= Power series of the Chebyshev differential equation
{title2=$a_{n+2}=\frac{n^2-k^2}{(n+2)(n+1)}a_n$}
For the <Chebyshev differential equation> $(1-x^2)y''-xy'+k^2y=0$, a <power series> about the <ordinary point> zero splits into even and odd solutions with the displayed <recurrence relation>. Every nonterminating parity series has <radius of convergence> one. If $k^2=m^2$ for a nonnegative integer $m$, the parity series containing $x^m$ terminates, giving a <polynomial> with infinite <radius of convergence>; the other independent series still has radius one. Normalized solutions are $\cos(k\arcsin x)$ and $\sin(k\arcsin x)/k$, with the latter replaced by $\arcsin x$ when $k=0$.
Back to article page
