Constructible Differentially Finite Algebraic Series in Several Variables

We extend the concept of CDF-series to the context of several variables, and show that the series solution of first order differential equations y′ = x(t, y) and functional equation y = x(t, y), with x CDF in two variables, are CDF-series. We also give many effective closure properties for CDF-series in several variables. 1. CDF-series in one variable We present in this paper, new properties and an extension to several variables of the concept of CDF-series introduced by C. Reutenauer and the first author in [4]. Throughout this text, the abbreviation CDF stands for “constructible differentially finite algebraic”. A subset of CDF-series in one variables is discussed in [2] from the standpoint of their involvement in the complexity analysis of algorithms on increasing trees. CDF-series appear naturally in the study of enumeration problems in a manner similar to D-finite series which have been shown to have great importance in enumerative combinatorics by Gessel [5], Stanley [8] and Zeilberger [9]. On the other hand, Zeilberger has underlined how the “holonomic paradigm” (D-finite in several variables) allows for the automatic derivation of identities. In this light, Salvy and Zimmermann, and Plouffe and the first author have shown how one can obtain explicit or implicit forms for series out of a limited knowledge of their coefficients (see [7] and [3]), supposing that the series in question lies in the class of D-finite series. The interest of considering the class of CDF-series for similar purposes is illustrated in [1] and [3]. However to fully exploit this approach it is essential to extend the concept of CDF-series to several variables. Recalling definitions of [4], we denote C(t) the class of CDF series in one variable t defined as follows. Definition 1. A formal power series y = y(t) ∈ C[[t]] is said to be CDF (in one variable t) if there exists k ≥ 1, and k series y1, . . . , yk ∈ C[[t]] with y1 = y satisfying y′ 1 = P1(y1, . . . , yk) y′ 2 = P2(y1, . . . , yk) .. y′ k = Pk(y1, . . . , yk) (1) with initial conditions yi(0) = yi0, for some polynomials P1, . . . , Pk ∈ C[y1, . . . , yk]. We further say that this system is of order k and degree m, where m is the maximum total degree of the polynomials Pi. Remark 2. In definition 1, we may substitute polynomials Qj ∈ C[t, y1, . . . , yk] for the polynomials Pi, without changing the class C(t), since we can always add the equation y′ k+1 = 1, with initial condition yk+1 = 0, to the system. Another useful characterization for series in C(t) given in [4] is the following. Lemma 3. A series is C(t) if and only if it is contained in some finitely generated subalgebra of C[[t]] which is closed for differentiation. Using this lemma, one can show that the class C(t) enjoys nice effective closure properties. They are closed for addition, Cauchy-Product, inversion (when defined), composition (when defined), compositional inversion (when defined) and integration. Furthermore, they contain the class of algebraic power series. To illustrate the effectiveness of these closure properties, let us suppose that x1 = x(t) and y1 = y(t) are in C(t), with systems y′ 1 = P1(y1, . . . , yk) x ′ 1 = Q1(x1, . . . , x ) .. .. .. .. y′ k = Pk(y1, . . . , yk) x ′ = Q (x1, . . . , x ) (2) for some polynomials Pi and Qj . If z1 = z(t) = x(y(t)), then adding the equations z′ 1 = Q1(z1, . . . , z )P1(y1, . . . , yk) z′ 2 = Q2(z1, . . . , z )P1(y1, . . . , yk) .. .. z′ k = Qk(z1, . . . , z )P1(y1, . . . , yk) to system (2) clearly gives a system of form (1) for z. A more general closure property is the following. Proposition 4. Let x(t) be any series in C(t). Then the power series solution y = y(t) of the differential equation y′ = x(y), with initial condition y(0) = 0, is a CDF-series. Proof . For x = x1 let xi = Pi(x1, . . . , xk), with 1 ≤ i ≤ k, be a system of form (1) for x. We construct the following system for y = y1: y′ 1 = y2 y′ 2 = P1(y2, . . . , yk+1) y2 .. .. y′ k = Pk(y2, . . . , yk+1) y2. hence y verifies a system of form (1). A similar proof gives the following proposition. Proposition 5. Let x(t) be any series in C(t). Then the power series solution y = y(t) of the functional equation y = t x(y) is a CDF-series. 2. Comparison of C(t) with the class of D-finite series Recall that a D-finite power series y = y(t) is a series satisfying some linear differential equation pn(t) y + pn−1(t) y(n−1) + · · · + p0(t) y = 0 where the pi’s are polynomials with coefficients in C. We denote D(t) the class of D-finite series. The properties of this class are discussed at length in a paper of Stanley [8]. As illustrated in [4], the classes C(t) and D(t) are incomparable. However, the following lemma shows that many D-finite series are in C(t).