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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 (a 1 ,,a p ;b 1 ,,b p ;z)·

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 ,,m p such that a 1 =b 1 +m 1 ,,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:
33C20Generalized hypergeometric series, p F q