Abstract
Closed-form formulas for general solutions to sixteen hyperbolic-cotangent-type systems of difference equations of interest are obtained, showing their practical solvability and completely solving a solvability problem for some concrete values of delays.
Highlights
1.1 A little on solvability Solvability of recurrence relations was started to be studied long time ago [1,2,3]
Difference equations has been known for a long time [4]
There we have shown that to some difference equations and systems of difference equations can be naturally associated some linear difference equations with constant coefficients such that some of their solutions can be used in representations of the general solutions to the equations and systems
Summary
1.1 A little on solvability Solvability of recurrence relations was started to be studied long time ago [1,2,3]. This means that for each of the sixteen systems in (3) there is a finite number of closed-form formulas representing its general solution (for some more explanations related to the notion, as well as for some examples, see [55]) This is done by considerable use of some methods and ideas on product-type difference equations and systems, which can be found, e.g., in the following recent papers: [33,34,35,36,37] (see the related references therein). Proof By a well-known result from theory of homogeneous linear difference equations with constant coefficients, a general solution to equation (1), in the case when the roots tk, k = 1, m, of the characteristic polynomial pm are distinct, has the following form: xn = c1t1n + c2t2n + · · · + cmtmn , n ≥ l – m,.
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