A determination of the number of real and imaginary roots of the hypergeometric series

Completely monotone sequences and universally prestarlike functions

“…One should remark that there are a number of theorems by various authors (for instance, Van Vleck [21], Hurwitz [8], Schafheitlin [17], Runckel [14] and Küstner [9]) dealing with the non-vanishing of hypergeometric functions in Λ. Our results above concern univalence in Λ, which, at least for the derivatives, imply non-vanishing statements as well, and the corresponding parameter sets {a, b, c} have large intersections.…”

On the Meromorphic Extension of the Spherical Functions on Noncompactly Causal Symmetric Spaces

“…K Remark 6.1. The result in Corollary 6.1 for the case \ Â N can be improved by using the explicit formula for the spherical functions and the classical results in [31] on the zeros x # (0, 1) of the hypergeometric func-…”

“…; 1&*; x) for * # R. It is in fact possible to deduce from [31] that, when \ Â N, the singularity of . * (a _ ) at *=k&\ is removable for exactly one value of _ in the following cases:…”

On the zeros of Jacobi polynomials, with applications

“…The following relations, being identities in a, , remain valid for the generalized J(x) (as well as for the orthogonal case), n 1, 2, .-.. (5) J,(x; a, ) -n(n + + fl 1)J_(x; a + 1, W 1) J,(-x; a, ) (-1)g(x; , (6) (7) J,(--i; a, fl)= 2".n'( n .a-1) ( n ;J(1;a,) (-2).n n +- Let Ni (j 1, 2, 3) denote the number of zeros of J(x; , fl) (assumed 0) in the intervals (-<x < -1),(-1 <x < 1), (1 <x < )respectively, and K denote the number of zeros of J,(x; a @ i, fl) (0) in (-< x < 1), where the smallest and largest such zeros (if they exist) are X, respectively.…”

“…Lawton [3] gave complete results for the closed interval (-1, 1) when n is sufficiently large. The results of Hilbert [4], Klein [5], Van Vleck [6], and Hurwitz [7] for the zeros of the hypergeometric function may also be applied to Jacobi polynomials.…”