Fibonacci numbers and Legendre polynomials (Q2774234)

Let \(U_n(x)\) be the second type of Chebyshev polynomials with generation function NEWLINE\[NEWLINE\sum_{n=0}^\infty U_n(x)t^n= \frac{1} {1-2xt+t^2},NEWLINE\]NEWLINE and let \(P_n(x)\) be the Legendre polynomials with generating function NEWLINE\[NEWLINE\sum_{n=0}^\infty P_n(x)t^n= \frac{1} {\sqrt{1-2xt+t^2}}.NEWLINE\]NEWLINE First the authors give a formula about the two kinds of polynomials and then establish several relations between the values of the Legendre polynomials.