Further results on Fermat type equations and a characterization of the Chebyshev polynomials (Q1086383)

This paper is concerned with a functional equation of the type \[ a(z)\Phi^ 2(z)+b(z)\Psi^ 2(z)=c(z), \] where a,b,c are given entire functions and with their meromorphic solutions. The discussion of polynomial solutions yields the special equation \[ (1)\quad (1-Q^ 2)\Phi^ 2(z)+\Psi^ 2(z)=1, \] where Q is any nonconstant polynomial. It is proved, that all polynomial solutions of (1) must have the representation \(\Phi =\pm U_ n(Q)\), \(\Psi =\pm T_ n(Q)\) where \(T_ n\) and \(U_ n\) denote the Chebyshev polynomials of the first and second kind respectively.