A generalization of Fourier trigonometric series

In this paper, by using the extended Sturm–Liouville theorem for symmetric functions, we introduce the differential equation as a generalization of the differential equation of trigonometric sequences and 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.