Power Series with Integral Coefficients
نویسنده
چکیده
Let f(z) be a function, meromorphic in \z\ < 1 , whose power series around the origin has integral coefficients. In [5], Salem shows that if there exists a nonzero polynomial p(z) such that p(z)f(z) is in H, or else if there exists a complex number a, such that l/(f(z)—a) is bounded, when |JS| is close to 1, then f(z) is rational. In [2], Chamfy extends Salem's results by showing that if there exists a complex number a and a nonzero polynomial p{z), such that p(z)/(f(z)—a) is in Hy then f(z) is rational. In this paper we show that if f(z) is of bounded characteristic in \z\ < 1 (i.e. the ratio of two functions, each regular and bounded in | z\ <1) , then f(z) is rational. Uf(z) is regular in \z\ < 1 , then, by [4], f(z) is of bounded characteristic in |2:| < 1 , if and only if
منابع مشابه
Hausdorff Moments, Hardy Spaces and Power Series
In this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the Hausdorff conditions, which play a relevant role in the moment problem theory. We prove that these series converge to functions analytic in cut domains. We are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the Pollacze...
متن کاملPower Series Coefficients for Probabilities in Finite Classical Groups
It is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in q. Moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equal modulo 2. The rate of stabilization of the finite dimensional coefficients as the dimension increases is di...
متن کاملTitle: Power series coefficients for probabilities in finite classical groups Running head: Power series coefficients for probabilities
It is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in q−1. Moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equal modulo 2. The rate of stabilization of the finite dimensional coefficients as the dimension increases is ...
متن کاملEigenvalue Decay of Integral Operators Generated by Power Series–like Kernels
We deduce decay rates for eigenvalues of integral operators generated by power serieslike kernels on a subset X of either Rq or Cq . A power series-like kernel is a Mercer kernel having a series expansion based on an orthogonal family { fα}α∈Zq+ in L 2(X ,μ) , in which μ is a complete measure on X . As so, we show that the eigenvalues of the integral operators are given by an explicit formula d...
متن کاملPower series for inverse Jacobian elliptic functions
The 12 inverse Jacobian elliptic functions are expanded in power series by using properties of the symmetric elliptic integral of the first kind. Suitable notation allows three series to include all 12 cases, three of which have been given previously. All coefficients are polynomials in the modulus k that are homogeneous variants of Legendre polynomials. The four series in each of three subsets...
متن کاملFamily of Formal Power Series with Unbounded Partial Quotients
There is a theory of continued fractions for formal power series in x−1 with coefficients in a field Fq. This theory bears a close analogy with classical continued fractions for real numbers with formal power series playing the role of real numbers and the sum of the terms of non-negative degree in x playing the role of the integral part. In this paper we give a family of formal power series in...
متن کامل