The Chebyshev polynomials are defined recursively: $T_0(x)=1$; $T_1(x)=x$; $T_n(x)=2xT_{n-1}(x)-T_{n-2}(x)$
I have been trying to find a closed form for the coefficient on the monomial $x^j$ of the $k$th polynomial. I denote this as $c_{k,j}$. I think it is clear that the $c_{k,j}$ satisfy the recurrence: $c_{k,j}=2c_{k-1,j-1}-c_{k-2,j}$, but I am unsure of how to evaluate this double-indiced recurrence. Can anyone come up with the closed form? Is this result already known?
The coefficients are given here. See the section on the formula:
a(n,m)=2^(m-1)*n*(-1)^[(n-m)/2]*[(n+m)/2-1]!/{[(n-m)/2]! m!} if n>0. - R. J. Mathar, Apr 20 2007