§16.9 Zeros

Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.9

Assume that $p=q$ and none of the $a_{j}$ is a nonpositive integer. Then $\mathop{{{}_{{p}}F_{{p}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many zeros if and only if the $a_{j}$ can be re-indexed for $j=1,\dots,p$ 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 $\mathop{{{}_{{p}}F_{{p}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many real zeros.

These results are proved in Ki and Kim (2000). For further information on zeros see Hille (1929).