Totally positive kernels, Pólya frequency functions, and generalized hypergeometric series (Q919499)

To indicate the author's results, let \(a_ 1,...,a_ p\in {\mathbb{R}}_+\), \(k_ 1,...,k_ p\in {\mathbb{N}}\), \(\gamma\geq 0\), \(\delta\in {\mathbb{R}}\), and let \({}_ pF_ q\) denote the classical hypergeometric function. The first main result on total positivity reads: The kernel \[ K_ p(x,y)=_ pF_ p(a_ 1,...,a_ p; a_ 1+k_ 1,...,a_ p+k_ p; xy) \] is \(STP_{\infty}\) on \({\mathbb{R}}^ 2\). A related result involving \(_{p+1}F_ p\) is also established. As to Pólya frequency functions of order r it is found that there exists a function f in this class such that \[ _ pF_ p(a_ 1+k_ 1,...,a_ p+k_ p; a_ 1,...,a_ p; z)\exp (-\gamma z^ 2+\delta z)=1/{\mathcal L}f(z), \] where \({\mathcal L}f\) denotes the Laplace transform and \(p>1\). There is a related result involving \({}_ pF_ q\) with \(q>p\).