On singularities, growth and domains of holomorphy of generating functions of P\´olya frequency sequences.
by
Maria Teresa Alzugaray
Universidade do Algarve, Portugal.

The Pólya frequency sequences, also called multiply positive sequences, were first introduced by Fekete in 1912. The sequence {c_k}_k=0^\infty is called a Pólya frequency sequence of order r, r in N \cup {\infty}, if all minors of order <= r (all minors if r=\infty) of the infinite matrix ||c_j-i||_{i, j=0}^\infty are non-negative. The class of these sequences is denoted by PF_r. We will use PF_r to denote the class of corresponding generating functions f(z)=\sum_k=0^\infty c_kz^k. The problem of the description of PF_\infty was exhaustively solved in 1953 by M. Aissen, A. Edrei, I.J. Schoenberg and A. Whitney. In 1955, Schoenberg set up the problem of the description of the class PF_r, r in N, and obtained results related to the zeros of polynomials belonging this class. O.M. Katkova and I.V. Ostrovski described the possible zero-sets and growth of entire functions from PF_r, r in N. Our study is devoted to the possible singularities of the functions from this class. We will show that the class PF_r, r in N, can be much richer than PF_\infty. Indeed, the only singularities a function of PF_\infty can possess are poles lying on the real positive axis which are "not too close" to each other. On the other hand, for any integer r >= 2 and any number \rho, 0 <= \rho <= \infty, there exists a function f(z) in PF_r which is analytic in the unit disc and whose order of growth in the disc is equal to \rho. Also, for any integer r >= 1 and for any domain G subset C satisfying the conditions (i) 0 in G, (ii) G is symmetric with respect to R, and (iii) dist(0, \partialG) in \partialG, there exists a function f in PF_r whose domain of holomorphy coincides with G. The problem of the complete description of PF_r, r in N, remains open.

Date received: April 26, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahk-14.