A characterization of the Chebyshev and Fibonacci polynomials (Q1389825)

Usual orthogonal polynomials \(\{P_n(x)\}\) \((n=0,1,2,\ldots)\) satisfy a hypergeometric type differential equation: \[ (\alpha x^2+\beta x+\gamma)P_n''+(\delta x+\varepsilon)P_n' -n[\delta+(n-1)\alpha]P_n=0. \tag{HGE} \] Theorem 1. If a sequence of polynomials \(\{P_n\}\) satisfy (HGE) and the 3-term recurrence relation: \[ P_n(x)=xP_{n-1}(x)-P_{n-2}(x)\;(n\geq 2), \quad P_0=h,\;P_1=x+k, \] then \(\alpha=-1,\beta=0,\gamma=4\) and \(\{P_n\}\) are essentially one of four types of Chebyshev polynomials. When the recurrence relation is \[ P_n(x)=xP_{n-1}(x)+P_{n-2}(x)\;(n\geq 2), \quad P_0=h,\;P_1=x+k, \] similar results are given, including Fibonacci polynomials. Finally, a necessary condition is given for \(\{P_n\}\) to satisfying a general recurrence relation: \[ P_n(x)=(Ax+B)P_{n-1}(x)\pm CP_{n-2}(x). \]