Abstract
The following discrete initial value problem xn+1=xn(xn−12−2)−x1,n∈N,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x_{n+1}=x_{n}\\bigl(x_{n-1}^{2}-2 \\bigr)-x_{1},\\quad n\\in {\\mathbb{N}}, $$\\end{document}x_{0}=2 and x_{1}=5/2, appeared at an international competition. It is known that the problem can be solved in closed form. Here we discuss the solvability of a more general initial value problem which includes the former one. We show that, in a sense, there are not so many solvable discrete initial value problems related to this one, showing its specificity, which is a bit surprising result.
Highlights
Throughout the paper we use the standard notations N, N0, Z, R, R+, R, C for the sets of natural numbers, nonnegative integers, integers, real numbers, positive real numbers, negative real numbers, and complex numbers, respectively
Solvability of recursive relations/difference equations has been studied for a long time
(2021) 2021:90 is one of the first which was solved in closed form
Summary
Throughout the paper we use the standard notations N, N0, Z, R, R+, R–, C for the sets of natural numbers, nonnegative integers, integers, real numbers, positive real numbers, negative real numbers, and complex numbers, respectively. One of the reasons for this was the fact that behind existence of a closed-form formula for solutions to nonlinear difference equations and systems of difference equations usually lies a method, along with some tricks, for solving them (as we have already mentioned many nonlinear difference equations are transformed to linear solvable ones by suitable changes of variables), and in the case of initial value problem (5)–(6) we have a closedform formula for the problem.
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