Complete systems of recursive integrals and Taylor series for solutions of Sturm–Liouville equations

Given a regular nonvanishing complex valued solution y₀ of the equation , x ∈ (a,b), assume that it is n times differentiable at a point x₀ ∈ [a,b]. We present explicit formulas for calculating the first n derivatives at x₀ for any solution of the equation . That is, a map transforming the Taylor expansion of y₀ into the Taylor expansion of u is constructed. The result is obtained with the aid of the representation for solutions of the Sturm‐Liouville equation in terms of spectral parameter power series. Copyright © 2012 John Wiley & Sons, Ltd.