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Given a regular nonvanishing complex valued solution  y  0  of the equationy    ′  ′    +   q   (   x   )   y   =   0  ,  x  ∈ ( a , b ), assume that it is  n  times differentiable at a point  x  0  ∈ [ a , b ]. We present explicit formulas for calculating the first  n  derivatives at  x  0  for any solution of the equationu    ′  ′    +   q   (   x   )   u   =   λu  . That is, a map transforming the Taylor expansion of  y  0  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.

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