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

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1039)

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.

Keywords

  • Rational Function
  • Arbitrary Number
  • Recurrence Formula
  • Polynomial Coefficient
  • Previous Part

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References

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Kovačeva, R.K. (1983). The roots of unity and the m-meromorphic extension of functions. In: Ławrynowicz, J. (eds) Analytic Functions Błażejewko 1982. Lecture Notes in Mathematics, vol 1039. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073370

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  • DOI: https://doi.org/10.1007/BFb0073370

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12712-3

  • Online ISBN: 978-3-540-38697-1

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