= Hypergeometric Frobenius recurrence
{title2=$(n+\sigma)(n+\sigma+\gamma-1)a_n=(n+\sigma+\alpha-1)(n+\sigma+\beta-1)a_{n-1}$}
For the <Gauss hypergeometric equation> with parameters $\alpha,\beta,\gamma$, a <Frobenius solution> $y=x^\sigma\sum_{n\geq0}a_nx^n$, $a_0\ne0$, has <indicial roots> $\sigma=0$ and $\sigma=1-\gamma$. Matching each successive power gives the displayed undivided recurrence. The usual ratio formula is valid only when its left coefficient is nonzero. At integer-separated roots, a zero coefficient requires a compatibility check as in an <undetermined coefficient at Frobenius resonance>; otherwise a logarithmic companion may be necessary. For example, $\gamma=0,\sigma=0$ gives $0=\alpha\beta a_0$ at $n=1$, obstructing that exponent-zero series when $\alpha\beta\ne0$.
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