Spectral parameter power series for Sturm–Liouville equations on time scales Lynn Erbe a , Raziye Mert a,b,⇑ , Allan Peterson a a Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA b Department of Mathematics and Computer Science, Çankaya University, 06810 Ankara, Turkey article info Keywords: Sturm–Liouville Spectral parameter Taylor monomials Time scale abstract We will derive formulas for finding two linearly independent solutions of the Sturm–Liouville dynamic equation. We will give several examples. In particular, the q-difference equation which has important applications in quantum theory will be presented. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction In this paper we obtain a spectral parameter power series representation for the solutions of the Sturm–Liouville dynamic equation ðpðtÞx D ðtÞÞ D þ qðtÞxðrðtÞÞ ¼ krðtÞxðrðtÞÞ ð1:1Þ in terms of a non-vanishing solution of the equation ðpðtÞx D ðtÞÞ D þ qðtÞxðrðtÞÞ ¼ 0; ð1:2Þ where p 2 CðT; KÞ and pðtÞ – 0 for all t 2 T; q; r 2 C rd ðT; KÞ (defined below), and k 2 K is a constant, where T denotes a time scale unbounded from above, i.e., sup T ¼1, and K is the set of complex numbers. Problems of this sort were considered for the differential equation ðpðtÞx 0 ðtÞÞ 0 þ qðtÞxðtÞ¼ krðtÞxðtÞ and for the difference equation DðpðnÞDxðnÞÞ þ qðnÞxðn þ 1Þ¼ krðnÞxðn þ 1Þ in several papers (see the very nice papers by Campos and Kravchenko [3] and Kravchenko and Porter [8]). Recall that a time scale T is an arbitrary nonempty closed subset of the real numbers R. The most well-known examples are T ¼ R; T ¼ Z, and T ¼ q Z :¼fq n : n 2 Zg S f0g, where q > 1. The forward and backward jump operators are defined by rðtÞ :¼ inf fs 2 T : s > tg and qðtÞ :¼ supfs 2 T : s < tg; respectively, where inf ; :¼ sup T and sup ; :¼ inf T. A point t 2 T is said to be left-dense if t > inf T and qðtÞ¼ t, right-dense if t < sup T and rðtÞ¼ t, left-scattered if qðtÞ < t, and right-scattered if rðtÞ > t. A function f that is defined on a time scale T is called rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists (finite) at every 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.037 ⇑ Corresponding author at: Department of Mathematics and Computer Science, Çankaya University, 06810 Ankara, Turkey. E-mail addresses: lerbe2@math.unl.edu (L. Erbe), raziyemert@cankaya.edu.tr (R. Mert), apeterson1@math.unl.edu (A. Peterson). Applied Mathematics and Computation 218 (2012) 7671–7678 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc