On the Zeros of Some Generalized Hypergeometric Functions

Let a1, . . . , ap, b1, . . . , bp be real constants with a1, . . . , ap/0,y1,y2, . . . and b1, . . . , bp)0, and letpFp z.spFp a1, . . . , ap; b1, . . . , bp; z.. It is shown that the following three conditions are equivalent to each other: i. pFp z. has only a finite number of zeros, ii. pFp z. has only real zeros, and iii. the aj’s can be re-indexed so that a1sb1qm1, . . . , apsbpqmpfor some nonnegative integers m1, . . . , mp.