Pólya frequency sequences and real zeros of some \(_3F_2\) polynomials
The authors prove that the zeros of three distinct classes of hypergeometric polynomials of \(_{3}F_{2}\) type are all real. More precisely, they show that the zeros of the polynomial \[ _{3}F_{2}\left(\begin{aligned} -n,\,-m,\,a-1/2 a,\,2a-1 \end{aligned} ;\,x \right),\quad a>0, \;m,\,n \in \mathbb{N}, \] are all real and negative. For \(n=m\) and \(a=1\) this implies that all zeros of the polynomial \[ _{3}F_{2}\left(\begin{aligned} -n,\,-n,\,1/2 1,\,1 \end{aligned} ;\,4x \right)=\sum_{k=0}^{n}\Big({n \atop k}\Big)^{2}\,\Big({2k \atop k}\Big)\,x^{k} \] are real and negative which settles a conjecture of C. Greene and H. Wilf. The same result is obtained for two other families of \(_{3}F_{2}\) hypergeometric polynomials. These results have connection with the theory of Pólya frequency sequences and functions. The authors also derive the asymptotic distribution (as \(n\rightarrow \infty\)) of the zeros of two of the classes of polynomials considered in the paper.
scientific article; zbMATH DE number 5161755