Calculation of a class determinant involving Chebyshev polynomials (Q2735422)

Chebyshev polynomials \(U_k(x)\) are defined by their generation function NEWLINE\[NEWLINE\frac{1}{1-2xt+t^2}=\sum^\infty_{k=0} U_k(x)t^k.NEWLINE\]NEWLINE In this paper, the author shows that if \(m\leq n-2\) and \(D\) is an \(n\)th order determinant such that its entries are \(m\)th powers of \(n^2\) continuous Chebyshev polynomials, then \(D=0\). As a special case, since \(U_k(\frac i2)=i^kF_{k+1}\), where \(i=\sqrt{-1}\) and \(F_{k+1}\) is the Fibonacci number, then the above result implies that the \(n\)th order determinant whose entries are \(m\)th \((\leq n-2)\) powers of continuous Fibonacci numbers equals zero.