= Frobenius recurrence for the Legendre equation
{c}
{title2=$a_{n+2}=\frac{(n+\sigma)(n+\sigma+1)-\lambda(\lambda+1)}{(n+\sigma+2)(n+\sigma+1)}a_n$}
Substitute $y=x^\sigma\sum_{n\geq0}a_nx^n$, $a_0\ne0$, into the <Legendre differential equation>. The lowest powers give $\sigma(\sigma-1)=0$ and $\sigma(\sigma+1)a_1=0$; subsequent powers give the recurrence in the title. Thus exponent zero has independent even and odd coefficients, while exponent one requires $a_1=0$ and represents the odd solution. Zero is an ordinary point of the equation, so the exponent-one choice simply selects a solution that vanishes there.
The singular points at $\pm1$ bound the generic radius of convergence about zero by one. Every nonterminating parity recurrence has successive-coefficient ratio tending to one in the variable $x^2$, giving radius exactly one. If $\lambda=N$ is a nonnegative integer, choosing the parity of $N$ makes the recurrence terminate at degree $N$, producing a <Legendre polynomial> with infinite radius. These are precisely the nonnegative parameters with a nonzero entire solution: even and odd parts of an entire solution would both have to be entire, so any nonzero nonterminating part is excluded.
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