Horn hypergeometric series

Remarks

The two conditions mean that the ratios of neighbouring coefficients are rational in $(m, n)$, as for the Gauss hypergeometric series in one variable. Example: Appell's $F_1(a; b, b'; c; x, y)$ has $A_{m, n} = \frac{(a)_{m + n} (b)_m (b')_n}{(c)_{m + n}\, m!\, n!}$. Then $f(m, n) = \frac{(a + m + n)(b + m)}{(c + m + n)(m + 1)}$ and $g(m, n) = \frac{(a + m + n)(b' + n)}{(c + m + n)(n + 1)}$, both rational, so $F_1$ is a Horn series of order $2$. Non-example: $A_{m, n} = 2^{m^2}$ gives $A_{m + 1, n} / A_{m, n} = 2^{2m + 1}$, which is not a rational function of $m$, so the first clause fails. Horn's list contains $34$ essentially distinct convergent series of order $2$: $14$ complete ones, among them the four Appell series, and $20$ confluent ones.