Differential Equation with One Turning Point and Singular Point

We consider the differential equation yq(x)y 22(x)y for xI :[0,1], (*) where the weight function f φ2 is real and has one zero in the open interval (0,1), so called turning point, and the assumption that is odd order. This turning point is admitted to be pole of the function q. Using the asymptotic solution as well as the distribution of positive and negative eigenvalue, we derive the canonical product of a particular solution of the Sturm-Liouville in one turning point case