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On the zeros of some generalized hypergeometric functions. (English) Zbl 0951.33006

The paper is concerned with the confluent hypergeometric function

$F={}_{p}{F}_{p}\left({a}_{1},\cdots ,{a}_{p};{b}_{1},\cdots ,{b}_{p};z\right)·$

It is assumed that the parameters are real, no numerator parameter equals zero or a negative integer, and the denominator parameters are positive. Then, the author establishes the equivalence of the following assertions: (i) $F$ has only a finite number of zeros; (ii) $F$ has only real zeros; and (iii) there exist nonnegative integers ${m}_{1},\cdots ,{m}_{p}$ such that ${a}_{1}={b}_{1}+{m}_{1},\cdots ,{a}_{p}={b}_{p}+{m}_{p}$. Some examples are given to show that the assumptions cannot be relaxed. Besides the classical theory of hypergeometric functions the author also applies the Pólya-Schur theorem on multiplier sequences.

##### MSC:
 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$