Functions for the General Solution of Parametric Word Equations

In this article we introduce the functions Fi(x1, x2)λ1,..., λs and Th(x1, x2, x3)iλ1,..., λ2s (i = 1, 2, 3), of the word variables xi and of the natural number variables λi, where s ≥ 0. By means of these functions, we give exactly the general solution (i.e. the set of all the solutions) of the first basic parametric equation: x1x2x3x4 = x3x1λx2x5, in a free monoid.