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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 $$F= {_pF_p}(a_1,\dots, a_p; b_1,\dots, b_p; z).$$ 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 $m_1,\dots, m_p$ such that $a_1= b_1+ m_1,\dots, a_p= b_p+ m_p$. 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.

33C20Generalized hypergeometric series, ${}_pF_q$
Full Text: DOI
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