Gauss hypergeometric series geometric tail bound

In the nonterminating case with $z\mathrm{\ne }0$, every term is nonzero. The Pochhammer product gives $h_{k+1}\mathrm{/}{h}_k=z(a+k)(b+k)\mathrm{/}((c+k)(k+1))$. For $k\ge N>\mathrm{\mid }c\mathrm{\mid }$, $\mathrm{\mid }a+k\mathrm{\mid }\le k(1+\mathrm{\mid }a\mathrm{\mid }\mathrm{/}N)$, $\mathrm{\mid }b+k\mathrm{\mid }\le k(1+\mathrm{\mid }b\mathrm{\mid }\mathrm{/}N)$, $\mathrm{\mid }c+k\mathrm{\mid }\ge k(1-\mathrm{\mid }c\mathrm{\mid }\mathrm{/}N)$, and $k+1\ge k$. These inequalities yield the stated $q_N$.

For fixed nonterminating parameters, $q_N\to \mathrm{\mid }z\mathrm{\mid }<1$, so a finite $N$ has $q_N<1$; summing the geometric majorant proves the stated tail bound. On a compact subset with $\mathrm{\mid }z\mathrm{\mid }\le r<1$, replacing $\mathrm{\mid }z\mathrm{\mid }$ by $r$ gives a uniform ratio bound below $1$ for all sufficiently large $N$, proving local uniform convergence. If $z=0$, the tail is exactly zero by the Statement. If $a$ or $b$ is a nonpositive integer, its Pochhammer product becomes zero after finitely many terms, so the series terminates and its tail after the last nonzero term is zero.