16 Generalized Hypergeometric Functions & Meijer G-FunctionGeneralized Hypergeometric Functions16.8 Differential Equations16.10 Expansions in Series of ${}_{p}{}^{}F_{q}^{}$ Functions

Assume that $p=q$ and none of the ${a}_{j}$ is a nonpositive integer. Then ${}_{p}{}^{}F_{p}^{}\beta \x81\u2018(\mathrm{\pi \x9d\x90\x9a};\mathrm{\pi \x9d\x90\x9b};z)$ has at most finitely many zeros if and only if the ${a}_{j}$ can be re-indexed for $j=1,\mathrm{\beta \x80\xa6},p$ in such a way that ${a}_{j}\beta \x88\x92{b}_{j}$ is a nonnegative integer.

Next, assume that $p=q$ and that the ${a}_{j}$ and the quotients ${\left(\mathrm{\pi \x9d\x90\x9a}\right)}_{j}/{\left(\mathrm{\pi \x9d\x90\x9b}\right)}_{j}$ are all real. Then ${}_{p}{}^{}F_{p}^{}\beta \x81\u2018(\mathrm{\pi \x9d\x90\x9a};\mathrm{\pi \x9d\x90\x9b};z)$ has at most finitely many real zeros.