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 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 composite formal power series

Let be a holomorphic map from to defined in a neighborhood of such that . If the Jacobian determinant of is not identically zero, P. M. Eakin et G. A. Harris proved the following result: any formal power series such that is analytic is itself analytic. If the Jacobian determinant of is identically zero, they proved that the previous conclusion is no more true.

The authors get similar results in the case of formal power series satifying growth conditions, of Gevrey type for instance. Moreover, the proofs here give, in the analytic case, a control of the radius of convergence of by the radius of convergence of .

RÉSUMÉ. Soit une application holomorphe de dans définie dans un voisinage de et vérifiant . Si le jacobien de n'est pas identiquement nul au voisinage de , P.M. Eakin et G.A. Harris ont établi le résultat suivant: toute série formelle telle que est analytique est elle-même analytique. Si le jacobien de est identiquement nul, ils montrent que la conclusion précédente est fausse.

Les auteurs obtiennent des résultats analogues pour les séries formelles à croissance contrôlée, du type Gevrey par exemple. De plus, les preuves données ici permettent, dans le cas analytique, un contrôle du rayon de convergence de par celui de .

     

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

M. . , . , ^p () (). , , .     

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 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 power series coefficients of certain quotients of Eisenstein series

In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series . In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.