A generalization of Fourier trigonometric series

Abstract

In this paper, by using the extended Sturm-Liouville theorem for symmetric functions, we introduce the differential equation Φn″ (t) + ((n + a (1 - (- 1)n) / 2)2 - frac(a (a + 1), cos2 t) (1 - (- 1)n) / 2) Φn (t) = 0, as a generalization of the differential equation of trigonometric sequences {sin n t}n = 1∞ and {cos n t}n = 0∞ for a = 0 and obtain its explicit solution in a simple trigonometric form. We then prove that the obtained sequence of solutions is orthogonal with respect to the constant weight function on [0, π] and compute its norm square value explicitly. One of the important advantages of this generalization is to find some new infinite series. A practical example is given in this sense. © 2008 Elsevier Ltd. All rights reserved.