Let a1,...,ap,b1,...,bp be real constants with a1,...,ap≠0,-1,-2,... and b1,...,bp0, and let pFp(z)=pFp(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 a1=b1+m1,...,ap=bp+mp for some nonnegative integers m1,...,mp.
Bibliographical noteFunding Information:
Professor Askey informed us that Theorem 3 of this paper is not known. We truly thank him for this. Professor Gasper conveyed many valuable facts to us. We thank him. We thank Professor Csordas for pointing out the correct theorem on the complex zero decreasing sequences. In the original version of this paper, we asserted that Rk( z)/ 0 for some k and z, and the referee pointed out that our proof of this may be invalid. We deeply thank the referee for this as well as for the valuable comments. The authors acknowledge the financial support of the Korea Research Foundation in the program year of (1998—2000). The second author was supported by the Korea Science and Engineering Foundation (KOSEF) through the Global Analysis Research Center (GARC) at Seoul National University.
All Science Journal Classification (ASJC) codes
- Applied Mathematics