Difference equations and inequalities

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Difference equations and inequalities

01 Jan 1992-

About: The article was published on 1992-01-01 and is currently open access. It has received 1574 citations till now. The article focuses on the topics: Summation equation & Rational difference equation.

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Dynamic Equations on Time Scales: An Introduction with Applications

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Martin Bohner, Allan Peterson

15 Jun 2001

TL;DR: The Time Scales Calculus as discussed by the authors is a generalization of the time-scales calculus with linear systems and higher-order linear equations, and it can be expressed in terms of linear Symplectic Dynamic Systems.

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Abstract: Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

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2,581 citations

Cites background from "Difference equations and inequaliti..."

...[5]) reads as follows: Let f : Z → R be a function, and let α ∈ Z....
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Book•

Jacobi Operators and Completely Integrable Nonlinear Lattices

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Gerald Teschl¹•Institutions (1)

University of Vienna¹

01 Jan 1999

TL;DR: In this paper, the Toda system and the Kac-van Moerbeke system are studied. But the initial value problem is not considered in this paper, as it is in the case of Jacobi operators with periodic coefficients.

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Abstract: Jacobi operators: Jacobi operators Foundations of spectral theory for Jacobi operators Qualitative theory of spectra Oscillation theory Random Jacobi operators Trace formulas Jacobi operators with periodic coefficients Reflectionless Jacobi operators Quasi-periodic Jacobi operators and Riemann theta functions Scattering theory Spectral deformations-Commutation methods Completely integrable nonlinear lattices: The Toda system The initial value problem for the Toda system The Kac-van Moerbeke system Notes on literature Compact Riemann surfaces-A review Hergoltz functions Jacobi difference equations with MathematicaR Bibliography Glossary of notations Index.

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782 citations

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...The Jacobi difference expression τ applied to a sequence gives the following result In[2]:= JacobiDE[f[n], n] Out[2]= a[−1 + n]f[−1 + n] + b[n]f[n] + a[n]f[1 + n] This can also be written as In[3]:= JacobiDE[f[n], n] == −BDiff[a[n]FDiff[f[n], n], n] + (a[n − 1] + a[n] + b[n])f[n]//Simplify Out[3]= True or In[4]:= JacobiDE[f[n], n] == −FDiff[a[n−1]BDiff[f[n], n], n]+(a[n−1]+a[n]+ b[n])f[n]//Simplify Out[4]= True The command In[5]:= SolutionJacobi[u] will tell Mathematica that u(z, n) satisfies the Jacobi equation τu(z) = z u(z)....
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...A book which contains numerous examples and special results is the one by Agarwal [4]....
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...Observe that SymbSum does not look for a closed form; which can be obtained by switching to the built-in Sum: In[4]:= S1[n] /....
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...In[2]:= Print[MatrixForm[Cm[3]], ” ”, MatrixForm[Dm[3]], ” ”, MatrixForm[Pm[z, 3]]]  1 m[1] m[2] m[1] m[2] m[3] m[2] m[3] m[4]   1 m[1] m[3] m[1] m[2] m[4] m[2] m[3] m[5]   1 m[1] m[2] m[1] m[2] m[3] 1 z z(2) ...
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...We refer the reader to, for instance, [4], [87], or [147] and return to (1....
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Journal Article•DOI•

Dynamic equations on time scales: a survey

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Ravi P. Agarwal¹, Martin Bohner², Donal O'Regan³, Allan Peterson⁴•Institutions (4)

National University of Singapore¹, Missouri University of Science and Technology², National University of Ireland³, University of Nebraska–Lincoln⁴

01 Apr 2002-Journal of Computational and Applied Mathematics

TL;DR: In this article, the authors give an introduction to the time scales calculus, and present various properties of the exponential function on an arbitrary time scale, and use it to solve linear dynamic equations of first order.

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575 citations

Journal Article•DOI•

Basic Calculus on Time Scales and some of its Applications

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Ravi P. Agarwal¹, Martin Bohner¹•Institutions (1)

National University of Singapore¹

01 Mar 1999-Results in Mathematics

TL;DR: In this article, the authors developed basic tools of calculus on time scales such as versions of Taylor's formula, l'Hospital's rule, and Kneser's theorem.

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Abstract: The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hospital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.

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527 citations

The history of q-calculus and a new method

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Thomas Ernst

01 Jan 2000

TL;DR: In this paper, the authors considered the problem of additive number theory for the Riemann surface and showed that the solution of the problem can be expressed as a polynomial series.

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Abstract: Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

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245 citations

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Difference equations and inequalities

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