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

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

such that

then f is m-meromorphic in a disk DR = {z, |z| <R}, where R satisfies the condition

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
These 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.
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References
GELFOND, A. O.: Die Differenzenrechnung, VEB Deutscher Verlag der Wissenschaften, Berlin, 1958.
ГОЛИЖИн, Г.М.: Геометрическая теория функций комплексного переменного, иэд. 2-е, Иэд. “Наука”. Москва 1966.
HARDY, H, and E. M. WRIGHT: An introduction to the theory of numbers, 5th ed., Clarendon Press, Oxford, 1979.
KOVACHEVA, R. K. [КОВАчЕВА, Р.К., KOVAČEVA, R. K.]: Die Einheits-wurzeln und die ihnen entsprechenden Basispolynome, Serdica, to appear.
MONTESSUS de BALLORE, R. de: Sur les fractions continues algebriques, Bull. Soc. Math. France 30 (1902), 28–36.
SAFF, E. B.: An extension of Montessus de Ballore's theorem on the convergence of interpolating rationalafunctions, J. Appr. Theory 6 (1972), 63–67.
— and J. KARLSSON: Singularities of functions determined by the poles of Padé approximants, in: Padé approximation and its applications, Amsterdam 1980, Proceedings (Lecture Notes in Math. 888), Springer-Verlag, Berlin — Heidelberg — New York 1981, pp. 238–254.
—, R. S. VARGA and A. SHARMA: An extension to rational functions of a theorem of J. L. Walsh on differences of interpolating polynomials, International Conference on Constructive Function Theory, Varna 1981, Proceedings, to appear.
WALSH, J. L.: Interpolation and approximation by rational functions in the complex domain, 5th ed. (Colloq. Publ. 20), American Math. Soc., Providence, RI 1969; Russian translation: Дж. Л. УОЛШ: Интерполяция и аппроксимация рационалъными Функциями в комплексной области, Иэд. Иностранной Литературы, Москва 1966.
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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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