A generalization of Fourier trigonometric series

In this paper, by using the extended Sturm-Liouville theorem for symmetric functions, we introduce the differential equation Phi n ''(t) + ((n + a(1 - (-1)(n)/2)(2) - a(a + 1)/cos(2)t (1 -(-1)(n)/2) Phi(n)(t) = 0, as a generalization of the differential equation of trigonometric sequences {sin nt}(n=1)(infinity) and {cos nt}(n=0)(infinity) 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, pi] 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. (C) 2008 Elsevier Ltd. All rights reserved.