Totally positive kernels, pólya frequency functions, and generalized hypergeometric series

Abstract Recently, K. I. Gross and the author [J. Approx. Theory 59:224–246 (1989)] analyzed the total positivity of kernels of the form K(x,y) = p F q (xy) , x,y ϵ R, wherep F q denotes a classical generalized hypergeometric series. There, we related the determinants which define the total positivity of K to the hypergeometric functions of matrix argument, defined on the space of n × n Hermitian matrices. In the first part of this paper, we apply these methods to derive the total positivity properties of the kernels K for more general choices of the parameters in these hypergeometric series. In particular, we prove that if ai > 0 and ki is a positive integer (i = 1,…,p) then K(x,y) = p F q (a 1 ,…,a p ;a 1 + k 1 ,…,a p + k p ; xy) is strictly totally positive on R2. In the second part, we use the theory of entire functions to derive some Polya frequency function properties of the hypergeometric series p F q(x). Some of these latter results suggest a curious duality with the total positivity properties described above. For instance, we prove that if p > 1 and the ai and ki are as above, then there exists a probability density function f on R, such that f is a strict Polya frequency function, and 1/ p F p (a 1 +k 1 ,…,a p +k p ;a 1 ,…,a p ;z= L f(z) , the Laplace transform of f.