The discrete Painleveequations are nonlinear

difference equations that arise as compatibility conditions of linear systems, says Joshi, then warns that the deceptive simplicity of this statement hides deep layers of mathematical properties, which she outlines here.

Implementing the Dynamic Back-Propagation algorithm requires the

difference equations of each parameter to be used in equations (14), (15), and (16).

The remaining part of the proof is similar to that of Theorem 0.1, by observing that Appell's O-functions satisfy the following

difference equations:

The general theory of linear quantum

difference equations was published in 1912 by Carmichael [3].

The following neutral difference equations/delay

difference equations are obtained as particular case of (2).

In the past few years, there is a great interest in studying the oscillatory and asymptotic behavior of solutions of higher order neutral type

difference equations, since such type of equations naturally arises in the applications including problems in population dynamics or in cobweb models in economics and so on.

Difference equations are powerful tool that describe the law of nature.

The initial system of

difference equations on one layer (2.2), (2.3) is equivalent to the newly obtained system (2.10)-(2.11) with formulas (2.3).

Drazin inverse is applied to (3) corresponding to the

difference equations; we will establish sufficient and necessary conditions for the solution of (3).

Differential and

difference equations are used for modeling, solving, and discussing many problems arising in engineering and natural sciences.

On the spectrum of eigenparameter-dependent quantum

difference equations. Appl.

For

difference equations there will be studied such issues as solvability, equilibrium existence and stability.