The paper is concerned with the confluent hypergeometric function
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) has only a finite number of zeros; (ii) has only real zeros; and (iii) there exist nonnegative integers such that . 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.