Lucas polynomials are Shabat polynomials ?

flip2004

A Shabat polynomial $P$ can be defined as a polynomial in one complex variable, with complex coefficients, having property that there are two distinct complex numbers A , B such that for every root $r$ of $P'$ we have $P(r)=A$ or $P(r)= B$ . (By $P'$ was denoted the derivative of $P$ ) .
Define the sequence $(L_n)_{n\ge 0}$ of Lucas polynomials by $L_0(x)=2 \; , \; L_1(x)=x , \;$ and $L_{n+1}(x) = xL_n(x) + L_{n-1}(x) ,\; ( n\ge 1) .$
Question: $L_n ,$ $(n>=2)$ , is a Shabat polynomial ?

**Posted:** Wed Aug 18, 2004 11:07 pm