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

In this paper we introduce recursive functions (Xi,X2)I' (S > O), Th(X1, X2, X3).A2 (i -1, 2,3; s > 0), Ro(X 1, X2, x3) z (i1, 2,3; s > O) of the word variables xI, X2, X3, natural nuniber variables Ak and variables Pk 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 X1X2X3X4 = ((xi, X2, X3)x5, where ((x1, X2, x3) is an arbitrary word in the alphabet xi, X2, X3, iri a free monoid.