scholarly journals Residue Reduced Form of a Rational Function as an Iterated Laurent Series

10.37236/2909 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Xin Guo Ce ◽  
Zhou Yue
Keyword(s):  
Functional Equation ◽  
Power Series ◽  
Rational Function ◽  
Efficient Algorithm ◽  
Laurent Series ◽  
Reduced Form ◽  
Finite Power

Lipshitz showed that the diagonal of a D-finite power series is still D-finite, but his proof seems hard to implement. This paper may be regarded as the first step towards an efficient algorithm realizing Lipshitz's theory. We show that the idea of a reduced form may be a big saving for computing the D-finite functional equation. For the residue in one variable of a rational function, we develop an algorithm for computing its minimal algebraic functional equation.

scholarly journals On some Lambert-like series

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
P Agarwal ◽  
S Kanemitsu ◽  
T Kuzumaki
Keyword(s):  
Power Series ◽  
Zeta Function ◽  
Rational Function ◽  
Laurent Series ◽  
Zeta Functions ◽  
Limit Function ◽  
Lambert Series ◽  
Radial Limits ◽  
Lerch Zeta Function ◽  
International Audience

International audience In this note, we study radial limits of power and Laurent series which are related to the Lerch zeta-function or polylogarithm function. As has been pointed out in [CKK18], there have appeared many instances in which the imaginary part of the Lerch zeta-function was considered by eliminating the real part by considering the odd part only. Mordell studied the properties of the power series resembling Lambert series, and in particular considered whether the limit function is a rational function or not. Our main result is the elucidation of the threshold case of b_n = 1/n studied by Mordell [Mor63], revealing that his result is the odd part of Theorem 1.1 in view of the identities (1.9), (1.5). We also refer to Lambert series considered by Titchmarsh [Tit38] in connection with Estermann's zeta-functions.

Some remarks on formal power series and formal Laurent series

2017 ◽  
Vol 67 (3) ◽  
Author(s):  
Dariusz Bugajewski ◽  
Xiao-Xiong Gan
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Formal Power

AbstractIn this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.—BUGAJEWSKI, D.:

scholarly journals One-dimensional game of life and its growth functions

1992 ◽  
Vol 15 (3) ◽  
pp. 499-508
Author(s):  
Mohammad H. Ahmadi
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Formal Power Series ◽  
Infinite Product ◽  
Growth Function ◽  
The Other ◽  
Arbitrary Integer ◽  
One Dimensional ◽  
Growth Functions ◽  
Formal Power

We start with finitely many1's and possibly some0's in between. Then each entry in the other rows is obtained from the Base2sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Defined1,jrecursively for1, a non-negative integer, andjan arbitrary integer by the rules:d0,j={1     for   j=0,k         (I)0   or   1   for   0<j<kd0,j=0   for   j<0   or   j>k              (II)di+1,j=di,j+1(mod2)   for   i≥0.      (III)Now, if we interpret the number of1's in rowias the coefficientaiof a formal power series, then we obtain a growth function,f(x)=∑i=0∞aixi. It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.

scholarly journals The diagonal of a D-finite power series is D-finite

1988 ◽  
Vol 113 (2) ◽  
pp. 373-378 ◽  
Author(s):  
L Lipshitz
Keyword(s):  
Power Series ◽  
Finite Power

scholarly journals Goldie twisted partial skew power series rings

2019 ◽  
Vol 18 (08) ◽  
pp. 1950151 ◽  
Author(s):  
Wagner Cortes ◽  
Simone Ruiz
Keyword(s):  
Power Series ◽  
Laurent Series ◽  
Prime Radical ◽  
Prime Ideals ◽  
Partial Action ◽  
Unital Ring ◽  
Power Series Rings

In this paper, we work with a unital twisted partial action of [Formula: see text] on a unital ring [Formula: see text]. We introduce the twisted partial skew power series rings and twisted partial skew Laurent series rings. We study primality, semi-primality and the prime ideals in these rings. We describe the prime radical in twisted partial skew Laurent series rings. We investigate the Goldie property in twisted partial skew power series rings and twisted partial skew Laurent series rings. Moreover, we describe conditions for the semiprimality in twisted partial skew power series rings.

scholarly journals REMARKS ON EISENSTEIN

2013 ◽  
Vol 94 (2) ◽  
pp. 158-180 ◽  
Author(s):  
YURI BILU ◽  
ALEXANDER BORICHEV
Keyword(s):  
Power Series ◽  
Number Field ◽  
Laurent Series ◽  
Quantitative Version ◽  
Algebraic Power Series

AbstractWe obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat ramified series and Laurent series, and we demonstrate some applications; for instance, we estimate the discriminant of the number field generated by the coefficients.

Transfer Function Reduction

1976 ◽  
Vol 190 (1) ◽  
pp. 643-651 ◽  
Author(s):  
R. Whalley
Keyword(s):  
Transfer Function ◽  
Series Expansion ◽  
Laurent Series ◽  
Hankel Matrix ◽  
Rational Functions ◽  
Reduced Form ◽  
Reduced Order Models ◽  
Minimum Phase ◽  
Reduced Order

SYNOPSIS A method of generating reduced order models from the Laurent series expansion of a transfer function is examined by means of the Hankel Matrix and its correspondence to the field of rational functions. The approach enables particularly simple results to be derived regarding the composition of the reduced form and the avoidance of non minimum phase characteristics therein.

On the Taylor coefficients of rational functions

1956 ◽  
Vol 52 (1) ◽  
pp. 39-48 ◽  
Author(s):  
K. Mahler ◽  
J. W. S. Cassels
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Rational Functions ◽  
Partial Solution ◽  
Convergent Power Series ◽  
Taylor Coefficients ◽  
Image Position

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power seriesThe problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.

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