scholarly journals Lindelöf Representations and (Non-)Holonomic Sequences

10.37236/275 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Philippe Flajolet ◽  
Stefan Gerhold ◽  
Bruno Salvy
Keyword(s):  
Finite Difference ◽  
Generating Functions ◽  
Complex Analysis ◽  
Integral Representations ◽  
Asymptotic Estimates ◽  
Linear Recurrences ◽  
Polynomial Coefficients ◽  
Difference Sequences ◽  
Analytic Expressions ◽  
Linear Differential

Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotic estimates of certain finite difference sequences come out as a byproduct of the Lindelöf approach.

A Recursion Formula for the Coefficients of Entire Functions Satisfying an ODE with Polynomial Coefficients

2004 ◽  
Vol 11 (3) ◽  
pp. 409-414
Author(s):  
C. Belingeri
Keyword(s):  
Differential Equations ◽  
Sum Rules ◽  
Entire Functions ◽  
Recursion Formula ◽  
Linear Differential Equations ◽  
Bell Polynomials ◽  
Polynomial Coefficients ◽  
Linear Differential

Abstract A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

Comparison Of Finite-Difference And Analytic Microwave Calculation Methods

1996 ◽  
Vol 430 ◽  
Author(s):  
F. I. Friedlander ◽  
H. W. Jackson ◽  
M. Barmatz ◽  
P. Wagner
Keyword(s):  
Finite Difference ◽  
Numerical Evaluation ◽  
Normal Modes ◽  
Materials Processing ◽  
Calculation Methods ◽  
Power Absorption ◽  
Microwave Cavities ◽  
Analytic Expressions ◽  
Lossy Dielectric ◽  
Electromagnetic Solver

AbstractNormal modes and power absorption distributions in microwave cavities containing lossy dielectric samples were calculated for problems of interest in materials processing. The calculations were performed both using a commercially available finite-difference electromagnetic solver and by numerical evaluation of exact analytic expressions. Results obtained by the two methods applied to identical physical situations were compared. Our studies validate the accuracy of the finite-difference electromagnetic solver. Relative advantages of the analytic and finitedifference methods are discussed.

On the Motion of Granular Materials Due to Shear

2000 ◽  
Author(s):  
Mehrdad Massoudi ◽  
Tran X. Phuoc
Keyword(s):  
Finite Difference ◽  
Granular Materials ◽  
Continuum Model ◽  
Theory Model ◽  
Governing Equations ◽  
Flat Plates ◽  
Material Parameters ◽  
Finite Difference Technique ◽  
Linear Differential ◽  
The Continuum

Abstract In this paper we study the flow of granular materials between two horisontal flat plates where the top plate is moving with a constant speed. The constitutive relation used for the stress is based on the continuum model proposed by Rajagopal and Massoudi (1990), where the material parameters are derived using the kinetic theory model proposed by Boyle and Massoudi (1990). The governing equations are non-dimensionalized and the resulting system of non-linear differential equations is solved numerically using finite difference technique.

scholarly journals A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems

2021 ◽  
Vol 8 (1) ◽  
pp. 1-11
Author(s):  
Abdul Abner Lugo Jiménez ◽  
Guelvis Enrique Mata Díaz ◽  
Bladismir Ruiz
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Stationary Regime ◽  
Heat Transfer Fluid ◽  
Mimetic Finite Difference Method ◽  
Linear Differential

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

scholarly journals Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1431
Author(s):  
Junesang Choi ◽  
Recep Şahin ◽  
Oğuz Yağcı ◽  
Dojin Kim
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Zeta Functions ◽  
Integral Representations ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
The One ◽  
Function Of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

Indefinite Green's Functions and Elementary Solutions

1963 ◽  
Vol 6 (1) ◽  
pp. 71-103 ◽  
Author(s):  
G. F. D. Duff ◽  
R. A. Ross
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Source Distribution ◽  
Asymptotic Estimates ◽  
Elementary Functions ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Superposition Of Solutions

Linear differential equations both ordinary and partial are often studied by means of Green's functions. One reason for this is that linearity permits superposition of solutions. A Green's function describes the "effect" of a point source, and the description of line, surface, or volume sources is achieved by superposing, that is to say, integrating, this function over the source distribution.For equations with constant coefficients the use of integral transforms permits the calculation of such source functions in the form of integrals. Only in the simplest cases is explicit evaluation by elementary functions possible, and this has perforce led to the use of asymptotic estimates, which so thoroughly pervade the domain of applied mathematics.

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