A generalization of Fourier trigonometric series

  • Authors:

  • Mohammad Masjed-Jamei;Mehdi Dehghan

  • Affiliations:

  • Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No 429, Hafez Ave, Tehran, Iran and Department of Applied Mathematics, K.N.Toosi ...;Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No 429, Hafez Ave, Tehran, Iran

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2008

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Abstract

In this paper, by using the extended Sturm-Liouville theorem for symmetric functions, we introduce the differential equation @F"n^''(t)+((n+a(1-(-1)^n)/2)^2-a(a+1)cos^2t(1-(-1)^n)/2)@F"n(t)=0, as a generalization of the differential equation of trigonometric sequences {sinnt}"n"="1^~ and {cosnt}"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,@p] 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.