To do this, we write any mixed state as a

formal power series in some variables [z.sub.1], ..., [z.sub.k], with the coefficient of

When (1) is linear, i.e., F = Af-b, where b = b([epsilon], z) and A = A([epsilon], z) are, respectively, a v-vector and a v x v matrix, whose entries are holomorphic in the polydisc [D.sub.R] x [D.sub.R], R > 0, such that A[(0, 0).sup.-1] exists, Balser and Kostov [1] have established the following: (a) there exists a unique formal solution in the ring O(r)[[[[epsilon]].sub.1] of

formal power seriesKirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, in Proceedings of the 5th Conference on

Formal Power Series and Algebraic Combinatorics (Florence, 1993), Discrete Math.

This last equation possesses

formal power series solutions [??](t,[epsilon]) = [S.sub.-1](t)/[epsilon] + [[summation].sub.n[greater than or equal to]0] [S.sub.n](t)[[epsilon].sup.n], where [S.sub.-1](t) satisfies the quadratic equation [S.sup.2.sub.-1](t) = Q(t).

The spherical growth series (SGS) of is the

formal power seriesIt is well known that the field of

formal power series over finite fields has a lot of properties in common to number fields (the finite extension of Q).

It is then natural to investigate the arithmetic properties of Z[[chi]], the ring of

formal power series with integer coefficients.

of Quebec, Montreal) show how

formal power series may be viewed as formal languages with coefficients, and how finite automata may be considered as linear representations of the free monoid.

(See Remark A.4.) Hence, (1.2) should be interpreted as a

formal power series, and in this sense (1.3) and (1.4) hold and determine A(z).

For T [member of] O we will refer to the

formal power series [[sigma].sub.T] as the symbol of the operator T.

Still, [??] is a

formal power series, assumed to be 1-summable in direction [theta] and we look for [??] in the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as previously.

However, in that case, to have existence and uniqueness of solutions, we were forced to consider formal solutions defined by

formal power series. As we consider here a more general situation of arbitrary time case, the difficulties that appeared in the continuous-time case must inevitably show up also in this paper.