scholarly journals Series in Le Roy Type Functions: A Set of Results in the Complex Plane—A Survey

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1361
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Power Series ◽  
Asymptotic Formula ◽  
Complex Plane ◽  
Special Functions ◽  
Convergence Theorems

This study is based on a part of the results obtained in the author’s publications. An enumerable family of the Le Roy type functions is considered herein. The asymptotic formula for these special functions in the cases of ‘large’ values of indices, that has been previously obtained, is provided. Further, series defined by means of the Le Roy type functions are considered. These series are studied in the complex plane. Their domains of convergence are given and their behaviour is investigated ‘near’ the boundaries of the domains of convergence. The discussed asymptotic formula is used in the proofs of the convergence theorems for the considered series. A theorem of the Cauchy–Hadamard type is provided. Results of Abel, Tauber and Littlewood type, which are analogues to the corresponding theorems for the classical power series, are also proved. At last, various interesting particular cases of the discussed special functions are considered.

scholarly journals On power series of products of special functions

2011 ◽  
Vol 24 (8) ◽  
pp. 1374-1378 ◽  
Author(s):  
Yu.A. Brychkov
Keyword(s):  
Power Series ◽  
Special Functions

Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

2011 ◽  
Vol 89 (11) ◽  
pp. 1083-1099
Author(s):  
Tam Do-Nhat
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Complex Plane ◽  
Branch Point ◽  
Upper Bound ◽  
Least Squares Method ◽  
Radius Of Convergence ◽  
New Method ◽  
Recursion Relations

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.

A note on fractional Bessel equation and its asymptotics

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Wojciech Okrasiński ◽  
Łukasz Płociniczak
Keyword(s):  
Power Series ◽  
Asymptotic Formula ◽  
Asymptotic Analysis ◽  
Fractional Derivative ◽  
Order Term ◽  
High Accuracy ◽  
Fast Convergence ◽  
Leading Order ◽  
Bessel Equation ◽  
Integer Order

AbstractIn this note we propose a fractional generalization of the classical modified Bessel equation. Instead of the integer-order derivatives we use the Riemann-Liouville version. Next, we solve the fractional modified Bessel equation in terms of the power series and provide an asymptotic analysis of its solution for large arguments. We find a leading-order term of the asymptotic formula for the solution to the considered equation. This behavior is verified numerically and shows high accuracy and fast convergence. Our results reduce to the classical formulas when the order of the fractional derivative goes to integer values.

scholarly journals U-Operators

2005 ◽  
Vol 78 (1) ◽  
pp. 59-90 ◽  
Author(s):  
L. Bernal-González ◽  
J. A. Prado-Tendero
Keyword(s):  
Power Series ◽  
Complex Plane ◽  
Entire Functions ◽  
Shift Operators ◽  
Compact Subsets

AbstractInspired by a statement of W. Luh asserting the existence of entire functions having together with all their derivatives and antiderivatives some kind of additive universality or multiplicative universality on certain compact subsets of the complex plane or of, respectively, the punctured complex plane, we introduce in this paper the new concept of U-operators, which are defined on the space of entire functions. Concrete examples, including differential and antidifferential operators, composition, multiplication and shift operators, are studied. A result due to Luh, Martirosian and Müller about the existence of universal entire functions with gap power series is also strengthened.

ON POLYNOMIAL SEQUENCES WITH RESTRICTED GROWTH NEAR INFINITY

2002 ◽  
Vol 34 (2) ◽  
pp. 189-199 ◽  
Author(s):  
J. MÜLLER ◽  
A. YAVRIAN
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Complex Plane ◽  
Partial Sums ◽  
Limit Function ◽  
Maximal Domain ◽  
Polynomial Sequences ◽  
Domain Of Existence ◽  
Similar Growth ◽  
Closed Plane

Let (Pn) be a sequence of polynomials which converges with a geometric rate on some arc in the complex plane to an analytic function. It is shown that if the sequence has restricted growth on a closed plane set E which is non-thin at ∞, then the limit function has a maximal domain of existence, and (Pn) converges with a locally geometric rate on this domain. If (snk) is a sequence of partial sums of a power series, a similar growth restriction on E forces the power series to have Ostrowski gaps. Moreover, the requirement of non-thinness of E at ∞ is necessary for these conclusions.

scholarly journals Stokes phenomena in discrete Painlevé II

2017 ◽  
Vol 473 (2198) ◽  
pp. 20160539 ◽  
Author(s):  
N. Joshi ◽  
C. J. Lustri ◽  
S. Luu
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Complex Plane ◽  
Painlevé Equation ◽  
Asymptotic Solutions ◽  
Series Expression ◽  
Painleve Equation ◽  
Rapid Switching ◽  
Power Series Methods ◽  
Stokes Phenomena

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.

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