On Some Solvable Difference Equations and Systems of Difference Equations

Abstract

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

Meromorphic functions that share a polynomial with their difference operators

“…For example, many solvable difference equations are essentially solved by using it (see [15][16][17]), and by using such obtained formulas the behavior of their solution can be studied (see, for example, recent papers [13,14]; see also many related references therein). As another simple application, by using a linear change of variables, the following corollary is obtained:…”

General k-Dimensional Solvable Systems of Difference Equations

“…First assume that (19) holds, that is, λ j = 0 for j = 1, k. By using the change of variables (15), system (8) is transformed into system (16). From the analysis preceding the formulation of the theorem we see that for every solution to system (16) which has no zero component (so, for almost all initial values) the sequence x…”

“…A related solvable system of difference equations was treated in [12]. Since that time various modifications of the method have been often used (see [13,14] and the references therein for some related difference equations, as well as [15][16][17] and the references therein for some related systems of difference equations). It should be pointed out that the systems are usually symmetric or close-to-symmetric, whose study was popularized by Papaschinopoulos and Schinas ( [18][19][20][21][22][23][24]).…”

Oscillation of solutions to non-linear difference equations with several advanced arguments

“…In the proof of the lemma we essentially follow the lines of a method for solving a difference equation, an area of considerable recent interest (see, for example [16][17][18] and the references therein, where closely related methods were used).…”