= Shifted Chebyshev equation and polynomial termination
{c}
{title2=$z(z+2)y''+(z+1)y'-\alpha^2y=0$}
At zero the <indicial equation> is $\sigma(2\sigma-1)=0$. The two <Frobenius series> have exponents zero and one half, with recurrence
$$
a_{n+1}=\frac{\alpha^2-(n+\sigma)^2}{(n+\sigma+1)(2n+2\sigma+1)}a_n.
$$
The denominators never vanish for these exponents. Unless a series terminates its coefficient ratio tends to $-1/2$, giving convergence within distance two of zero. If $\alpha=m$ is a <positive integer>, the exponent-zero series terminates at degree $m$. The substitution $t=z+1$ gives the standard <Chebyshev differential equation>, and this solution is a multiple of $T_m(z+1)$. For half-integer $\alpha$, termination can instead occur in the power-series factor of the exponent-one-half solution.
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