On the convergence of generalized power series satisfying an algebraic ODE

Abstract

Here we propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.

Keywords

algebraic ODE formal solution generalized power series majorant method convergent series

1. Introduction

In the paper we study some properties of generalized power series $φ = \sum_{k = 0}^{\infty} c_{k} x^{s_{k}}, c_{k} \in C, s_{0} ≺ s_{1} ≺ \dots \in C, lim_{k \to \infty} Re s_{k} = + \infty,$ (1) formally satisfying an ordinary differential equation $F (x, y, δ y, \dots, δ^{n} y) = 0$ (2) of order n with respect to the unknown y, where $F (x, y_{0}, y_{1}, \dots, y_{n}) ≢ 0$ is a polynomial of $n + 2$ variables and $δ = x \frac{d}{d x}$ . Here ≺ is a usual ordering by first difference: $s_{k} ≺ s_{k + 1}$ iff $Re s_{k} < Re s_{k + 1}$ or $Re s_{k} = Re s_{k + 1}$ , $Im s_{k} < Im s_{k + 1}$ .

Note that substituting the series (1) into Eq. (2) makes sense, as only a finite number of terms in φ contribute to any term of the form $c x^{s}$ in the expansion of $F (x, φ, δ φ, \dots, δ^{n} φ)$ in powers of x. Indeed, $δ^{j} φ = \sum_{k = 0}^{\infty} c_{k} s_{k}^{j} x^{s_{k}}$ and an equation $s = s_{k_{0}} + s_{k_{1}} + \dots + s_{k_{m}}$ has a finite number of solutions $(s_{k_{0}}, s_{k_{1}}, \dots, s_{k_{m}})$ , since $Re s_{k} \to + \infty$ . Furthermore, for any integer N an inequality $s_{k_{0}} + s_{k_{1}} + \dots + s_{k_{m}} ⪯ N$ has also a finite number of solutions, so that powers of x in the expansion of $F (x, φ, δ φ, \dots, δ^{n} φ)$ are well ordered with respect to ≺.

Earlier in the paper [8] generalized power series of the form (1), with $s_{0} < s_{1} < \dots \in R$ , were studied. There was proved (not assuming $s_{k} \to + \infty$ ) that they form a differential ring, and if the series (1) satisfies Eq. (2), then ${lim}_{k \to \infty} s_{k} = + \infty$ . Furthermore, the exponents $s_{k} \in R$ of the formal solution (1) generate a $Z$ -module of finite type. Here we prove this fact in the case of complex exponents $s_{k} \in C$ (Lemma 2).

In a mathematical literature series of the form (1) as formal solutions of nonlinear ordinary differential equations are known long ago (in particular, one can meet them in papers concerning the Painlevé equations; for example, see [4,7,14]). Here we are mainly interested in general conditions of the convergence of such series.

For the generalized power series (1) one may naturally define the valuation $val φ = s_{0},$ and this is also well defined for any polynomial in $φ, δ φ, \dots, δ^{n} φ$ with coefficients of the form $α x^{β}$ , $α, β \in C$ .

The main result of the paper is the following sufficient condition of the convergence of (1).

Theorem 1.

Let the generalized power series ( 1 ) formally satisfy Eq.( 2 ) , $\frac{\partial F}{\partial y_{n}} (x, φ, δ φ, \dots, δ^{n} φ) \neq 0$ and for each $i = 0, 1, \dots, n$ one has $\frac{\partial F}{\partial y_{i}} (x, φ, δ φ, \dots, δ^{n} φ) = B_{i} x^{λ} + C_{i} x^{λ_{i}} + \dots, Re λ_{i} > Re λ, B_{n} \neq 0 .$ (3) Then for any sector S of sufficiently small radius with the vertex at the origin and of the opening less than $2 π$ , the series φ converges uniformly in S.

This theorem in a somewhat different form has been formulated in [3, Theorem 3.4] for the case of real powers $s_{k} \in R$ . In the case of integer powers our theorem reduces to the case of convergence in Malgrange’s theorem [10] concerning formal solutions $\hat{φ} = \sum_{k = 0}^{\infty} c_{k} x^{k} \in C ⟦ x ⟧$ of (2). An idea of our proof is based on the construction of a majorant algebraic equation and was used in [6] for estimating the radius of convergence of a formal solution $\hat{φ} \in C ⟦ x ⟧$ satisfying the conditions of the corresponding part of Malgrange’s theorem. It originally comes from [4, Chapter 1, Section 7], similar ideas are seemed to be already appeared in [5] for studying some properties of divergent formal solutions $\hat{φ} \in C ⟦ x ⟧$ of (2).

The paper is organized as follows. In the next section we prove some auxiliary lemmas preceding the proof of Theorem 1, which is given in Section 3. In Section 4 we illustrate the statement of Theorem 1 with an example of generalized power series solutions of the third Painlevé equation. The last section contains some concluding remarks concerning further possible investigations of generalized formal power series solutions of (2).

2. Auxiliary lemmas

The first auxiliary lemma is an analogue of the classical linearization lemma (see, for example, [10]) adapted to the case of generalized power series. Here we mainly use technical tools of the paper [10].

Lemma 1.
Under the assumptions of Theorem 1, there exists an integer $μ^{'} ⩾ 0$ such that for any integer $μ ⩾ μ^{'}$ a transformation $y = \sum_{k = 0}^{μ} c_{k} x^{s_{k}} + x^{s_{μ}} u$ reduces Eq.(2) to an equation of the form $L (δ) u = N (x, u, δ u, \dots, δ^{n} u),$ (4) where L is a polynomial of degree n, and N is a finite sum of monomials of the form $α x^{β} u^{q_{0}} {{(δ u)}^{q_{1}} \dots (δ^{n} u)}^{q_{n}}, α, β \in C, Re β > 0, q_{i} \in Z_{+} .$
Proof.
Making a transformation $y = x^{s_{0}} v$ and taking into consideration the equality $δ (x^{s_{0}} v) = x^{s_{0}} (δ + s_{0}) v$ , we have the relations $δ^{i} y = x^{s_{0}} {(δ + s_{0})}^{i} v, i = 1, \dots, n,$ and therefore come from (2) to an equation $H (x, v, δ v, \dots, δ^{n} v) = 0,$ (5) where the function $H (x, v_{0}, v_{1}, \dots, v_{n})$ is a finite sum of monomials of the form $c x^{l + m s_{0}} v_{0}^{q_{0}} v_{1}^{q_{1}} \dots v_{n}^{q_{n}}, c \in C, l, m, q_{i} \in Z_{+} .$ We also may assume that $l + m s_{0} ⪰ 0$ (multiplying, if necessary, Eq. (5) by a corresponding $x^{K}$ , $K \in N$ ). The obtained Eq. (5) has a formal solution $\tilde{φ} = \sum_{k = 0}^{\infty} c_{k} x^{s_{k} - s_{0}}, 0 ≺ s_{1} - s_{0} ≺ s_{2} - s_{0} ≺ \dots .$

For any $μ \in Z_{+}$ , the formal series $\tilde{φ}$ can be represented in the form $\tilde{φ} = \sum_{k = 0}^{μ} c_{k} x^{s_{k} - s_{0}} + x^{s_{μ} - s_{0}} ψ = φ_{μ} + x^{s_{μ} - s_{0}} ψ, val ψ = s_{μ + 1} - s_{μ} ≻ 0 .$ Then denoting $Φ = (\tilde{φ}, δ \tilde{φ}, \dots, δ^{n} \tilde{φ}) = Φ_{μ} + x^{s_{μ} - s_{0}} Ψ$ and applying the Taylor formula to the relation $H (x, Φ) = 0$ , we have $\begin{array}{rclr} 0 & = & H (x, Φ_{μ} + x^{s_{μ} - s_{0}} Ψ) \\ = & H (x, Φ_{μ}) + x^{s_{μ} - s_{0}} \sum_{i = 0}^{n} \frac{\partial H}{\partial v_{i}} (x, Φ_{μ}) ψ_{i} + \frac{1}{2} x^{2 (s_{μ} - s_{0})} \sum_{i, j = 0}^{n} \frac{\partial^{2} H}{\partial v_{i} \partial v_{j}} (x, Φ_{μ}) ψ_{i} ψ_{j} + \dots, & (6) \end{array}$ where $ψ_{i} = {(δ + s_{μ} - s_{0})}^{i} ψ$ .

One can easily check that the assumption $\frac{\partial F}{\partial y_{n}} (x, φ, δ φ, \dots, δ^{n} φ) \neq 0$ of Theorem 1 implies $\frac{\partial H}{\partial v_{n}} (x, Φ) \neq 0$ , and assumption (3) provides the analogous form of the series $\frac{\partial H}{\partial v_{i}} (x, Φ)$ : $\frac{\partial H}{\partial v_{i}} (x, Φ) = b_{i} x^{θ} + c_{i} x^{θ_{i}} + \dots, i = 0, 1, \dots, n,$ where $b_{i} \in C$ ( $b_{n} \neq 0$ ), $c_{i} \in C^{}$ and $Re θ_{i} > Re θ$ . Define a polynomial $L (ξ) = b_{0} + b_{1} (ξ + s_{μ} - s_{0}) + \dots + b_{n} {(ξ + s_{μ} - s_{0})}^{n}$ (7) of degree n and choose a number μ such that the following three conditions hold: $(1) Re (s_{μ} - s_{0}) > Re θ, (2) Re (s_{μ + 1} - s_{μ}) > 0, (3) L (ξ) \neq 0 \forall ξ ≻ 0 .$ Now we show that such a number μ is from the statement of the lemma.

Let us note that $\begin{array}{rcl} val (\frac{\partial H}{\partial v_{i}} (x, Φ) - \frac{\partial H}{\partial v_{i}} (x, Φ_{μ})) & = & val (x^{s_{μ} - s_{0}} \sum_{j = 0}^{n} \frac{\partial^{2} H}{\partial v_{i} \partial v_{j}} (x, Φ_{μ}) ψ_{j} + \dots) \\ = & θ + α_{i}, Re α_{i} > 0, i = 0, 1, \dots, n, \end{array}$ since $Re (s_{μ} - s_{0}) > Re θ$ , and the real parts of the numbers $val \frac{\partial^{2} H}{\partial v_{i} \partial v_{j}} (x, Φ_{μ})$ are nonnegative (the same is true for the valuations of the other partial derivatives of the function H). Therefore, $\frac{\partial H}{\partial v_{i}} (x, Φ_{μ}) = b_{i} x^{θ} + {\tilde{c}}_{i} x^{{\tilde{θ}}_{i}} + \dots, Re {\tilde{θ}}_{i} > Re θ .$ Now from relation (6) and conditions $Re (s_{μ + 1} - s_{μ}) > 0$ , $Re (s_{μ} - s_{0}) > Re θ$ , it follows that $Re (val H (x, Φ_{μ}) - (s_{μ} - s_{0} + θ)) > 0 .$ Hence relation (6) can be divided by $x^{s_{μ} - s_{0} + θ}$ , and we obtain the equality of the form $L (δ) ψ - N (x, ψ, δ ψ, \dots, δ^{n} ψ) = 0,$ where the polynomial L is defined by formula (7), and N is such as in the statement of the lemma. Thus, the transformation $v = \sum_{k = 0}^{μ} c_{k} x^{s_{k} - s_{0}} + x^{s_{μ} - s_{0}} u$ reduces Eq. (5) to the equation $L (δ) u = N (x, u, δ u, \dots, δ^{n} u),$ with a formal solution $u = ψ$ . □
Remark 1.
The condition $L (ξ) \neq 0$ , for all $ξ ≻ 0$ , is not used in the proof of Lemma 1, but we add it from the beginning as it will be used in the proof of the next lemma and in the sequel.
Remark 2.
Like in the case of integer power series, Lemma 1 plays an important role in the proof of the convergence in our case of generalized power series. The difference between the statements of the lemma in these two cases is that coefficients $α x^{β}$ of the monomials of the sum N in (4) have complex powers β, while in the case of integer power series β are certainly integer (and strictly positive). Here we have the real parts of β are strictly positive, which is essentially used in the proof of the next lemma. For the proof of the convergence it is also important that the order of the linear differential operator $L (δ)$ is exactly n (though we deal with generalized power series having complex power exponents, note that $L (δ)$ is a linear differential operator with constant coefficients (with respect to δ) like in the case of integer power series).

Let us define an additive semi-group Γ generated by a (finite) set of power exponents of the variable x containing in $N (x, u, δ u, \dots, δ^{n} u)$ , and let $r_{1}, \dots, r_{ν}$ be generators of this semi-group, that is, $Γ = {m_{1} r_{1} + \dots + m_{ν} r_{ν} | m_{i} \in Z_{+}, \sum_{i = 1}^{ν} m_{i} > 0}, Re r_{i} > 0 .$ The second auxiliary lemma is a consequence of the first one and describes a structure of the set of power exponents $s_{k} \in C$ of the formal solution (1).
Lemma 2.
All the numbers* $s_{k} - s_{μ}$ , $k ⩾ μ + 1$ , belong to an additive semi-group Γ.
Proof.
We use the fact that the generalized formal power series $ψ = \sum_{k = μ + 1}^{\infty} c_{k} x^{s_{k} - s_{μ}}$ satisfies the relation $L (δ) ψ = N (x, ψ, δ ψ, \dots, δ^{n} ψ) .$ (8) The first term of the left-hand side of (8) is $L (s_{μ + 1} - s_{μ}) c_{μ + 1} x^{s_{μ + 1} - s_{μ}}$ (note that $L (s_{μ + 1} - s_{μ}) \neq 0$ , as $s_{μ + 1} - s_{μ} ≻ 0$ ), while the first term of the right-hand side of (8) is a monomial $α x^{β}$ , since other monomials $α x^{β} ψ^{q_{0}} {{(δ ψ)}^{q_{1}} \dots (δ^{n} ψ)}^{q_{n}}$ , $\sum_{i = 0}^{n} q_{i} > 0$ , have higher valuations. Therefore, $L (s_{μ + 1} - s_{μ}) c_{μ + 1} x^{s_{μ + 1} - s_{μ}} = α x^{β},$ and $s_{μ + 1} - s_{μ} = β \in Γ$ .

Further, the second term of the left-hand side of (8) is $L (s_{μ + 2} - s_{μ}) c_{μ + 2} x^{s_{μ + 2} - s_{μ}}$ , while the second term of the right-hand side of (8) is of the form $α x^{β + m (s_{μ + 1} - s_{μ})}$ , $m \in Z_{+}$ , $β \in Γ$ . Therefore, $L (s_{μ + 2} - s_{μ}) c_{μ + 2} x^{s_{μ + 2} - s_{μ}} = α x^{β + m (s_{μ + 1} - s_{μ})},$ and $s_{μ + 2} - s_{μ} = β + m (s_{μ + 1} - s_{μ}) \in Γ$ . In the analogous way, one obtains that all the numbers $s_{k} - s_{μ}$ , $k > μ$ , belong to the semi-group Γ. □

We may assume that the generators $r_{1}, \dots, r_{ν}$ of the semi-group Γ are linearly independent over $Z$ . This is provided by the following lemma. Lemma 3.
There are complex numbers $ρ_{1}, \dots, ρ_{τ}$ linearly independent over $Z$ , such that all $Re ρ_{i} > 0$ and an additive semi-group $Γ^{'}$ generated by them contains the above semi-group Γ generated by $r_{1}, \dots, r_{ν}$ .
Proof.
Let $r_{1}, \dots, r_{τ}$ be a maximal system of linearly independent over $Z$ elements from the set ${r_{1}, \dots, r_{ν}}$ . It is sufficient to prove that if we add any number b with $Re b > 0$ such that $r_{1}, \dots, r_{τ}, b$ , become linearly dependent over $Z$ , then the semi-group G generated by $r_{1}, \dots, r_{τ}, b$ , is contained in some semi-group $Γ^{'}$ generated by τ linearly independent over $Z$ complex numbers $ρ_{1}, \dots, ρ_{τ}$ having positive real parts. We may assume that $b = m_{1} r_{1} + \dots + m_{τ - j} r_{τ - j} - m_{τ - j + 1} r_{τ - j + 1} - \dots - m_{τ} r_{τ}, m_{i} \in N, 1 ⩽ j ⩽ τ - 1,$ (9) and we prove the existence of such a semi-group $Γ^{'}$ by the induction with respect to the number j of the signs “−” before the coefficients in the linear combination (9). (We assume that all $m_{i} \neq 0$ . In general, if some coefficients $m_{i}$ are equal to zero, then the corresponding generators $r_{i}$ are included without changes into the set of generators of a new semi-group $Γ^{'}$ , i.e., $ρ_{i} = r_{i}$ for such $r_{i}$ . In this case the reader can easily make the corresponding changes in the reasonings below.)

For $j = 1$ we have $b = m_{1} r_{1} + \dots + m_{τ - 1} r_{τ - 1} - m_{τ} r_{τ} .$ There are rational positive numbers $p_{1} / q_{1}, \dots, p_{τ - 1} / q_{τ - 1}$ such that $p_{1} / q_{1} + \dots + p_{τ - 1} / q_{τ - 1} = 1$ and $\frac{p_{i}}{q_{i}} < \frac{m_{i} Re r_{i}}{m_{τ} Re r_{τ}}, i = 1, \dots, τ - 1 .$ (10) Indeed, the intersection of the box ${(x_{1}, \dots, x_{τ - 1}) \in R^{τ - 1} ∣ 0 < x_{i} < m_{i} Re r_{i} / m_{τ} Re r_{τ}}$ with the hyperplane $π = {x_{1} + \dots + x_{τ - 1} = 1}$ is a non-empty open subset of π, since $\sum_{i = 1}^{τ - 1} m_{i} Re r_{i} / m_{τ} Re r_{τ} > 1$ in view of the condition $Re b > 0$ . Hence, we can choose a point in this intersection that has rational coordinates.

Generators $ρ_{1}, \dots, ρ_{τ}$ of a semi-group $Γ^{'}$ that are linearly independent over $Z$ now can be defined as follows: $\begin{matrix} ρ_{1} = r_{1} - \frac{p_{1}}{q_{1} m_{1}} m_{τ} r_{τ}, \dots, ρ_{τ - 1} = r_{τ - 1} - \frac{p_{τ - 1}}{q_{τ - 1} m_{τ - 1}} m_{τ} r_{τ}, \\ ρ_{τ} = \frac{1}{(q_{1} m_{1}) \dots (q_{τ - 1} m_{τ - 1})} r_{τ} . \end{matrix}$ Then $Re ρ_{i} > 0$ according to (10). Furthermore $G \subset Γ^{'}$ , as $b = m_{1} ρ_{1} + \dots + m_{τ - 1} ρ_{τ - 1}$ and $\begin{matrix} r_{1} = ρ_{1} + \frac{p_{1}}{q_{1} m_{1}} m_{τ} r_{τ} = ρ_{1} + n_{1} ρ_{τ}, n_{1} \in N, \\ \dots \\ r_{τ - 1} = ρ_{τ - 1} + \frac{p_{τ - 1}}{q_{τ - 1} m_{τ - 1}} m_{τ} r_{τ} = ρ_{τ - 1} + n_{τ - 1} ρ_{τ}, n_{τ - 1} \in N, \\ r_{τ} = (q_{1} m_{1}) \dots (q_{τ - 1} m_{τ - 1}) ρ_{τ} . \end{matrix}$

For an arbitrary $j > 1$ we write the number b in the form $b = b^{'} - m_{τ} r_{τ}$ , where $b^{'} = m_{1} r_{1} + \dots + m_{τ - j} r_{τ - j} - m_{τ - j + 1} r_{τ - j + 1} - \dots - m_{τ - 1} r_{τ - 1}$ has $j - 1$ signs “−” in its representation of the form (9). Thus, we may apply an inductive assumption to the number $b^{'}$ and write it as follows: $b^{'} = m_{1} ρ_{1} + \dots + m_{τ - j} ρ_{τ - j},$ where the numbers $ρ_{1}, \dots, ρ_{τ - j}, \dots, ρ_{τ - 1}$ are linearly independent over $Z$ and expressed via linear combinations of $r_{1}, \dots, r_{τ - 1}$ with rational coefficients (conversely, $r_{1}, \dots, r_{τ - 1}$ are expressed via linear combinations of $ρ_{1}, \dots, ρ_{τ - 1}$ with positive integer coefficients). Hence, $r_{τ}$ and $ρ_{1}, \dots, ρ_{τ - j}, \dots, ρ_{τ - 1}$ are linearly independent over $Z$ , and we conclude for $b = m_{1} ρ_{1} + \dots + m_{τ - j} ρ_{τ - j} - m_{τ} r_{τ}$ as in the case $j = 1$ . □

3. Proof of Theorem 1

For the simplicity of presentation, we give the proof in the case of two generators of the semi-group Γ: $Γ = {l r_{1} + m r_{2} ∣ l, m \in Z_{+}, l + m > 0}, Re r_{1, 2} > 0 .$ As will be shown further, in this case we deal with functions of two variables. In the case of an arbitrary number ν of generators, as can be easily seen, the proof is analogous, only functions of higher number of variables are involved.

We should prove the convergence of the generalized formal power series $ψ = \sum_{k = μ + 1}^{\infty} c_{k} x^{s_{k} - s_{μ}},$ which satisfies the equality $L (δ) ψ = N (x, ψ, δ ψ, \dots, δ^{n} ψ)$ (11) obtained in Lemma 1. According to Lemma 2, all the exponents $s_{k} - s_{μ}$ belong to the semi-group Γ: $s_{k} - s_{μ} = l r_{1} + m r_{2}, (l, m) \in M \subset Z_{+}^{2} ∖ {0}$ for some subset M such that the map $k \mapsto (l, m)$ is a bijection from $N ∖ {1, \dots, μ}$ to M. Then $\begin{array}{rcl} L (δ) ψ & = & \sum_{k = μ + 1}^{\infty} L (s_{k} - s_{μ}) c_{k} x^{s_{k} - s_{μ}} = \sum_{(l, m) \in M} L (l r_{1} + m r_{2}) c_{l, m} x^{l r_{1} + m r_{2}} \\ = & \sum_{(l, m) \in Z_{+}^{2} ∖ {0}} L (l r_{1} + m r_{2}) c_{l, m} x^{l r_{1} + m r_{2}} \end{array}$ (in the last series one puts $c_{l, m} = 0$ , if $(l, m) \notin M$ ).

Without lost of generality we may assume that all $| s_{k} - s_{μ} | = | l r_{1} + m r_{2} | ⩾ 1$ , since ${lim}_{k \to \infty} (s_{k} - s_{μ}) = \infty$ . (In the opposite case we make the transformation $u = \sum_{k = μ + 1}^{ν} c_{k} x^{s_{k} - s_{μ}} + w$ , where ν is such that all $| s_{k} - s_{μ} | ⩾ 1$ for $k > ν$ . This transformation reduces Eq. (4) to an equation of the same form with respect to the unknown w, with the formal solution $w = \sum_{k = ν + 1}^{\infty} c_{k} x^{s_{k} - s_{μ}}$ .)

The function $N (x, u_{0}, u_{1}, \dots, u_{n})$ determining the right-hand side of equality (11) is a finite sum of the form $N (x, u_{0}, u_{1}, \dots, u_{n}) = \sum_{p, q, Q} α_{p, q, Q} x^{p r_{1} + q r_{2}} u_{0}^{q_{0}} u_{1}^{q_{1}} \dots u_{n}^{q_{n}},$ where $α_{p, q, Q} \in C$ , $(p, q) \in Z_{+}^{2} ∖ {0}$ , $Q = (q_{0}, q_{1}, \dots, q_{n}) \in Z_{+}^{n + 1}$ . Thus equality (11) is written as follows: ${{\sum_{(l, m) \in Z_{+}^{2} ∖ {0}} L (l r_{1} + m r_{2}) c_{l, m} x^{l r_{1} + m r_{2}} = \sum_{p, q, Q} α_{p, q, Q} x^{p r_{1} + q r_{2}} ψ^{q_{0}} (δ ψ)}^{q_{1}} \dots (δ^{n} ψ)}^{q_{n}},$ (12) where $δ^{j} ψ = \sum_{k = μ + 1}^{\infty} c_{k} {(s_{k} - s_{μ})}^{j} x^{s_{k} - s_{μ}} = {\sum_{(l, m) \in Z_{+}^{2} ∖ {0}} (l r_{1} + m r_{2})}^{j} c_{l, m} x^{l r_{1} + m r_{2}} .$ Therefore, we have the coincidence of the following two formal power series of two independent variables $z_{1}$ , $z_{2}$ : $\sum_{(l, m) \in Z_{+}^{2} ∖ {0}} L (l r_{1} + m r_{2}) c_{l, m} z_{1}^{l} z_{2}^{m} = \sum_{p, q, Q} α_{p, q, Q} z_{1}^{p} z_{2}^{q} ψ_{0}^{q_{0}} ψ_{1}^{q_{1}} \dots ψ_{n}^{q_{n}},$ (13) where $ψ_{j} = {\sum_{(l, m) \in Z_{+}^{2} ∖ {0}} (l r_{1} + m r_{2})}^{j} c_{l, m} z_{1}^{l} z_{2}^{m} \in C ⟦ z_{1}, z_{2} ⟧, j = 0, 1, \dots, n .$ Indeed, the coefficient $a_{l, m}$ of a monomial $z_{1}^{l} z_{2}^{m}$ in the right-hand side of (13) coincides with the coefficient of the corresponding monomial $x^{l r_{1} + m r_{2}}$ in the right-hand side of equality (12), since for each pair $(l, m) \in Z_{+}^{2} ∖ {0}$ there is no another pair $(l^{'}, m^{'})$ such that $l^{'} r_{1} + m^{'} r_{2} = l r_{1} + m r_{2}$ (in view of the linear independence of the numbers $r_{1}$ , $r_{2}$ over $Z$ ). Hence, $a_{l, m} = L (l r_{1} + m r_{2}) c_{l, m}$ .

To prove the convergence of $ψ_{0} \in C ⟦ z_{1}, z_{2} ⟧$ in some neighbourhood of the origin, we construct an equation $σ W = \sum_{p, q, Q} | α_{p, q, Q} | z_{1}^{p} z_{2}^{q} W^{q_{0}} W^{q_{1}} \dots W^{q_{n}},$ (14) whose right-hand side is obtained from that of equality (13) by the change of the coefficients $α_{p, q, Q}$ to their absolute values and all the $ψ_{j}$ to the one variable W. The number σ is defined by the formula $σ = inf_{k > μ} \frac{| L (s_{k} - s_{μ}) |}{| s_{k} - s_{μ} |^{n}} = inf_{(l, m) \in M} \frac{| L (l r_{1} + m r_{2}) |}{| l r_{1} + m r_{2} |^{n}}$ and is a positive real number, since $L (ξ) \neq 0$ for all $ξ ≻ 0$ , and ${lim}_{k \to \infty} | L (s_{k} - s_{μ}) | / | s_{k} - s_{μ} |^{n} = | b_{n} | > 0$ (recall that $L (ξ) = b_{0} + b_{1} (ξ + s_{μ} - s_{0}) + \dots + b_{n} {(ξ + s_{μ} - s_{0})}^{n}$ , $b_{n} \neq 0$ ). Equation (14) possesses a unique holomorphic near the origin solution $W = \sum_{(l, m) \in Z_{+}^{2} ∖ {0}} A_{l, m} z_{1}^{l} z_{2}^{m}$ satisfying the condition $W (0, 0) = 0$ . This follows from the theorem on implicit function. One can write the coefficients $A_{l, m}$ in the form $A_{l, m} = C_{l, m} | l r_{1} + m r_{2} |^{n}, | C_{l, m} | ⩽ | A_{l, m} | .$ Further we prove that the convergent near the origin power series $\tilde{W} = \sum_{(l, m) \in Z_{+}^{2} ∖ {0}} C_{l, m} z_{1}^{l} z_{2}^{m}$ is majorant for the formal power series $ψ_{0}$ , that is, $C_{l, m} \in R_{+}, | c_{l, m} | ⩽ C_{l, m} \forall (l, m) \in Z_{+}^{2} ∖ {0},$ which will imply the convergence of $ψ_{0}$ in some neighbourhood of the origin.

First we use equality (13) to obtain recursive expressions for the coefficients $c_{l, m}$ . Denote by ϕ the formal power series from the right-hand side of this equality, $ϕ = \sum_{p, q, Q} α_{p, q, Q} z_{1}^{p} z_{2}^{q} ψ_{0}^{q_{0}} ψ_{1}^{q_{1}} \dots ψ_{n}^{q_{n}} \in C ⟦ z_{1}, z_{2} ⟧,$ then (13) implies $L (l r_{1} + m r_{2}) c_{l, m} = \frac{\partial_{z_{1}}^{l} \partial_{z_{2}}^{m} ϕ}{l! m!} |_{z_{1} = z_{2} = 0},$ (15) where $\partial_{z_{1}}$ is the partial derivative with respect to $z_{1}$ , and $\partial_{z_{2}}$ is that with respect to $z_{2}$ . To express $\partial_{z_{1}}^{l} \partial_{z_{2}}^{m} ϕ (0, 0)$ , let us apply the formulae for the derivation of a product, $\begin{matrix} \partial_{z_{1}}^{l} (f_{0} \dots f_{n}) = \sum_{l_{0} + \dots + l_{n} = l} \frac{l!}{l_{0}! l_{1}! \dots l_{n}!} (\partial_{z_{1}}^{l_{0}} f_{0}) \dots (\partial_{z_{1}}^{l_{n}} f_{n}), f_{i} \in C ⟦ z_{1}, z_{2} ⟧, \\ \partial_{z_{1}}^{l} \partial_{z_{2}}^{m} (f_{0} \dots f_{n}) = \sum_{\begin{array}{c} l_{0} + \dots + l_{n} = l \\ m_{0} + \dots + m_{n} = m \end{array}} l! m! \frac{\partial_{z_{1}}^{l_{0}} \partial_{z_{2}}^{m_{0}} f_{0}}{l_{0}! m_{0}!} \dots \frac{\partial_{z_{1}}^{l_{n}} \partial_{z_{2}}^{m_{n}} f_{n}}{l_{n}! m_{n}!} . \end{matrix}$ Thus, we have $\frac{1}{l! m!} \partial_{z_{1}}^{l} \partial_{z_{2}}^{m} (z_{1}^{p} z_{2}^{q} ψ_{0}^{q_{0}} \dots ψ_{n}^{q_{n}}) |_{z_{1} = z_{2} = 0} = \sum_{\begin{array}{c} l_{0} + \dots + l_{n} = l - p \\ m_{0} + \dots + m_{n} = m - q \end{array}} \frac{\partial_{z_{1}}^{l_{0}} \partial_{z_{2}}^{m_{0}} ψ_{0}^{q_{0}}}{l_{0}! m_{0}!} \dots \frac{\partial_{z_{1}}^{l_{n}} \partial_{z_{2}}^{m_{n}} ψ_{n}^{q_{n}}}{l_{n}! m_{n}!} |_{z_{1} = z_{2} = 0}$ (16) for any $l ⩾ p$ , $m ⩾ q$ (and $= 0$ , if $l < p$ or $m < q$ ). Every partial derivative in this sum is expressed as follows: $\begin{array}{rclr} \frac{\partial_{z_{1}}^{l_{j}} \partial_{z_{2}}^{m_{j}} ψ_{j}^{q_{j}}}{l_{j}! m_{j}!} |_{z_{1} = z_{2} = 0} & = & \sum_{\begin{array}{c} λ_{1} + \dots + λ_{q_{j}} = l_{j} \\ μ_{1} + \dots + μ_{q_{j}} = m_{j} \end{array}} \frac{\partial_{z_{1}}^{λ_{1}} \partial_{z_{2}}^{μ_{1}} ψ_{j}}{λ_{1}! μ_{1}!} \dots \frac{\partial_{z_{1}}^{λ_{q_{j}}} \partial_{z_{2}}^{μ_{q_{j}}} ψ_{j}}{λ_{q_{j}}! μ_{q_{j}}!} |_{z_{1} = z_{2} = 0} \\ = & \sum_{\begin{array}{c} λ_{1} + \dots + λ_{q_{j}} = l_{j} \\ μ_{1} + \dots + μ_{q_{j}} = m_{j} \end{array}} c_{λ_{1}, μ_{1}} {{(λ_{1} r_{1} + μ_{1} r_{2})}^{j} \dots c_{λ_{q_{j}}, μ_{q_{j}}} (λ_{q_{j}} r_{1} + μ_{q_{j}} r_{2})}^{j} & (17) \end{array}$ (note that this expression is equal to zero, if $l_{j} + m_{j} < q_{j}$ ). Combining formulae (16), (17) we obtain $\frac{\partial_{z_{1}}^{l} \partial_{z_{2}}^{m} ϕ}{l! m!} |_{z_{1} = z_{2} = 0} = \sum_{p, q, Q} α_{p, q, Q} \sum_{\begin{array}{c} l_{0} + \dots + l_{n} = l - p \\ m_{0} + \dots + m_{n} = m - q \end{array}} \frac{\partial_{z_{1}}^{l_{0}} \partial_{z_{2}}^{m_{0}} ψ_{0}^{q_{0}}}{l_{0}! m_{0}!} \dots \frac{\partial_{z_{1}}^{l_{n}} \partial_{z_{2}}^{m_{n}} ψ_{n}^{q_{n}}}{l_{n}! m_{n}!} |_{z_{1} = z_{2} = 0},$ (18) where $\frac{\partial_{z_{1}}^{l_{j}} \partial_{z_{2}}^{m_{j}} ψ_{j}^{q_{j}}}{l_{j}! m_{j}!} |_{z_{1} = z_{2} = 0} = \sum_{\begin{array}{c} λ_{1} + \dots + λ_{q_{j}} = l_{j} \\ μ_{1} + \dots + μ_{q_{j}} = m_{j} \end{array}} {(λ_{1} r_{1} + μ_{1} r_{2})}^{j} \dots {(λ_{q_{j}} r_{1} + μ_{q_{j}} r_{2})}^{j} c_{λ_{1}, μ_{1}} \dots c_{λ_{q_{j}}, μ_{q_{j}}} .$ (19)

The summands in the right-hand side of (19) do not contain the coefficient $c_{l, m}$ . Indeed, if $(λ_{i}, μ_{i}) = (l, m)$ in some summand, this would necessary imply $(l_{j}, m_{j}) = (l, m)$ and, therefore, $(λ_{s}, μ_{s}) = (0, 0)$ for the other $(λ_{s}, μ_{s})$ in this summand. Thus, formula (15) can be written in the form $L (l r_{1} + m r_{2}) c_{l, m} = p_{l, m} ({α_{p, q, Q}}, {c_{λ, μ}}),$ where $p_{l, m}$ is the polynomial of the variables ${α_{p, q, Q}}$ , ${c_{λ, μ}}$ (with $(λ, μ) \in M$ , $λ ⩽ l$ , $μ ⩽ m$ , $(λ, μ) \neq (l, m)$ ) determined by formulae (18), (19).

Now we similarly use equality (14) to obtain recursive expressions for the coefficients $C_{l, m}$ . Denote by Φ the power series from the right-hand side of this equality, $Φ = \sum_{p, q, Q} | α_{p, q, Q} | z_{1}^{p} {z_{2}}^{q} W^{q_{0}} W^{q_{1}} \dots W^{q_{n}} \in C {z_{1}, z_{2}}$ for $W = \sum_{(l, m) \in Z_{+}^{2} ∖ {0}} | l r_{1} + m r_{2} |^{n} C_{l, m} z_{1}^{l} z_{2}^{m}$ . Then (14) implies $σ | l r_{1} + m r_{2} |^{n} C_{l, m} = \frac{\partial_{z_{1}}^{l} \partial_{z_{2}}^{m} Φ}{l! m!} |_{z_{1} = z_{2} = 0} .$ (20) Keeping in mind the analogy and difference between the series $ϕ \in C ⟦ z_{1}, z_{2} ⟧$ and $Φ \in C {z_{1}, z_{2}}$ , we obtain $\frac{\partial_{z_{1}}^{l} \partial_{z_{2}}^{m} Φ}{l! m!} |_{z_{1} = z_{2} = 0} = \sum_{p, q, Q} | α_{p, q, Q} | \sum_{\begin{array}{c} l_{0} + \dots + l_{n} = l - p \\ m_{0} + \dots + m_{n} = m - q \end{array}} \frac{\partial_{z_{1}}^{l_{0}} \partial_{z_{2}}^{m_{0}} W^{q_{0}}}{l_{0}! m_{0}!} \dots \frac{\partial_{z_{1}}^{l_{n}} \partial_{z_{2}}^{m_{n}} W^{q_{n}}}{l_{n}! m_{n}!} |_{z_{1} = z_{2} = 0},$ (21) where $\frac{\partial_{z_{1}}^{l_{j}} \partial_{z_{2}}^{m_{j}} W^{q_{j}}}{l_{j}! m_{j}!} |_{z_{1} = z_{2} = 0} = \sum_{\begin{array}{c} λ_{1} + \dots + λ_{q_{j}} = l_{j} \\ μ_{1} + \dots + μ_{q_{j}} = m_{j} \end{array}} | λ_{1} r_{1} + μ_{1} r_{2} |^{n} \dots | λ_{q_{j}} r_{1} + μ_{q_{j}} r_{2} |^{n} C_{λ_{1}, μ_{1}} \dots C_{λ_{q_{j}}, μ_{q_{j}}} .$ (22) Thus, formula (20) can be written in the form $σ | l r_{1} + m r_{2} |^{n} C_{l, m} = P_{l, m} ({| α_{p, q, Q} |}, {C_{λ, μ}}),$ (23) where $P_{l, m}$ is the polynomial of the variables ${| α_{p, q, Q} |}$ , ${C_{λ, μ}}$ ( $λ ⩽ l$ , $μ ⩽ m$ , $(λ, μ) \neq (l, m)$ ) with the real positive coefficients determined by formulae (21), (22). Since for $(l, m)$ equal to $(1, 0)$ and $(0, 1)$ we have $σ | r_{1} |^{n} C_{1, 0} = \partial_{z_{1}} Φ (0, 0) = | α_{1, 0, 0} |, σ | r_{2} |^{n} C_{0, 1} = \partial_{z_{2}} Φ (0, 0) = | α_{0, 1, 0} |,$ all the coefficients $C_{l, m}$ are real nonnegative numbers.

Finally we come to a conclusive part of the proof, the estimates $| c_{l, m} | ⩽ C_{l, m} \forall (l, m) \in Z_{+}^{2} ∖ {0} .$ We prove them by the induction with respect to the sum $l + m$ of the indices.

For $l + m = 1$ according to (15) we have $L (r_{1}) c_{1, 0} = \partial_{z_{1}} ϕ (0, 0) = α_{1, 0, 0}, L (r_{2}) c_{0, 1} = \partial_{z_{2}} ϕ (0, 0) = α_{0, 1, 0},$ hence1

Note that if $(1, 0) \in M$ , then $| L (r_{1}) | / | r_{1} |^{n} ⩾ σ$ , and in the opposite case $c_{1, 0} = 0$ . The similar is true for the index $(0, 1)$ .

| c_{1, 0} | = \frac{| α_{1, 0, 0} |}{| L (r_{1}) |} = \frac{σ | r_{1} |^{n} C_{1, 0}}{| L (r_{1}) |} ⩽ C_{1, 0}, | c_{0, 1} | = \frac{| α_{0, 1, 0} |}{| L (r_{2}) |} = \frac{σ | r_{2} |^{n} C_{0, 1}}{| L (r_{2}) |} ⩽ C_{0, 1} .

Further, by the construction of the polynomials

p_{l, m}

and

P_{l, m}

, for any

(l, m) \in M

we have

| p_{l, m} ({α_{p, q, Q}}, {c_{λ, μ}}) | ⩽ P_{l, m} ({| α_{p, q, Q} |}, {| c_{λ, μ} |})

(here we use the estimate

| λ r_{1} + μ r_{2} | ⩾ 1

for all

(λ, μ) \in M

), and the inductive assumption (the second inequality below) implies

\begin{array}{rclr} | L (l r_{1} + m r_{2}) | | c_{l, m} | & = & | p_{l, m} ({α_{p, q, Q}}, {c_{λ, μ}}) | ⩽ P_{l, m} ({| α_{p, q, Q} |}, {| c_{λ, μ} |}) \\ ⩽ & P_{l, m} ({| α_{p, q, Q} |}, {C_{λ, μ}}) ⩽ σ | l r_{1} + m r_{2} |^{n} C_{l, m},^{2} & (24) \end{array}

whence the required estimates follow:

In the left-hand side of the last inequality in (24) the indices $(λ, μ)$ of the variables $C_{λ, μ}$ belong to the set M, while in equality (23) they belong to $Z_{+}^{2} ∖ {0}$ . Therefore we write in (24) the inequality instead of the equality.

| c_{l, m} | ⩽ \frac{σ | l r_{1} + m r_{2} |^{n}}{| L (l r_{1} + m r_{2}) |} C_{l, m} ⩽ C_{l, m}

(for

(l, m) \in M

we have

| L (l r_{1} + m r_{2}) | / | l r_{1} + m r_{2} |^{n} ⩾ σ

, and

c_{l, m} = 0

for

(l, m) \notin M

Now it remains to note that for any sector S with the vertex at $0 \in C$ and of the opening less than $2 π$ , the terms of the series (1) are regarded as holomorphic single-valued functions in S, and to pass from the convergence of $ψ_{0} = \sum_{(l, m) \in Z_{+}^{2} ∖ {0}} c_{l, m} z_{1}^{l} z_{2}^{m}$ to the convergence of $φ = \sum_{k = 0}^{\infty} c_{k} x^{s_{k}}$ . Let the power series $ψ_{0}$ converge in a neighbourhood of a closed polydisk ${| z_{1} | ⩽ R, | z_{2} | ⩽ R}$ . Then there is a positive constant C such that $| c_{l, m} | ⩽ C / R^{l + m}$ for all $(l, m) \in Z_{+}^{2} ∖ {0}$ . If $x \in S$ is small enough for the inequalities $| x^{r_{1}} | = | x |^{Re r_{1}} e^{- Im r_{1} \cdot arg x} ⩽ R_{1} < R, | x^{r_{2}} | = | x |^{Re r_{2}} e^{- Im r_{2} \cdot arg x} ⩽ R_{1} < R$ to be held (recall that $Re r_{1, 2} > 0$ ), then $| c_{k} x^{s_{k} - s_{μ}} | = | c_{l, m} | \cdot {{{{| x^{r_{1}} |}^{l} \cdot | x^{r_{2}} |}^{m} ⩽ C (\frac{R_{1}}{R})}^{l} (\frac{R_{1}}{R})}^{m} = C h^{l + m}, l = l (k), m = m (k) .$ As $0 < h < 1$ , the series $\sum_{(l, m) \in Z_{+}^{2}} h^{l + m}$ converges, hence the series $φ = \sum_{k = 0}^{\infty} c_{k} x^{s_{k}}$ converges uniformly in S for sufficiently small $| x |$ .

4. An example

Let us consider the third Painlevé equation $\frac{d^{2} y}{d x^{2}} = \frac{1}{y} {(\frac{d y}{d x})}^{2} - \frac{1}{x} \frac{d y}{d x} + \frac{a y^{2} + b}{x} + c y^{3} + \frac{d}{y},$ (25) where a, b, c, d are complex parameters. Rewritten with respect to the operator $δ = x \frac{d}{d x}$ it has the form $y δ^{2} y = {(δ y)}^{2} + x y (a y^{2} + b) + c x^{2} y^{4} + d x^{2},$ or $F (x, y, δ y, δ^{2} y) = 0$ , where $F (x, y_{0}, y_{1}, y_{2}) = y_{0} y_{2} - y_{1}^{2} - x y_{0} (a y_{0}^{2} + b) - c x^{2} y_{0}^{4} - d x^{2} .$

As known [7], Eq. (25) has a two-parametric family of formal solutions $φ = c_{r} x^{r} + \sum_{s \in K} c_{s} x^{s},$ where $c_{r}$ is an arbitrary complex number, r is any complex number with $- 1 < Re r < 1$ . The other coefficients $c_{s}$ are determined uniquely by $c_{r}$ and the set K of power exponents is of the form $K = {r + l (1 - r) + m (1 + r) ∣ l, m \in Z_{+}, l + m > 0} .$ We see that $val φ = r$ (if $c_{r} \neq 0$ ) and the exponents s can be ordered with respect to ≺, so that φ is a generalized power series. This follows from the fact that there is only a finite number of exponents $s = r + l (1 - r) + m (1 + r)$ with a fixed real part $Re s = Re r + l (1 - Re r) + m (1 + Re r)$ , since $1 - Re r$ and $1 + Re r$ are positive.

To prove the convergence of φ in sectors of small radius it is sufficient to find the partial derivatives $\frac{\partial F}{\partial y_{0}}$ , $\frac{\partial F}{\partial y_{1}}$ , $\frac{\partial F}{\partial y_{2}}$ along φ and verify assumption (3). One has $\frac{\partial F}{\partial y_{0}} = y_{2} - x (3 a y_{0}^{2} + b) - 4 c x^{2} y_{0}^{3}, \frac{\partial F}{\partial y_{1}} = - 2 y_{1}, \frac{\partial F}{\partial y_{2}} = y_{0} .$ Hence, $\begin{matrix} \frac{\partial F}{\partial y_{2}} (x, φ, δ φ, δ^{2} φ) = φ = c_{r} x^{r} + C_{2} x^{λ_{2}} + \dots, c_{r} \neq 0, Re λ_{2} > Re r, \\ \frac{\partial F}{\partial y_{1}} (x, φ, δ φ, δ^{2} φ) = - 2 δ φ = - 2 r c_{r} x^{r} + C_{1} x^{λ_{1}} + \dots, Re λ_{1} > Re r, \\ \frac{\partial F}{\partial y_{0}} (x, φ, δ φ, δ^{2} φ) = δ^{2} φ - x (3 a φ^{2} + b) - 4 c x^{2} φ^{3} = r^{2} c_{r} x^{r} + C_{0} x^{λ_{0}} + \dots, Re λ_{0} > Re r, \end{matrix}$ whence the convergence follows (the inequality $Re λ_{0} > Re r$ holds as $val (3 a x φ^{2}) = 2 r + 1$ , $val (b x) = 1$ and $val (4 c x^{2} φ^{3}) = 3 r + 2$ , whose real parts exceed $Re r$ ).

5. Concluding remarks

A majorant method we use here to prove Theorem 1 also allows to estimate the radius of a sector S of convergence. For this it is sufficient to estimate the radius R of a polydisk where the series W satisfying (14) converges. One can do this by constructing an equation majorant for (14) but we would like to postpone details for future possible investigations not to complicate a presentation of this paper.

A sufficient condition of the convergence of generalized power series proposed in Theorem 1 looks like an attempt to study problems concerning a summation of such series. Among questions that naturally arise further in this direction we may highlight the following:

to obtain a Maillet type theorem estimating the growth of coefficients of a divergent generalized power series formally satisfying (2) (this was Maillet [9] who has first obtained such estimates for integer power series, then they were precised by Ramis [11] in the linear case, and in the general case by Malgrange [10] and finally by Sibuya [15, Appendix 2]);

to investigate the possibility of generalization of the Ramis–Sibuya fundamental theorem [12] on asymptotic Gevrey expansions in sectors of small opening, as well as generalization of k-summability and multisummability (as known, divergent integer power series formally satisfying (2) possess the multisummability property, which is obtained by Ramis in the linear case and by Braaksma in the nonlinear one, see [1,2,13]).

Footnotes

Acknowledgement

We are very thankful to Anton A. Vladimirov who has kindly proved Lemma for us and to the referee for significant improving remarks.

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