Further results on Fermat type equations and a characterization of the Chebyshev polynomials
type: Item
instance of: scholarly article
title: Further results on Fermat type equations and a characterization of the Chebyshev polynomials
zbMATH Open document ID: 0608.30029
DOI: 10.1016/0022-247X(87)90250-2
author: Charles F. Osgood
author: Fred Gross
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 3983569
zbMATH Keywords: functional equation
zbMATH Keywords: polynomial solutions
zbMATH Keywords: Chebyshev polynomials
MaRDI profile type: Publication
cites work: Q3658242
cites work: On the equation 𝑓ⁿ+𝑔ⁿ=1
cites work: Q5683031
cites work: Sums of Squares
description: scientific article; zbMATH DE number 3983569