A generalization of Fourier trigonometric series

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−a(a+1)cos2t(1−(−1)n)/2)Φn(t)=0, as a generalization of the differential equation of trigonometric sequences {sinnt}n=1∞ and {cosnt}n=0∞ for a=0a=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][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.