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).

16.8 Differential Equations 16.10 Expansions in Series of

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

Functions

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§16.9 Zeros

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