If we use the change of variables, the third equation in ( 3.2) becomes The third equations in systems ( 3.1) and ( 3.3) are particular cases of the following difference equation (up to the shifting indices): Now we discuss systems of difference equations given in ( 3.1)–( 3.3).įrom the third equation in ( 3.1), we getįrom the third equation in ( 3.3), we get The joint feature for all three cases is that can be solved in closed form. Since we know solutions for and, it is only needed to find explicit solutions for in the third equations in systems ( 3.1)–( 3.3), that is, in all three equations, the only unknown sequence is. , have been studied recently (see and the references therein).Īs is directly seen, the first two equations in systems ( 3.1)–( 3.3) are the same, and they form a particular case of system ( 1.1) which is solved in. The following third-order systems of difference equations Some Third-Order Systems of Difference Equations Related to ( 1.1) This method can be applied to any equation or system of difference equations, and one can get papers with putative “new” results. Systems of difference equations in ( 2.1) are artificially obtained in this way. In this way it can be obtained countable many, at first sight different, systems of difference equations. įor example, a natural extension of the systems in ( 2.1) is obtained for taking, ,, , and in ( 2.5), that is, the system becomes This means that, , are (independent) solutions of system ( 1.1), and solutions of system ( 2.5) are obtained by interlacing solutions of systems ( 2.7). Then system ( 2.5) is reduced to the following systems of difference equations: Systems ( 2.1) can be extended as follows: Hence, formulae for the solutions of systems ( 2.1) given in follow directly from those in. However, all the systems of difference equations in ( 2.4) are particular cases of system ( 1.1). This means that, , are two (independent) solutions of the systems of difference equations Then the systems in ( 2.1) are reduced to the next systems Indeed, if we use the change of variables Now we show that the results regarding system ( 2.1) easily follow from known ones. In the recent paper were given some formulae for the solutions of the following systems of difference equations: Where and are real numbers, was completely solved in, that is, we found formulae for all well-defined solutions of system ( 1.1). Our aim here is to give theoretical explanations for some of the formulae recently appearing in the literature, as well as to give some extensions of their equations.īefore we formulate our results, we would like to say that the system of difference equations Our explanation of such a formula that we gave in has re-attracted attention to solvable difference equations. īeside the above-mentioned papers, there are some papers which give formulae of some very particular equations and systems which are proved by induction, but without any explanation how these formulae are obtained and how these authors came across the equations and systems. For some classical results in the topic see, for example. Recently, there has been a great interest in difference equations and systems (see, e.g., ), and among them in those ones which can be solved explicitly (see, e.g., and the related references therein). 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.
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