Functions for parametrization of solutions of an equation in a free monoid

Abstract:

In this paper we introduce recursive functions \begin{align*} &{}^{\mathbf {Fi}}(x_1,x_2)^{\lambda _1,\dotsc ,\lambda _s}\qquad (s\ge 0),\ &{}^{\mathbf {Th}}(x_1,x_2,x_3)_i^{\lambda _1,\dotsc ,\lambda _{2s}} \qquad (i=1,2,3;s\ge 0),\ &{}^{\mathbf {Ro}}(x_1,x_2,x_3)_i^{\mu _1,\dotsc ,\mu _s}\qquad (i=1,2,3;s\ge 0) \end{align*} of the word variables $x_1,x_2,x_3$, natural number variables $\lambda _k$ and variables $\mu _k$ whose values are finite sequences of natural number variables. By means of these functions we give finite expressions for the family of solutions of the equation \[ x_1x_2x_3x_4=\zeta (x_1,x_2,x_3)x_5,\] where $\zeta (x_1,x_2,x_3)$ is an arbitrary word in the alphabet $x_1,x_2,x_3$, in a free monoid.