On products of power series

Ohne Zusammenfassung

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On products of power series

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Work performed under the auspices of the U.S. Atomic Energy Commission.     

On integrability of power series

This paper deals with the integrability of a power series. Our results generalize certain results of Ram, and Askey and Karlin.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

On computing reciprocals of power series

It is shown that root-finding iterations can be used in the field of power series. As a consequence, we obtain a class of new algorithms for computing reciprocals of power series. In particular, we show that the recent sieveking algorithm for computing reciprocals is just Newton iteration. Moreover, ifL _n is the number of non scalar multiplications needed to compute the firstn+1 terms of the reciprocal of a power series, we show that $$n + 1 \leqq L_n \leqq 4n - \log _2 n$$ and conjecture that $$L_n = 4n - lowerorderterms.$$      

On analytic continuation of a hypergeometric series using the Euler-Knopp transformation

The Euler-Knopp transformation is considered in terms of the problems of regularity and acceleration of the rate of convergence. The object of study is the hypergeometric series

On power series with positive coefficients

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On classes of everywhere divergent power series

We prove the everywhere divergence of series $$ \sum_{n=0}^\infty a_n e^{i\rho_n}e^{inx}, \quad\text{and}\quad \sum_{n=0}^\infty {(-1)}^{[\rho_n]}a_n \cos nx, $$ for sequences a _n and ρ _n satisfying some extremal conditions. These results generalize some well known examples of everywhere divergent power and trigonometric series.     

On the absolute convergence of power series

We obtain a two-dimensional analog of the Hardy-Littlewood result on the absolute convergence of power series in the case of multiple series on the boundary of a unit polydisk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 594–602, May, 1999.     

On power series rings over valuation domains

Let V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one.     

On power series of products of special functions

Some methods are considered for obtaining power series of products and powers of elementary and special functions