Zeros of \(_3F_2\left(\begin{smallmatrix}-n,b,c\\ d,e\end{smallmatrix};z\right)\) polynomials
Let \(_pF_q\) denote the generalized hypergeometric function with \(p\) numerator and \(q\) denominator parameters defined, for \(|z|<1\), by \[ _pF_q\left( \begin{matrix} a_1,a_2,\dots,a_p\\ b_1,b_2,\dots,b_q \end{matrix}; z\right) =1+\sum^\infty_{k=1} {(a_1)_k(a_2)_k \dots (a_p)_k\over (b_1)_k(b_2)_k \dots (b_q)_k} {z^k\over k!}, \] where, as usual, \((a)_k=a(a+1) \dots (a+k-1)\), \(k=1,2, \dots\) It is easy to see that if one of the numerator parameters is equal to a negative integer, say \(a_1=-n\), \(n=1,2,\dots\), then the above series terminates and it is a polynomial of degree \(n\). In this paper the authors establish the location of zeros of six types of \(_3F_2\) hypergeometric polynomials that admit representations as various kinds of products involving \(_2F_1\) polynomials. They categorize the \(_3F_2\) polynomials considered according to whether they are well-poised or \(k\)-balanced. Details about these definitions and concepts can be found in \textit{G. Andrews}, \textit{R. Askey} and \textit{R. Roy}'s book [Special functions Encyclopedia of Mathematics and Its Applications. 71. Cambridge: Cambridge University Press (1999; Zbl 0920.33001)].
scientific article; zbMATH DE number 1845401