The paper under review introduces the notion of a finitely \(n\)-reducible difference field extension, uses this concept for the study of solutions of first-order difference equations, and presents applications of the obtained results to the analysis of some classical functions.
An ordinary difference field extension \(\mathcal{L}/\mathcal{K}\) is said to be finitely \(n\)-reducible (\(n\) is a positive integer) if there exists a chain of difference field extensions \(\mathcal{K} = \mathcal{K}_{0}\subset \mathcal{K}_{1}\subset\dots\subset \mathcal{K}_{m}= \mathcal{L}\) such that each extension \(\mathcal{K}_{i}/\mathcal{K}_{i-1}\) satisfies one of the following: (i) \(K_{i}/K_{i-1}\) is algebraic (\(K\), \(L\), etc. denote the underlying fields of \(\mathcal{K}\), \(\mathcal{L}\), etc.). (ii) Both \(\mathcal{K}_{i}\) and \(\mathcal{K}_{i-1}\) are inversive and \(K_{i}/K_{i-1}\) is a finitely generated field extension whose transcendence degree is \(\leq n\).
The main result of the paper is the following statement (Theorem 1.2): Let \(\mathcal{K}\) be an ordinary difference field and \(f\) a solution of a difference equation over \(\mathcal{K}\) of the form \(B(y)y_{1} = A(y)\), where \(A\) and \(B\) are relatively prime nonzero univariate polynomials such that \(\max\{\deg A, \deg B\}\geq 2\). If \(\mathcal{K}\langle f\rangle\subset \mathcal{L}\) for some finitely \(n\)-reducible difference field extension \(\mathcal{L}/\mathcal{K}\), then \(f\) is algebraic over \(K\). (\(y_{i}\) denotes the \(i\)th translation of \(y\); the underlying field of \(\mathcal{K}\langle f\rangle\) is \(K(f, f_{1}, f_{2},\dots)\).)
Applying this result, the author shows that the exponential function \(e^{z}\), the trigonometric functions \(\cos z\) and \(\sin z\), as well as the Weierstrass elliptic function, cannot be built up by using the variable \(z\) and constants, together with repeated algebraic operations and taking solutions of linear difference equations on the multiplication \(z\mapsto 2z\) and solutions of algebraic difference equations such as difference Riccati equation on \(z\mapsto 2z\).