The roots of unity and the m-meromorphic extension of functions

Abstract

For a function f analytic on the closed unit disk and a positive integer m, denote by Rn,m = Rn,m(f) the rational function pn,m/qn,m, where qn,m ≢ 0, deg pn,m ≤ n, deg qn,m ≤ m, and the function (1) is analytic. Let Rn,m = Pn,m/Qn,m, where Pn,m and Qn,m have no common divisor and the polynomial Qn,m is monic. Denote by ∥ ∥ the norm of the (m + 1)-dimensional space of polynomial coefficients. As a further generalization of a generalized theorem of Montessus de Ballore (1902), due to E. B. Saff (1972), the author proves that if there exists a polynomial (2) such that (3) then f is m-meromorphic in a disk DR = {z, |z| <R}, where R satisfies the condition (4) and all the zeros of Q (including their multiplicities) are poles of f in DR. As a consequence the author obtains a criterion for the m-meromorphic extensibility of f onto DR with R > 1.KeywordsRational FunctionArbitrary NumberRecurrence FormulaPolynomial CoefficientPrevious PartThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.