16 Generalized Hypergeometric Functions and 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 ${}_{p}F_{p}\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many zeros if and only if the ${a}_{j}$ can be re-indexed for $j=1,\mathrm{\dots},p$ in such a way that ${a}_{j}-{b}_{j}$ is a nonnegative integer.

Next, assume that $p=q$ and that the ${a}_{j}$ and the quotients ${\left(\mathbf{a}\right)}_{j}/{\left(\mathbf{b}\right)}_{j}$ are all real. Then ${}_{p}F_{p}\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many real zeros.