On the zeros of some generalized hypergeometric functions
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.
scientific article; zbMATH DE number 1439361