scholarly journals Integrability of Double Power Series with Nonnegative Coefficients

1998 ◽  
Vol 219 (2) ◽  
pp. 229-245
Author(s):  
S.M Mazhar ◽  
F Móricz
Keyword(s):  
Power Series ◽  
Nonnegative Coefficients ◽  
Double Power Series

scholarly journals On the convergence criterion for branched continued fractions with independent variables

2018 ◽  
Vol 9 (2) ◽  
pp. 120-127 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Special Form ◽  
Absolute Convergence ◽  
Convergence Criterion ◽  
Complex Numbers ◽  
Independent Variables ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Multiple Power Series

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

scholarly journals Is the Sibuya distribution a progeny?

2019 ◽  
Vol 56 (01) ◽  
pp. 52-56
Author(s):  
Gérard Letac
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Branching Process ◽  
Power Series Expansion ◽  
Nonnegative Coefficients ◽  
Positive Integers

AbstractFor 0 < a < 1, the Sibuya distribution sa is concentrated on the set ℕ+ of positive integers and is defined by the generating function $$\sum\nolimits_{n = 1}^\infty s_a (n)z^{{\kern 1pt} n} = 1 - (1 - z)^a$$. A distribution q on ℕ+ is called a progeny if there exists a branching process (Zn)n≥0 such that Z0 = 1, such that $$(Z_1 ) \le 1$$, and such that q is the distribution of $$\sum\nolimits_{n = 0}^\infty Z_n$$. this paper we prove that sa is a progeny if and only if $${\textstyle{1 \over 2}} \le a < 1$$. The main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.

RATIO TESTS FOR DOUBLE POWER SERIES

1940 ◽  
Vol os-11 (1) ◽  
pp. 183-192 ◽  
Author(s):  
P. J. DANIELL
Keyword(s):  
Power Series ◽  
Double Power Series

The diagonal of a double power series

1971 ◽  
Vol 38 (2) ◽  
pp. 229-235 ◽  
Author(s):  
M. L. J. Hautus ◽  
D. A. Klarner
Keyword(s):  
Power Series ◽  
Double Power Series

scholarly journals WIMAN’S TYPE INEQUALITY FOR SOME DOUBLE POWER SERIES

2021 ◽  
Vol 9 (1) ◽  
pp. 56-63
Author(s):  
O. Skaskiv ◽  
A. Kuryliak
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Type Inequality ◽  
Double Power Series

By $\mathcal{A}^2$ denote the class of analytic functions of the formBy $\mathcal{A}^2$ denote the class of analytic functions of the form$f(z)=\sum_{n+m=0}^{+\infty}a_{nm}z_1^nz_2^m,$with {the} domain of convergence $\mathbb{T}=\{z=(z_1,z_2)\in\mathbb C^2\colon|z_1|<1,\ |z_2|<+\infty\}=\mathbb{D}\times\mathbb{C}$ and$\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}.$ In this paper we prove some analogue of Wiman's inequalityfor analytic functions $f\in\mathcal{A}^2$. Let a function $h\colon \mathbb R^2_+\to \mathbb R_+$ be such that$h$ is nondecreasing with respect to each variables and $h(r)\geq 10$ for all $r\in T:=(0,1)\times (0,+\infty)$and $\iint_{\Delta_\varepsilon}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}=+\infty$ for some $\varepsilon\in(0,1)$, where $\Delta_{\varepsilon}=\{(t_1, t_2)\in T\colon t_1>\varepsilon,\ t_2> \varepsilon\}$.We say that $E\subset T$ is a set of asymptotically  finite $h$-measure on\ ${T}$if $\nu_{h}(E){:=}\iint\limits_{E\cap\Delta_{\varepsilon}}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}<+\infty$ for some $\varepsilon>0$. For $r=(r_1,r_2)\in T$ and a function $f\in\mathcal{A}^2$ denote\begin{gather*}M_f(r)=\max \{|f(z)|\colon  |z_1|\leq r_1,|z_2|\leq r_2\},\\mu_f(r)=\max\{|a_{nm}|r_1^{n} r_2^{m}\colon(n,m)\in{\mathbb{Z}}_+^2\}.\end{gather*}We prove the following theorem:{\sl Let $f\in\mathcal{A}^2$. For every $\delta>0$ there exists a set $E=E(\delta,f)$ of asymptotically  finite $h$-measure on\ ${T}$ such that for all $r\in (T\cap\Delta_{\varepsilon})\backslash E$ we have \begin{equation*} M_f(r)\leq\frac{h^{3/2}(r)\mu_f(r)}{(1-r_1)^{1+\delta}}\ln^{1+\delta} \Bigl(\frac{h(r)\mu_f(r)}{1-r_1}\Bigl)\cdot\ln^{1/2+\delta}\frac{er_2}{\varepsilon}. \end{equation*}}

ON THREE INTEGRALS ARISING IN THE THEORY OF ANISOTROPIC CRITICALITY

1996 ◽  
Vol 10 (09) ◽  
pp. 377-383
Author(s):  
YONKO T. MILLEV
Keyword(s):  
Power Series ◽  
Coupling Constant ◽  
Elementary Functions ◽  
Reduced Dimensionality ◽  
Series Representations ◽  
Anisotropic Dispersion ◽  
By Products ◽  
Double Power Series

Three integrals encountered in earlier studies of anisotropic criticality (anisotropic dispersion laws characterised by the coupling constant f) are solved in terms of elementary functions. The nature of the singularity for the case f→1 which corresponds to an effectively reduced dimensionality has been revealed. Double power-series representations for the integrals are also found and some mathematical implications and by-products are discussed. The advance relates to a prospective full-scope analysis of dipolar criticality.

On the distribution of multiple power series regularly varying at the boundary point

2019 ◽  
Vol 29 (6) ◽  
pp. 409-421 ◽  
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Limit Theorem ◽  
Power Series ◽  
Boundary Point ◽  
Local Version ◽  
Random Vectors ◽  
Integral Limit ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Regularly Varying ◽  
Multiple Power Series

Abstract Let B(x) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1)n and diverges at the point 1 = (1, …, 1). Random vectors (r.v.)ξx such that ξx has distribution of the power series B(x) type is studied. The integral limit theorem for r.v. ξx as x ↑ 1 is proved under the assumption that B(x) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B(x) are one-sided weakly oscillating at infinity.

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