Legendre,Chebychev polynomials

From: Alex.Lupas (alexandru.lupas_at_ulbsibiu.ro)
Date: 03/26/05 

Date: 25 Mar 2005 20:40:54 -0800

In the following, let us denote :
SUM{A_k}:=A_1+A_2+...+A_n ; (n >= 1 )
P_n(x)=k_n*[(x^2-1)^n]^{(n)} , P_n(1)=1 , be the Legendre polynomial
        having the roots a_1,a_2,...,a_n ;
U_n(x)=sin((n+1)*arccos(x))/((n+1)*sqrt(1-x^2)) , U_n(1)=1 , be the Chebychev
       polynomial of second kind with the roots
       b_k=cos(k*pi/(n+1)) , k in {1,2,...,n} .

[A.] I need elementary proofs of following identities /if true/:

(1) P_n(b_k)=SUM{(b_k)^n *U_n((b_k)^2)}

(2) U_n(a_k)=-SUM{(b_k)^n *U_n(a_k*b_k)}

It's true that
       =======================================
(1-2) P_n(x)-U_n(x)=SUM{(b_k)^n*U_n(x*b_k)} ??
       =======================================

[B.] Let c_k=cos((2k-1)pi/(2n)) , k in {1,2,...,n}, and denote
     
       Q_{2n}(x):= SUM{ P_n(x^2+(1-x^2)*c_k) } .

It's true that Q_{2n}(x) has only double roots ??