16 Generalized Hypergeometric Functions & Meijer G-Function Generalized Hypergeometric Functions 16.8 Differential Equations 16.10 Expansions in Series of

{{}_{p}F_{q}}

§16.9 Zeros

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Keywords:: generalized hypergeometric functions, zeros
Permalink:: http://dlmf.nist.gov/16.9
See also:: Annotations for Ch.16

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 $j=1,\dots,p$ 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.

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

{{}_{p}F_{q}}

Functions