# MIXED SPECIES AND SYSTEMS OF COMBINATORIAL DIFFERENTIAL EQUATIONS

Benjamin Randrianirina

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Benjamin Randrianirina. MIXED SPECIES AND SYSTEMS OF COMBINATORIAL DIFFERENTIAL EQUATIONS. 2025. hal-05097963

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# MIXED SPECIES AND SYSTEMS OF COMBINATORIAL DIFFERENTIAL EQUATIONS

BENJAMIN RANDRIANIRINA

ABSTRACT. The goal of this paper is to give some complements to the Leroux-Viennot theory of combinatorial differential equations. In particular, we specify the context under which the initial conditions can take non-zero values in a ring. Also, we establish a clear connection between combinatorial differential operators and Chen grammars. Since the species we consider are weighted, we begin with a study of infinite summable weighted sets. We then introduce the concept of mixed species, which is the main tool in our study of combinatorial differential equations and systems of combinatorial differential equations. An application is given to hyperelliptic species.

## 1. INTRODUCTION.

This article is the English translation of a paper we published in Annales des sciences mathématiques du Québec (see [23]). We take this opportunity to make a few corrections and modifications to the presentation. In particular, Sections 5, 6, and 7 have been merged into a single section.

The goal of this article is, on the one hand, to provide certain extensions to the Leroux-Viennot theory for combinatorial differential equations (see [17], [18], [16]), in particular by including initial conditions that are non-zero constants belonging to a ring; and on the other hand, to establish a connection between combinatorial differential systems, combinatorial differential operators, and W. Chen's grammar (see [6]).

Let us consider the combinatorial differential equation

$$Y' = F(Y), \quad Y(0) = Z, \tag{1}$$

where $F$ is a species. Leroux-Viennot's method shows that the solution is the species $A = A_F(T, Z)$ of $F$-enriched increasing trees (see Section 5). If $F$ is a $\mathbb{B}$-species, then $A = A_F(T, Z)$ is an $\mathbb{L}$-species with respect to the sort $T$, and a $\mathbb{B}$-species with respect to the sort $Z$. In this case, we say that $A = A_F(T, Z)$ is a mixed species (see Section 4).

On the other hand, we consider the combinatorial differential equation

$$Y' = F(Y), \quad Y(0) = \alpha \tag{2}$$

where $F$ is a $\mathbb{B}$-species and $\alpha$ a constant. We show that the combinatorial solution to equation (2) is the $\mathbb{L}$-species $T_{Z:\alpha}A_F(T)$, of the types with respect to the sort $Z$ of the $A_F(T, Z)$-structures (see Subsection 5.2), in which $\alpha$ is a counter of elements of sort $Z$. Note that the exponential generating series verify $y = T_{Z:\alpha}A_F(t) = Z_F(y; \alpha, \alpha^2, \alpha^3, \dots)$, where $Z_F$ is the cycle index series of $F$. These results extend naturally to systems of differential equations (see Subsection 5.3).

We note that for each totally ordered finite set $l$ the weighted sets $T_{Z:\alpha}A_F F[l]$ may be infinite. It is therefore necessary to reformulate certain fundamental principles of enumerative

<footnote>Email: benjamin randrianirina@univ-fianarantsoa.mg.</footnote>
combinatorics, as demonstrated in the case of finite sets. In particular, we prove an infinite version of the Garsia-Milne involution principle [11] (see Section 2).

We apply these results to study the relationships between isomorphism and equipotence, and to prove a decomposition theorem for $\mathbb{L}$-species, which can be viewed, in a certain sense, as the molecular decomposition of weighted $\mathbb{L}$-species (see Section 3).

We also generalize a theorem by Bergeron and Reutenauer (see [4]) concerning upper bounds on the coefficients of the solutions to differential systems (see Section 6), and provide a new and simpler combinatorial proof of a theorem by Ginocchio (see [12]) regarding powers of a differential operator (see Section 7).

Moreover, under certain specific conditions, specifying a Chen grammar is equivalent to specifying a particular differential system (see Section 8). We apply these results to the study of so-called hyperelliptic species (see Section 9), which are solutions of the corresponding combinatorial differential system

## 2. $\mathbb{A}$-WEIGHTED SETS

**Definition 2.1.** Given a topological ring $\mathbb{A}$ (cf. Bourbaki [5]), an $\mathbb{A}$-weighted set (or simply weighted set if there is no risk of confusion) is a pair $(E, v)$, where $E$ is an arbitrary set and $v: E \to \mathbb{A}$ is a map. The weighted set $(E, v)$ is said to be $\mathbb{A}$-summable (or simply summable) if there exists an element $s \in E$ such that for every neighborhood $V$ of $0$ in $\mathbb{A}$, there exists a finite subset $E_0 \subset E$ such that for every finite subset $B \subset E$ such that $E_0 \subset B$, we have : $s - \sum_{x \in B} v(x) \in V$. The element $s$ is then called the total weight of $E$, and we write $|E|_v := s = \sum_{x \in E} v(x)$.

It is easy to verify that if $E$ is finite then $E$ is $\mathbb{A}$-summable, and if $(E_i)_{i \in I}$ is a finite family of $\mathbb{A}$-summable weighted sets, then the disjoint union $\sum_{i \in I} E_i$ is $\mathbb{A}$-summable. In the general case, we use the concept of a weighting ring introduced by Lévesque in his dissertation (cf. [20]).

**Definition 2.2.** A weighting ring is a topological ring $\mathbb{A}$ for which the following two properties hold :

(1) If a set $E$ is $\mathbb{A}$-summable, then every subset $B \subset E$ is also $\mathbb{A}$-summable.

(2) If $(A_i)_{i \in I}$ is a finite family of $\mathbb{A}$-summable weighted sets, then the cartesian product $\prod_{i \in I} E_i$ is $\mathbb{A}$-summable. In this case, the weighting on $\prod_{i \in I} E_i$ is defined by : $w((e_i)_{i \in I}) = \sum_{i \in I} v_i(e_i)$, and the total weight is given by : $|\prod_{i \in I} E_i|_w = \sum_{i \in I} |E_i|_{v_i}$.

Examples of weighting rings include the field of real numbers $\mathbb{R}$, as well as rings of formal series (cf. [20]). In general, $\mathbb{A}$ will be a ring of formal series $\mathbb{A} = \mathbb{K}[[X]]$, where $X$ is a set of formal variables, and $\mathbb{K}$ is an integral domain of characteristic zero.

**Definition 2.3.** A morphism of $\mathbb{A}$-weighted sets is a map $f : (E, v) \to (F, w)$ such that for every $x \in E$, we have $w(f(x)) = v(x)$.

The $\mathbb{A}$-summable weighted sets and their morphisms form a category, denoted $\mathcal{E}_{\mathbb{A}}$. An isomorphism of weighted sets is then a bijective morphism.

**Definition 2.4.** A weighted involution of a weighted set $(E, v)$ is a bijection $\varphi : E \to E$, such that :

(1) $\varphi \circ \varphi = \mathrm{Id}_E$

(2) For every $x \in E$, if $x$ is not a fixed point of $\varphi$, then $v(\varphi(x)) = -v(x)$.
**Definition 2.5.** A weighted set, $(E, v)$ is said to be reduced if for every pair $(x, y) \in E^2$ with $x \neq y$, we have $v(x) \neq -v(y)$. We also say that $E$ is reduced.

Let $\psi$ be a weighted involution of $E$. Denote by $\text{Fix}(\psi)$ the set of fixed points of $\psi$. If $E$ is summable, then $\text{Fix}((\psi))$ is also summable, and :

$$|E|_v = |\text{Fix}((\psi)|_v. \qquad (3)$$

The following proposition is straightforward.

**Proposition 2.6.** A weighted set $(E, v)$ is reduced if and only if the identity map is the only weighted involution of $(E, v)$.

**Definition 2.7.** A weighted involution $\psi$ is said to be reductive if its set of fixed points $\text{Fix}(\psi)$ is reduced.

**Proposition 2.8.** Every weighted set $(E, v)$ admits a reductive weighted involution.

*Proof:* Indeed, let $\mathcal{F}$ be the set of pairs $(B, \psi_B)$, where $B \subset E$ and $\psi_B$ is a reductive weighted involution of $B$ such that for every $x \in \text{Fix}(\psi_B)$ and every $y \in E \setminus B$, we have $v(x) \neq -v(y)$. Ordered by $(B, \phi_B) \leq (B', \phi_{B'})$ if $B \subset B'$ and $\phi_B = \phi_{B'}|_B$, this set is inductive (cf. P. Halmos [13]). If $(E_0, \phi_0)$ is a maximal element of $\mathcal{F}$, then $E_0 = E$. ◇

The following proposition is a variant and generalization of the involution principle of Garsia and Milne (cf. [11], or Stanton and White [26]).

**Proposition 2.9.** Let $(E, v)$ be an $\mathbb{A}$-summable weighted set with only finitely many elements of zero weight. If $\phi$ and $\psi$ are two reductive weighted involutions of $(E, v)$, then there exists an isomorphism of weighted sets : $\varphi : \text{Fix}(\phi) \to \text{Fix}(\psi)$.

*Proof:* Consider the graph $\Gamma$ on $E$, formed by the superposition of the graphs associated with $\phi$ and $\psi$ (see Figure 1). Since each vertex has degree at most 2, the connected components of $\Gamma$ are either isolated points, chains, or cycles.

<figure><img src="image_2.png" /><figcaption>FIGURE 1 – Graph $\Gamma$</figcaption></figure>

— The isolated points are the elements $x \in \text{Fix}(\psi) \cap \text{Fix}(\psi)$. In this case, we define $\varphi(x) = x$.

— All chains are finite and of even length. Indeed, the existence of an infinite chain would imply the existence of a non-summable subfamily of $E$, which is impossible since $\mathbb{A}$ is a weighting ring. Moreover, a chain of odd length would connect two fixed points $x$ and $y$ of $\phi$ (or of $\psi$) such that $v(x) = -v(y)$, which is impossible because both $\text{Fix}(\psi)$ and $\text{Fix}(\psi)$ are reduced. A chain of even length connects a fixed point $x$ of $\phi$ to a fixed point $y$ of $\psi$. In this case, we define $\varphi(x) = y$.

— A cycle contains no fixed points.

◇

**Remark 2.10.** If $E$ contains an infinite number of elements of zero weight, the existence of a weight-preserving bijection is no longer guaranteed. Indeed, the set of natural numbers $\mathbb{N}$,
weighted by $v(2i + 1) = x^i$ and $v(2i) = 0$, is $\mathbb{Z}[[x]]$-summable. The maps $\phi$ and $\psi$ defined by
$\phi(2i + 1) = \psi(2i + 1) = 2i + 1$, $\phi(4i) = 4i + 2$, $\phi(4i + 2) = 4i$, $\psi(0) = 0$, $\psi(4i + 2) = 4(i + 1)$,
$\psi(4(i + 1)) = 4i + 2$ are both reductive weighted involutions. However, $\mathrm{Fix}(\psi) = \{2i + 1 | i \in \mathbb{N}\} \cup \{0\}$ and $\mathrm{Fix}(\phi) = \{2i + 1 | i \in \mathbb{N}\}$ are not isomorphic.

### 3. REDUCTIONS OF $\mathbb{L}$-SPECIES

**3.1. Background.** For more details on species, we refer the reader to the books by Bergeron, Labelle, and Leroux (cf. [1] or [2]). A $\mathbb{L}$-weighted species $F = F_v$ is a functor (cf. Mac Lane [21]) from the category $\mathbb{L}$ of finite totally ordered sets and order-preserving bijections, to the category $\mathcal{E}_A$ of summable $A$-weighted sets, where $A$ is a weighting ring. In other words, it is a rule that assigns : to every totally ordered finite set $l$, a weighted set $F_v[l]$, and to every isomorphism of totally ordered sets $\sigma : l \to l'$, a morphism of weighted sets $F[\sigma] : F_v[l] \to F_v[l']$, such that : $F[\mathrm{Id}_l] = \mathrm{Id}_{F[l]}$ and $F[\sigma\sigma'] = F[\sigma] \circ F[\sigma']$.

A morphism of $\mathbb{L}$-species $\varphi : F_v \to G_w$ is a natural transformation, that is, a family $\varphi = (\varphi_l)$ of morphisms of weighted sets $\varphi_l : F_v[l] \to G_w[l]$, such that for any isomorphism $\tau : l \to m$, we have : $G[\tau]\varphi_l = \varphi_m F[\tau]$. The morphism $\varphi = (\varphi_l)$ is an isomorphism if each $\varphi_l$ is an isomorphism of weighted sets. In this case, we say that $F_v$ and $G_w$ are isomorphic, and we write $F_v \simeq G_w$, or simply $F_v = G_w$.

Given an $\mathbb{L}$-species $F = F_v$, the (exponential) generating series of $F$ is the series :

$$F(t) = \sum_{n \ge 0} f_n \frac{t^n}{n!},$$

where $f_n = |F[n]|_v$ is the total weight of $F$-structures on $[n] = \{1, 2, \dots, n\}$.

Two weighted $\mathbb{L}$-species $F$ and $G$ are said to be equipotent if $F(t) = G(t)$. This relation is denoted by $F \equiv G$. If $F \simeq G$, then $F \equiv G$, however, the converse is generally not true.

We have the following result, proven in [22].

**Proposition 3.1.** Two $\mathbb{L}$-species $F = F_v$ and $G = G_w$ are isomorphic if and only if, for each $n$, there exists an isomorphism of weighted sets $\varphi_n : F[n] \to G[n]$.

In particular, two trivially weighted $\mathbb{L}$-species (that is, such that for every structure $s$, $v(s) = 1$), $F$ and $G$, are isomorphic if and only if $F(t) = G(t)$.

**Remark 3.2.** (1) More generally, an $\mathbb{L}$-species of $k$ sorts is a functor $F : \mathbb{L}^k \to \mathcal{E}_A$.

(2) The class of $\mathbb{L}$-species of structures is equipped with the usual operations - sum, product, substitution, differentiation, and integration - which are compatible with the transition to generating series. If $F$ and $G$ are two $\mathbb{L}$-species, then for all $l$ :

(a) $(F + G)[l] = F[l] + G[l]$ (disjoint union)

(b) $(F \cdot G)[l] = \sum_{A \subseteq [l]} F[A] \times G([n] \setminus A)$

(c) if $G[\emptyset] = \emptyset$, then $(F \circ G)[l] = \sum_{\pi \in \mathrm{Par}[l]} F[\pi] \times (\prod_{b \in \pi} G[b])$, where $\mathrm{Par}[l]$ is the set of partitions of $l$.

(d) $F'[l] = F[1 \oplus l]$, where $1 \oplus l = \{1\} \cup l$, whiff $1 \notin l$ and $1 < x$ for all $x \in l$,

(e) $(\int F)[l] = F[l \setminus \min l]$ if $l \neq \emptyset$ and $(\int F)[\emptyset] = \emptyset$.

**3.2. Reduction of $\mathbb{L}$-species.** In the following, we will study the relationship between isomorphism and equipotence.
**Proposition 3.3.** Two $\mathbb{L}$-species $F = F_v$ and $G = G_w$ are isomorphic if and only if, for each $n$, there exists an isomorphism of weighted sets $\varphi_n : F[n] \to G[n]$. In particular, two trivially weighted $\mathbb{L}$-species (that is, for every structure $s$, $v(s) = 1$), $F$ and $G$ are isomorphic if and only if $F(t) = G(t)$.

*Proof:* If $l$ is a totally ordered set such that $|l| = n$, $\varphi_l = G[\sigma] \circ \varphi_n \circ F[\sigma]^{-1}$ is an isomorphism of weighted sets. ◇

**Definition 3.4.** A weighted involution of a weighted $\mathbb{L}$-species $F$ is a family $\phi = (\phi_l)$, where each $\phi_l : F[l] \to F[l]$ is a weighted involution, compatible with transport of structures. The weighted involution $\phi$ is said to be reductive if for all $l$, $\phi_l$ is reductive, i.e., if $\mathrm{Fix}(\phi_l)$ is reduced.

**Lemma 3.5.** Every weighted $\mathbb{L}$-species admits a reductive weighted involution.

*Proof:* For every $n$, let $\phi_n : F[n] \to F[n]$ be the reductive involution defined by Proposition 2.8. Then, for each $l$ such that $|l| = n$, define $\phi_l := F[\sigma_l]\phi_n F[\sigma_l]^{-1}$, where $\sigma_l : [n] \to l$ is the unique increasing bijection. This yields a reductive weighted involution of $F$. ◇

**Definition 3.6.** (1) The $\mathbb{L}$-species $G$ is said to be a sub-species of $F$ if, for each $l$ :

(a) $G[l] \subset F[l]$,

(b) For every morphism $\sigma : l \to m$, $G[\sigma] := F[\sigma]|_{G[l]}$,

(c) For each $s \in G[l]$, $w(s) = v(s)$, where $v$ and $w$ are the weight functions of $F$ and $G$, respectively.

(2) A weighted $\mathbb{L}$-species $F = F_v$ is said to be reduced if, for every $l$, the weighted set $F[l]$ is reduced.

**Proposition 3.7.** Every weighted species is equipotent to a reduced sub-species.

*Proof:* Given a reductive weighted involution $\psi$ of $F = F_v$, the subspecies $F_{\mathrm{red}}$ defined by $F_{\mathrm{red}}[l] = \mathrm{Fix}(\psi_l)$ is reduced. Moreover, $|F_{\mathrm{red}}(\psi)[l]|_v = |\mathrm{Fix}[l]|_v = |F[l]|_v$. ◇

**Proposition 3.8.** Let $F = F_v$ be an $\mathbb{L}$-species such that, for every integer $n$, the set $\{s \in F_v[n] | v(s) = 0\}$ is finite. If $\psi$ and $\phi$ are two reductive weighted involutions of $F$, then $F_{\mathrm{red}(\psi)} \simeq F_{\mathrm{red}(\phi)}$.

*Proof:* Since $\psi$ and $\phi$ reductive, Proposition 2.9 shows that for each $n$, there exists a weighted set isomorphism $\varphi_n : F_{\mathrm{red}(\psi)}[n] \to F_{\mathrm{red}(\phi)}[n]$. Proposition 3.3 then yields the result. ◇

A representative of this isomorphism class is denoted by $F_{\mathrm{red}}$ and is called the reduction of $F$.

**Definition 3.9.** The weight $v$ of a weighted set $(E, v)$ is said to be monomial if, for every $s \in E$, the weight of the form $v(s) = (-1)^{\epsilon(s)} x^{\rho(s)}$ where $x = (x_1, \cdots, x_k)$ are independent formal variables, $\rho(s) = (\rho_1(s), \cdots, \rho_k(s)) \in \mathbb{N}^k$, and $x^{\rho(s)} = x_1^{\rho_1(s)} \cdots x_k^{\rho_k(s)}$.

We say that $F$ is a monomially weighted $\mathbb{L}$-species if, for every finite set $l$, the weight $v_l$ of $F[l]$ is monomial.

It is clear that the class of monomially weighted $\mathbb{L}$-species is stable under elementary operations : addition, multiplication, composition, differentiation, and integration.

**Theorem 3.10.** If $F = F_v$ and $G = G_w$ are two monomially weighted $\mathbb{L}$-species such that $F \equiv G$, then $F_{\mathrm{red}} \simeq G_{\mathrm{red}}$. In particular, if $F$ and $G$ are reduced monomially weighted species, then the equivalence $F \equiv G \Leftrightarrow F \simeq G$ holds.
**Proof:** Note first that if $F$ is monomially weighted and $s_1, s_2 \in F_{\text{red}}[l]$, then $\rho(s_1) = \rho(s_2) \Leftrightarrow v(s_1) = v(s_2)$. For $n \in \mathbb{N}$ and $a = (a_1, \dots, a_k) \in \mathbb{N}^k$, we define $A_{n,a}(F) = \{s \in F_{\text{red}}[n] | \rho(s) = a\}$. Then if $s, s' \in A_{n,a}(F)$, we have $v(s) = v(s')$.

Suppose $F \equiv G$. Then $|A_{n,a}(F)| = |A_{n,a}(G)|$, and for all $s \in A_{n,a}(F)$, $s' \in A_{n,a}(G)$, we have $v(s) = w(s')$. Indeed, $F_{\text{red}}(t) = G_{\text{red}}(t)$, so if $(-1)^{k(a)}n![x^at^n]F_{\text{red}} = c_{n,a}$ then $c_{n,a} = |A_{n,a}(F)| = |A_{n,a}(G)|$. Let $\varphi_{n,a} : A_{n,a}(F) \to A_{n,a}(G)$ be a bijection. $\varphi_{n,a}$ can be extended to a bijection $\varphi_n : F_{\text{red}}[n] \to G_{\text{rmred}}[n]$ preserving the weights. The result follows from Proposition 3.3. $\diamond$

**Remark 3.11.** The previous theorem states that a reduced monomially weighted $\mathbb{L}$-species is characterized by its generating series, in the sense that equipotence $\Leftrightarrow$ isomorphism. This equivalence is not true if the assumption of monomial weighting is removed. For instance, the species $T_{-ax} + T_{-bx} + T_{(a+b)x}$, whose total weight is zero, is not isomorphic to the empty species. Here, $T_y$ denotes the species of singletons of weight $y$.

**3.3. Canonical decomposition of $\mathbb{L}$-species.** Each natural number $k$ can be regarded as an $\mathbb{L}$-species by setting $\mathbf{k}=\mathbf{1}+\mathbf{1}+\cdots+\mathbf{1}$, that is, we have $k$ structures on the empty set, which we may denote by $\emptyset_1, \emptyset_2, \cdots, \emptyset_k$.

Given a natural number $k$ and an $\mathbb{L}$-species $F = F_v$, we define the species of $k$-colored $\mathbb{L}$-structures, denoted $\mathbf{k} \cdot F$, as the product of the species $k$ and the species $F$. A $\mathbf{k} \cdot F$-structure on a set $l$ is thus a pair $(\emptyset_j, s)$ where $1 \le j \le k$ and $s \in F[l]$, with weight defined by $w(\emptyset_j, s) = v(s)$, and transport of structure defined by $(\mathbf{k} \cdot F)[\tau](\emptyset_j, s) = (\emptyset_j, F[\tau](s))$.

Let $F$ be a weighted $\mathbb{L}$-species, $\alpha \in \mathbb{A}$, and $n \in \mathbb{N}$. The set $\{s \in F_v[n] | v(s) = \alpha\}$ is finite. We define $c_{n,\alpha}(F) = |\{s \in F_v[n] | v(s) = \alpha\}|$.

Let us denote by $E_n$ the species of sets of cardinality $n$, weighted trivially. We have the following decomposition theorem, which is the analogue of the molecular decomposition theorem in the case of $\mathbb{B}$-species.

**Theorem 3.12.** For any weighted $\mathbb{L}$-species $F = F_v$, we have :

$$F \approx \sum_{n \ge 0} \sum_{\alpha \in \mathbb{A}} \mathbf{c}_{n,\alpha}(\mathbf{F}) \cdot E_{n,\alpha} \qquad (4)$$

**Proof:** Consider a bijection $j = j_{n,\alpha} : \{s \in F_v[n] | v(s) = \alpha\} \to [c_{n,\alpha}(F)]$, where $[c_{n,\alpha}(F)] = \{1, 2, \cdots, c_{n,\alpha}\}$. For each pair $(n, \alpha)$, let $\varphi_{n,\alpha} : \{s \in F_v[n] | v(s) = \alpha\} \to \mathbf{c}_{n,\alpha} \cdot E_{n,\alpha}$ be the bijection defined by $\varphi_{n,\alpha}(s) = (\emptyset_{j(s)}, [n])$. where $j(s)$ is the index assigned to $s$ via the bijection $j_{n,\alpha}$. It is clear that $\varphi_{n,\alpha}$ preserves weights This bijection $f$ can be uniquely extended to a weight-preserving bijection $\varphi_n : F_v[n] \to \sum_{\alpha \in \mathbb{A}} \mathbf{c}_{n,\alpha}(\mathbf{F}) \cdot E_{n,\alpha}$. $\diamond$

## 4. MIXED SPECIES.

In this section, we develop the necessary tools for the study of initial conditions of the form $Y(0) = \alpha$, where $\alpha$ is a constant, in combinatorial differential equations and systems.

**4.1. Generals definitions.** Let us first recall that a $\mathbb{A}$-weighted $\mathbb{B}$-species is a functor $F : \mathbb{B} \to \mathcal{E}_{\mathbb{A}}$, where $\mathbb{B}$ is the category of finite sets and bijection, and $\mathcal{E}_{\mathbb{A}}$ is the category of summable $\mathbb{A}$-weighted sets.

**Definition 4.1.** A mixed species is a functor $F : \mathbb{L} \times \mathbb{B} \to \mathcal{E}_{\mathbb{A}}$.

That is, a rule which :
(1) to each pair $(l, U)$, where $l$ is a finite totally ordered set and $U$ is an arbitrary finite set, assigns a summable weighted set $F[l, U]$,

(2) to each pair $(\phi, \psi) : (l, U) \to (l', U')$, where $\phi : l \to l'$ is an order isomorphism and $\psi : U \to U'$ is any bijection, assigns an isomorphism of weighted sets $F[\phi, \psi] : F[l, U] \to F[l', U']$, called the transport of structures, satisfying : $F[\phi, \psi] \circ F[\phi', \psi'] = F[\phi \circ \phi', \psi \circ \psi']$ and $F[1_l, 1_U] = 1_{F[l, U]}$.

Two mixed species $F$ and $G$ are isomorphic if there exists a natural family of isomorphisms $\varphi = (\varphi_{l,U})$ of weighted sets such that, for every morphism $(\phi, \psi) : (l, U) \to (l', U')$, $G[\phi, \psi] \circ \varphi_{l,U} = \varphi_{l',U'} \circ F[\phi, \psi]$.

A mixed species $F$ is also denoted by $F(T, Z)$. The elements of the totally ordered set $l$ are called points of sort $T$ (or sort $\mathbb{L}$), and the elements of the unordered set $U$ are points of sort $Z$ (or sort $\mathbb{B}$). Note that, usual operations, sum, product, partitioned composition, derivation with respect to either variable, and integration with respect to the first variable, can be defined for mixed species.

**Definition 4.2.** Two structures $s \in F[l, U]$ and $s' \in F[l, V]$ are isomorphic if there exists a bijection $\tau : U \to V$ such that $s' = F[1_l, \tau](s)$. In this case, we write $s \sim s'$. The equivalence class of $s$ modulo $\sim$, denoted by $\tilde{s}$, is called the isomorphism type of $s$, or simply the type of $s$, with respect to points of sort $\mathbb{B}$.

We denote by $\overline{F[n,k]}$ the quotient of $F[n, k]$ by $\sim$. Note that if $s \sim t$ then $v(s) = v(t)$, and we define $v(\tilde{s}) = v(\tilde{s})$.

To a mixed species $F = F_v$, we can associate the following series :

(1) The exponential generating series :

$$F(t, z) = \sum_{n,k \ge 0} f_{n,k} \frac{t^n z^k}{n! k!}$$

where $f_{n,k} = |F[n, k]|_v$;

(2) The generating series of isomorphism types :

$$\tilde{F}(t, z) = \sum_{n,k \ge 0} \tilde{f}_{n,k} z^k \frac{t^n}{n!}$$

where $\tilde{f}_{n,k} = |\overline{F[n,k]}|_v$;

(3) The cycle index series :

$$Z_F(t; z_1, z_2, z_3, \dots) = \sum_{n,k \ge 0} \frac{t^n}{n!} \left( \sum_{\sigma \in S_k} |\mathrm{Fix}F[1, \sigma]|_v z^{\sigma_1} z_2^{\sigma_2} z_3^{\sigma_3} \dots \right)$$

where $\mathrm{Fix}F[1, \sigma]$ is the set of structures fixed by $F[1, \sigma]$ and $(\sigma_1, \sigma_2, \sigma_3, \dots)$ the cycle type of $\sigma$.

**Remark 4.3.** With respect to the variable $z$, these series behave exactly like the corresponding series in the case of $\mathbb{B}$-species (cf. [1] or [2]). For example, using Burnside's lemma, we have $F(t; z) = Z_F(t; z, z^2, z^3, \dots)$.
### 4.2. The $\mathbb{L}$-species of types with respect to points of sort $\mathbb{B}$. In this subsection, we will study the $\mathbb{L}$-species of types of $\mathbb{F}$-structures with respect to the sort $\mathbb{B}$, for any mixed species $F = F_v$.

**Definition 4.4.** Let $l$ be a totally ordered set. We define

$$T_{Z:z}F[l] := \bigcup_{k \in \mathbb{N}} \frac{F[l, k]}{\sim}.$$

If $\tilde{s} \in T_{Z:z}F[l]$, with $s \in F[l; [k]]$, then we define the weight of $\tilde{s}$ as $w(s) = v(s)z^k$. Furthermore, if $\tau: l \to l'$ is an order-preserving bijection and $s_1 \in F[l; k]$ such that $F[\tau, \mathrm{Id}_{[k]}](s_1) = s_2$, then we define $T_{Z:z}F[\tau](\tilde{s}_1) := \tilde{s}_2$. The resulting $\mathbb{L}$-species $T_{Z:z}F(T)$ is called the $\mathbb{L}$-species of types with respect to $Z$ of the $\mathbb{F}$-structures.

<figure><img src="image_3.png" /><figcaption>FIGURE 2 - A $T_{Z:z}F$-structure on $l = \{1, 2, 3, 4, 5\}$</figcaption></figure>

Figure 2 represents a generic $T_{Z:z}F$-structure over $l = \{1, 2, 3, 4, 5\}$. Note that in a $T_{Z:z}F$-structure, the variable $z$ is as a counter of indistinguishable points of type $Z$. This counter ensures the summability of the set $T_{Z:z}F[l]$ weighted by $w$.

**Proposition 4.5.** The generating series $T_{Z:z}F(t)$ satisfies :

$$T_{Z:z}F(t) = \sum_{n \ge 0} \left( \sum_{k \ge 0} \tilde{f}_{n,k} z^k \right) \frac{t^n}{n!} \quad (5)$$

*Proof:* By definition, we have :

$$\left| \frac{F[n, k]}{\sim} \right|_w = \sum_{\tilde{s} \in \frac{F[n,k]}{\sim}} w(\tilde{s}) = z^k \sum_{\tilde{s} \in \frac{F[n,k]}{\sim}} v(s) = \tilde{f}_{n,k} z^k$$

Taking the exponential generating series with respect to the variable $t$ and considering Remark 4.3, we have the result. ◇

**Proposition 4.6.** If $F = F(T, Z)$ and $G = G(T, Z)$ are two mixed species, then : $T_{Z:z}(F + G) = T_{Z:z}(F) + T_{Z:z}(G)$, $T_{Z:z}(\frac{\partial F}{\partial T}) = (T_{Z:z}F)'$, and $T_{Z:z}(F \cdot G) = T_{Z:z}(F) \cdot T_{Z:z}(G)$.

*Proof:* The first two equalities are obvious. For the product, it suffices to observe that :

$$\frac{(F \cdot G)[l, k]}{\sim} = \sum_{l_1+l_3=l, k_1+k_2=k} \left( \frac{F[l_1, k_1]}{\sim} \right) \times \left( \frac{C[l_2, k_2]}{\sim} \right)$$

◇

The case of substitution is more delicate. Let $F = F_w$ be a mixed species such that $F(\emptyset, \emptyset) = \emptyset$, and $G = G_v$ a $\mathbb{B}$-species. We set :

$$G(F)[l, U] = \sum_{\pi \in \mathrm{Par}[l+U]} G[\pi] \times \prod_{b \in \pi} F[l \cap b, U \cap b],$$

where $\mathrm{Par}[l+U]$ denotes the set of partitions of the set $l+U$. Thus we obtain the mixed species $G(F)$, weighted by $u(\pi, g_\pi, (f_b)_{b \in \pi}) = v(g_\pi) \prod_{b \in \pi} w(f_b)$
**Theorem 4.7.** If $G_v = G_v(Z)$ is a $\mathbb{B}$-species and $F_w = F_w(T, Z)$ is a mixed species, then :

$$T_{Z:z}G_v(F_w)(t) = Z_{G_v}(T_{Z:z}F_w(t), T_{Z:z}F_w(0), T_{Z:z^2}F_{w^2}(0), T_{Z:z^3}F_{w^3}(0), \dots), \quad (6)$$

where $Z_{G_v}(t; x_1, x_2, \dots)$ is the cycle index series of $G_v$. More generally :

$$Z_{G_v}(Fw)(t; z_1, z_2, \dots) = Z_{G_v}(Z_{F_w}(t; z_1, z_2, \dots), Z_{F_{w^2}}(0; z_2, z_4, \dots), \dots).$$

*Proof:* Proof of this theorem is similar to the classical proofs of the plethystic formula (see, for example, [1], [2] or [7]) and requires several steps.

To the mixed species $F_w$, we associate the auxiliary mixed species $\tilde{F}_w$, defined by : $\tilde{F}_w[l, U] = \{|s \in F_w[l; U], \sigma \in \mathbb{S}[U], (1_l, \sigma)$ an automorphism of $s\}$, weighted by $w(s, \sigma) = w(s)$. The transport of structures is defined by : $\tilde{F}_w[\rho, \beta](s, \sigma) = (F_w[\rho, \beta](s), \beta^{-1} \circ \sigma \circ \beta)$, if $\rho : l \to l$, and $\beta : U \to U$ is a bijection, and $(s, \sigma) \in \tilde{F}_w[l; U]$.

**Lemma 4.8.** The exponential generating series of $\tilde{F}_w$ satisfies :

$$\tilde{F}_w(t, z) = (Tz_{Z:z}F)(t) = Z_{F_w}(t; z, z^2, z^3, \dots).$$

*Proof:* We define an action $\mathbb{S}_n \times F[k, n] \to F[k, n]$ by setting $\sigma \cdot s = F[1_k, \sigma](s)$ if $\sigma : [n] \to [n]$ is a bijection and $s \in F[k, n]$. Using Burnside's lemma, we obtain : $|\tilde{F}_w[k, n]| = |\{(s, \sigma)|\sigma \cdot s = s\}|_w = \sum_{\sigma \in \mathbb{S}_n} \{w(s); s \in F_w[k, n], \sigma \cdot s = s\} = n!|\frac{F_w[k, n]}{\sim}|_w = n!\tilde{f}_{n,k}$. Therefore, $\tilde{F}_w(t, z) = \sum_{k \ge 0} \frac{t^k}{k!}(\sum_{n \ge 0} \tilde{f}_{k,n}z^n) = T_{Z:z}F(t). \diamond$

Defining the weight by $(w \cdot \vec{x})(s, \sigma) = w(s)x_1^{c_1(\sigma)}x_2^{c_2(\sigma)} \dots$, where $c_i(\sigma)$ is the number of cycles of length $i$ in $\sigma$, we obtain a weighted species denoted $\tilde{F}_{w \cdot \vec{x}}$.

**Lemma 4.9.** We have : $\tilde{F}_{w \cdot \vec{x}}(t, z) = Z_{F_w}(t; zx_1, z^2x_2, z^3x_3, \dots)$.

*Proof:* Indeed,

$$\begin{aligned} \tilde{F}_{w \cdot \vec{x}} &= \sum_{(s, \sigma) \in F[n, k] \times \mathbb{S}_k, \sigma \cdot s = s} (w \cdot \vec{x})(s, \sigma) = \sum_{\sigma \in \mathbb{S}_k} \sum_{s \in \text{Fix}[1_n, \sigma]} w(s) x_1^{c_1(\sigma)} x_2^{c_2(\sigma)} \dots \\ &= \sum_{\sigma \in \mathbb{S}_k} |\text{Fix}[1_n, \sigma] w(s) x_1^{c_1(\sigma)} x_2^{c_2(\sigma)} \dots| \diamond \end{aligned}$$

Now consider a $G_v(F_w)$-structure $s = (\pi, g_\pi, (f_b)_{b \in \pi})$ on $(l, U)$. Any automorphism $\tau = (1_l, \sigma)$ of $s$ induces a bijection $\tau_\pi : \pi \to \pi$, which is an automorphism of the $G$-structure $g_\pi$. Note that if $b \in \pi$, then $\sigma_\pi(b) = \{x|x \in b \cap l\} \cup \{\sigma(x)|x \in b \cap U\}$ Thus, if $b \cap l \ne \emptyset$, then $\tau_\pi(b) \cap b \ne \emptyset$, i.e., $\tau_\pi(b) = b$ (see Figure 3(A)). Now let $N = 1^{n_1}2^{n_2}3^{n_3}\dots$ be the cycle type

<figure><img src="image_4.png" /><figcaption>FIGURE 3 – (A) a $G_v(F_w)$-structure; (B) a crown of $F_w$-structure</figcaption></figure>

of $\tau_\pi$, and let $\tilde{G}_v(\tilde{F}_w)_N$ denote the subspecies of $\tilde{G}_v(\tilde{F}_w)$-structures where the permutation $\tau_\pi$ has cycle type $N$. It is clear that : $\tilde{G}_v(\tilde{F}_w) = \sum_N \tilde{G}_v(\tilde{F}_w)_N$.

We call a crown of $F_w$-structures an $\mathcal{E}(F_w)$-structure whose permutation $\tau_\pi$ (see [14]) has a single cycle, and we denote by $K_m(F_w)$ the species of crowns of $F_w$-structures in
which the cycle has length $m$. Note that an $\widetilde{E(Fw)}$-structure is a triple $(\pi, (f_b)_{b \in \pi}, \tau)$, where $\tau = (1, \sigma)$, such that for each $b \in \pi$, we have $f_{\tau_{\pi}(b)} = F[1_{l \cap b}, \tau|_{U \cap b}](f_b)$. Thus, for $m=1$, a $K_1(F_w)$-structure can be identified with a pair $(f, \tau)$ such that $F[\tau](f) = f$. In other words, $K_1(F_w)[l, U] = \widetilde{F_w}[l, U]$, and for $m \ge 2$, the $K_m(F_w)$-structures are defined only on points of sort $Z$. Figure 3(B) shows a $(K_4F_w)(0, Z)$-structure.

Consider a crown $(\pi, (f_b)_{b \in \pi}, \tau)$ such that $\tau_{\pi}$ is a cycle of length $m \ge 2$. Then $\tau$ is actually a bijection on $U$. By pointing a block $b_0$, we obtain a list of isomorphic $F(0, Z)$-structures $L = (f_{b_0}, F_w[\tau](f_{b_0}), \dots, F_w[\tau^{m-1}](f_{b_0}))$ whose total weight is $\Pi_{b \in \pi} w(f_b) = w^m m(f_{b_0})$ (see Figure 4).

<figure><img src="image_5.png" /><figcaption>FIGURE 4 – Isomorphism between a pointed crown and a list of isomorphic $F(0, Z)$-structures.</figcaption></figure>

Note also that $\tau^m$ is an automorphism of $f_{b_0}$, and that $(f_{b_0}, \tau^m)$ is a $K_1(F(0, Z))$-structure. Moreover, each element $x$ of $(\tau_{\pi})^i(b_0)$ has the form $x = \tau^i(x_0)$, with $x_0 \in b_0$. Thus, the list $L$ can be identified with a $K_1(F^{w^m})(0, Z^m)$-structure. Conversely, consider a $K_1(F_{w^m}(0, Z^m))$-structure $((\pi', f, (l_b)_{b \in \pi'}), \rho)$, where $l_b = (x_{1,b}, x_{2,b}, \dots, x_{m,b})$. Let $\pi = \{b_i | b_i = \{x_{i,b}, b \in \pi'\}, 1 \le i \le m\}$. For each $i$, we define a $K_1(F_w(0, Z))$-structure $(f_{b_i}, \sigma_i)$ on each $b_i$ by considering the $K_1(F_{w^m}(0, Z))$-structure $f$, and replacing each block $b$ with $x_{i,b}$, the weight $w^m$ with $w$, and setting $\tau_i(x_{i,b}) = x_{i,\rho(b)}$. We thus obtain a list of $F_w(0, Z)$-structures that can be identified with a pointed $K_m(F_w)$-structure. Thus, the species of pointed $K_m(F_w)$-structures is isomorphic to the species $K_1(F_{w^m}(0, Z)) \circ Z^m$ (see Figure 4). It follows that $mK_m(F_w)(0, z) = K_1(F_{w^m})(0, z^m)$. We obtain the following lemma, with the second part following from Lemma 4.9.

**Lemma 4.10.** The generating series of the species $K_m(F_w)$ satisfies $K_1(F_w)(t, z) = F_w(t, z)$, an for $m \ge 2$, $K_m(F_w)(0, z) = \widetilde{F_w^m}(0, z^m)/m$. Moreover, denoting by $K_m(F_{w \cdot \bar{x}})$ the species of crowns of $F_{w \cdot \bar{x}}$-structures whose cycle has length $m$, we have $K_1(F_{w \cdot \bar{x}})(t, z) = Z_{F_w}(t; zx_1, z^2x_2, z^3x_3, \dots)$, and for $m \ge 2$, $mK_m(F_{w \cdot \bar{x}})(t, z) = Z_{F_{w^m}}(0; z^mx_m, z^{2m}x_{2m}, \dots)$.

*Continuation of the proof of Theorem 4.7.* Observe that, for a given type $N$, a $\widetilde{G_v(F_w)}_N$-structure is entirely determined by a $G_v$-structure fixed by a permutation of type $N$, together with an assembly of crowns of $G_v$-structures of the form $\mathcal{E}_{n_1}(K_1(G))\mathcal{E}_{n_2}(K_2(G))\dots$. Therefore,

$$ (\widetilde{G_v(F_w)})_N(t, z) = |\text{FixG}[N]|_v \frac{(K_1(F_w)(t, z))^{n_1}}{n_1!} \frac{(K_2(F_w)(0, z))^{n_2}}{n_2!} \frac{(K_3(F_w)(0, z))^{n_3}}{n_3!} \dots $$

Then we have :
$$
\begin{align*}
T_{Z=z}(F_w)(t) &= \widetilde{G_v(F_w)}(t, z) = \sum_{N} (\widetilde{G_v(F_w)})_N(t, z) \\
&= \sum_{N} |\text{FixG}[N]|_v \frac{(K_1(F_w)(t, z))^{n_1}}{n_1!} \frac{(K_2(F_w)(0, z))^{n_2}}{n_2!} \frac{(K_3(F_w)(0, z))^{n_3}}{n_3!} \dots \\
&= \sum_{N} |\text{FixG}[N]|_v \frac{(\widetilde{F}_w(t, z))^{n_1}}{1^{n_1} n_1!} \frac{(\widetilde{F}_{w^2}(0, z))^{n_2}}{2^{n_2} n_2!} \frac{(\widetilde{F}_{w^3}(0, z))^{n_3}}{n_3!} \dots \\
&= \sum_{N} |\text{FixG}[N]|_v \frac{(T_{Z:z}F_w(t, z))_{n_1}}{1^{n_1} n_1!} \frac{(T_{Z:z}F_{w^2}(0, z))^{n_2}}{2^{n_2} n_2!} \frac{(T_{Z:z}F_{w^3}(0, z))^{n_3}}{n_3!} \dots \\
&= \widetilde{Z}_{G_v}(T_{Z:z}(F_w)(t, z), T_{Z:z^2}(F_w)(0), T_{Z:z^3}(F_w)(0), \dots).
\end{align*}
$$

Similarly, Lemmas 4.9 and 4.10 show that :

$$
\begin{align*}
Z_{G_v(F_w)}(t; zx_1, z^2 x_2, z^3 x_3, \dots) &= \widetilde{G_v(F_{w \cdot \vec{x}}(t, z))} = \sum_{N} (\widetilde{G_v(F_{w \cdot \vec{x}})}_N(t, z) \\
&= \sum_{N} |\text{FixG}[N]|_v \frac{(K_1(F_{w \cdot \vec{x}})(t, z))^{n_1}}{n_1!} \frac{(K_2(F_{w \cdot \vec{x}})(0, z))^{n_2}}{n_2!} \frac{(K_3(F_{w \cdot \vec{x}})(0, z))^{n_3}}{n_3!} \dots \\
&= \sum_{N} |\text{FixG}[N]|_v \frac{Z_{F_w}(t; zx_1, z^2 x_2, \dots)}{1^{n_1} n_1!} \frac{Z_{F_{w^2}}(0; z^2 x_2, z^4 x_4, \dots)}{2^{n_2} n_2!} \frac{Z_{F_{w^3}}(0; z^3 x_3, z^6 x_6, \dots)}{3^{n_3} n_3!} \dots \\
&= Z_{(F_w)}(Z_{G_v}(t; zx_1, z^2 x_2, \dots), Z_{G_v}(0; z^2 x_2, z^4 x_4, \dots), Z_{G_v}(0; z^3 x_3, z^6 x_6, \dots), \dots)
\end{align*}
$$

Setting $z_i = z^i x_i$, we obtain the result. ◇

**Corollary 4.11.** If $F = F_w(T, Z)$ is a mixed species such that $F(0, Z) = Z$, and $G$ is a $\mathbb{B}$-species, then :

$$ T_{Z:z}G_v(F_w)(t) = Z_{G_v}(T_{Z:z}F_w(t); z^2, z^3, \dots). \quad (7) $$

In particular, if $G$ is asymmetric, in which case $Z_G(x_1, x_2, x_3, \dots) = G(x_1)$, then

$$ T_{Z=z}G(F)(t) = G(T_{Z:z}F(t)). $$

5. THE LEROUX-VIENNOT THEORY FOR SYSTEM OF COMBINATORIAL DIFFERENTIAL EQUATIONS.

In this section, we recalls the Leroux-Viennot theory (see [17]) for system of combinatorial differential equations.

5.1. **Combinatorial differential equation.** Let us first note that theorem 3.12 shows that if $\mathbb{H} = \mathbb{Z}[[\vec{x}, t]]$ then every element of $\mathbb{H}$ is the generating series of a weighted $\mathbb{L}$-species. In other words, every element $f(t) \in \mathbb{H}$ admits a combinatorial interpretation (not unique), which we denote by $F(T)$. Thus, to any analytic equation $y' = f(y)$), with $f \in \mathbb{H}$, we may associate a combinatorial equation $Y' = F(Y)$, where $F = F(T)$ is a species whose generating series is $f(t)$.

In general, one can associate to any $\mathbb{L}$-species or $\mathbb{B}$-species $F$ a combinatorial differential equation;

$$ Y' = F(Y), Y(0) = Z; \qquad (8) $$

where $Z$ is a sort of point representing the initial condition. Intuitively, to solve (8), the idea is to find a species $Y = G = G(T, Z)$ satisfying $\frac{\partial G}{\partial T} = F(G)$ and $G(0, Z) = Z$. In other words, equation (8) can also be written in the form

$$ \frac{\partial Y}{\partial T} F(Y(T, Z)), Y(0, Z) = Z. \qquad (9) $$
The initial condition $Y(0) = Z$ in (8) translates as $Y(0, Z) = Z$, which means that $Y[\emptyset, U] = Z[U] = \{U\}$ if $|U| = 1$, and $Y[\emptyset, U] = \emptyset$ if $|U| \neq 1$.

**Definition 5.1.** A solution of equation (8) (or (9)) is a pair $(G, \phi)$ where :

(1) $G = (T, Z)$ is a $\mathbb{L}$-species if $F$ is a $\mathbb{L}$-species and a mixed species if $F$ is a $\mathbb{B}$-species. such that $G(0, Z) = Z$.

(2) $\phi: \frac{\partial G}{\partial T} \to F(G)$ is an isomorphism of $\mathbb{L}$-species or of mixed species.

Equation (9) can be written in integral form as :

$$Y(T, Z) = Z + \int_{0}^{T} F(Y(X, Z))dX.$$

<figure><img src="image_6.png" /><figcaption>FIGURE 5 – (A) Integral equation; (B) an increasing $F$-enriched tree</figcaption></figure>

This integral equation, as interpreted by Figure 5(A), constitutes an iterative process for the construction of combinatorial solutions. We obtain the species $A_F$ of increasing $F$-enriched trees. Figure 5(B) shows a increasing $F$-enriched tree on the pair $(l, U)$, where $l = [19]$ and $U = \{a, b, \dots, k\}$. Note that the fiber of each internal vertex (vertex of sort $T$) is equipped with an $F$-enrichment, and the points of sort $Z$ appear at the leaves.

An increasing $F$-enriched tree on a pair $(l, U)$, with $l \neq \emptyset$, consists of the minimal element of $l$, to which is attached an $F(A_F)$-structure. The isomorphism $\phi: \frac{\partial A_F}{\partial T} \to F(A_F)$ is defined canonically by forgetting the root, as indicated in Figure 7. We obtain the following theorem.

<figure><img src="image_7.png" /><figcaption>FIGURE 6 – Isomorphism $\phi$</figcaption></figure>

The complete proof of this is given in [17] (see also [1] or [2]).

**Theorem 5.2.** The equation (8) admits as a canonical solution the pair $(A_F, \phi)$, where :

(1) $A_F$ is the two-sorted $\mathbb{L}$-species of growing $F$-enriched trees, if $F$ is an $\mathbb{L}$-species, and $A_F$ is the mixed species of growing $F$-enriched trees if $F$ is a $\mathbb{B}$-species

(2) $\phi: \frac{\partial A_F}{\partial T} \to F(A_F)$ is the isomorphism that forgets the root of the tree.

Moreover, this solution is unique up to isomorphism, that is, for any solution $(B, \psi)$, there exists a unique species isomorphism $\Phi: A_F \to B$, such that the following diagram commutes : Furthermore, the generating series $y = A_F(t, z)$ is a solution of the analytic equation :

$$\frac{\partial y}{\partial t} = F(y(t, z)), y(0, z) = z \qquad (10)$$
$$ \begin{array}{ccc} A_F & \frac{\partial A_F}{\partial T} & \phi \\ \Big| & \Phi & \Big| \frac{\partial \Phi}{\partial T} \\ \Big| & & F(\Phi) \\ B & \frac{\partial B}{\partial T} & \psi \end{array} \longrightarrow F(B) $$

5.2. **Analysis of Initial Conditions.** In Leroux-Viennot's theory, the initial conditions are of the form $Y(0) = 0$, or $Y(0) = 1$, or $Y(0) = Z$, where Z represents a sort of points. In this subsection, we analyze initial conditions of the form $Y(0) = \alpha$, where $\alpha$ is either a constant or a formal variable.

5.2.1. Initial condition Z = 0. This initial condition corresponds to the combinatorial differential equation:

$$ Y' = F(Y), Y(0) = 0. \tag{11} $$

It means that the solutions contain no leaf of type Z. These solutions are the L-species $A_F(T) = A_F(T, 0)$, whose generating series $y(t) = A_F(t)$ is a solution of the differential equation:

$$ y' = F(y), y(0) = 0. \tag{12} $$

5.2.2. Initial condition Z = 1_α. Recall that 1_α is the species 1 of weighted by α. The condition Z = 1_α corresponds to an analytic initial condition of the form $y(0) = \alpha$.

Consider the combinatorial differential equation:

$$ Y' = F(Y), Y(0) = 1_{\alpha}. \tag{13} $$

To interpret $A_F(T, 1_α)$, we must first give a general definition of $H(T, 1_α)$, that is, the substitution of the species $1_α$ for Z in a given species $H(T, Z)$.

I). Suppose that $H = H_v$ is a mixed species and $\alpha$ is a formal variable.

In this case, we set $H(T, 1_α) = (T_{Z:α}H)(T)$ (definition 4.4). An $H(T, 1_α)$-structure is thus an isomorphism type of structures with respect to the sort Z, and the variable $\alpha$ is a counter of Z-type points.

II). Suppose $H = H_v$ is a mixed species and $\alpha$ is a constant.

We consider the L-species $T_{Z:z}H(T)$, whose generating series is:

$$ T_{Z:z}H(t) = \sum_{n \ge 0} \sum_{k \ge 0} \tilde{h}_{n,k} z^k \frac{t^n}{n!}. $$

If, for every integer $n$, the sum $\sum_{k \ge 0} \tilde{h}_{n,k} z^k$ exists, then the disjoint union $\sum_{k \ge 0} \frac{H[n,k]}{\sim}$ is A-summable for the weight function $w$ defined by $w(\tilde{s}) = z^k v(s)$, where $\tilde{s}$ is the class of $s$ in $\frac{H[n,k]}{\sim}$. We define the new species $T_{Z:α}H(T)$, weighted by $w$, by setting $T_{Z:α}H[l] = T_{Z:z}H[l]|_{z=α}$.

Note that if $H$ is trivially weighted, then the sum $\sum_{k \ge 0} \tilde{h}_{n,k} z^k$ exists if and only if it is a convergent series. In particular, if $z = 1$, and since $\tilde{h}_{n,k}$ are integers, the series converges if and only if there exists an integer $k_0 > 0$ such that for all $k \ge k_0$, $\tilde{h}_{n,k} = 0$. Necessarily, $h_{n,k} = 0$ for all $k \ge k_0$.

**Definition 5.3.** (1) A species $F$ is said to be polynomial if there exists $k_0 \in \mathbb{N}$ such that for all $n \ge k_0$, $F[n] = \emptyset$.
(2) A mixed species $H = H(T, Z)$ is polynomial in $Z$ if, for every totally ordered set $l$, the $\mathbb{B}$-species $H[l, -]$ is polynomial.

**Proposition 5.4.** (1) If $H = H(T, Z)$ is trivially weighted, then the $\mathbb{L}$-species $T_{Z,1}H(T)$ is defined if and only if $H$ is polynomial in $Z$.

(2) If $F = F(Z)$ is a polynomial $\mathbb{B}$-species, then the species $A_F(T, Z)$ is polynomial in $Z$.

*Proof:* 1) Knowing that $\alpha = 1$ and that $H$ is trivially weighted, the set is $\mathbb{A}$-summable if and only if it is finite.

2) By induction on $|l|$. ◇

III). Suppose $H$ is an $\mathbb{L}$-species of two sorts. In this case, the notion of isomorphism type has no meaning. It is therefore impossible to give an interpretation to $H(T, \mathbf{1}_\alpha)$ in this case.

Suppose that in (8), $F$ is a $\mathbb{B}$-species. In this case, $A_F(T, Z)$ is a mixed species. We consider the two following hypotheses :

$H_1$: $\alpha$ is a formal variable;

$H_2$: $\alpha$ is a numeric constant, in which case we assume $F$ is polynomial.

Consider the $\mathbb{L}$-species $G = G(T) = (T_{Z:\alpha}A_F)(T)$ of types of $A_F$-structures with respect to $Z$. Since $A_F(0, Z) = Z$, it is clear that $G(0) = \mathbf{1}_\alpha$. On the other hand, $G' \simeq F(G)$ because : $G' = \frac{d}{dT}(T_{Z:\alpha}A_F)(T) = T_{Z:\alpha}\frac{\partial}{\partial T}A_F(T, Z) \simeq T_{Z:\alpha}F(A_F(T, Z)) \simeq F((T_{Z:\alpha}A_F)(T)) = F(G)$. Using Corollary 4.11 of Theorem 4.7, we obtain the following result :

**Theorem 5.5.** If $F$ is a $\mathbb{B}$-species and $\alpha \in \mathbb{A}$, then under hypotheses $H_1$ or $H_2$, equation (13) admits the $\mathbb{L}$-species $(T_{Z:\alpha}A_F)(T)$ as its solution. In this case, the generating series $y(t) = (T_{Z:\alpha}A_F)(t)$ satisfies the differential equation :

$$y'(t) = Z_F(y(t); \alpha^2, \alpha^3, \dots), y(0) = \alpha, \qquad (14)$$

where $Z_F$ is the cycle index series of $F$.

**Remark 5.6.** If $F$ is asymmetric, then $Z_F(x_1, x_2, x_3, \dots) = F(x_1)$, and equation (14) becomes : $y' = F(y), y(0) = \alpha$. Moreover, for any $\mathbb{B}$-species $F$, $Z_F(x, 0, 0, \dots) = F(x)$. Thus, setting $\alpha = 0$ in equation (14) we have equation (12).

**Example 5.7.** Increasing $(1-n)$-trees. Let's consider the equation

$$Y' = E_2(Y), Y(0) = Z, \qquad (15)$$

where $E_2$ is the $\mathbb{B}$-species of 2-element sets. The canonical solution is the species $A_{E_2}(T, Z)$ of increasing $E_2$-enriched trees. Observing that each node's children form an $E_2$-structure, we adopt the convention that these children are placed from left to right in increasing order. Figure 7 represent an $A_{E_2}$-structure on [8], weighted by $z^9$. The exponential generating func-

<figure><img src="image_8.png" /><figcaption>FIGURE 7 - An $A_{E_2}$-structure on [8],</figcaption></figure>

tion of this species, $y(t, z) = A_{E_2}(t, z)$, is the solutions of the differential equation $\frac{dy}{dt} = \frac{y^2}{2}$, $y(0, z) = z$, is :

$$y(t, z) = \frac{2z}{2 - zt}.$$
The solution of the similar equation $Y' = E_2(Y)$, $Y(0) = 1$, obtained by taking types with respect to $Z$, $Y = T_{Z:1}A_{E_2}^\dagger$, is isomorphic to the $\mathbb{L}$-species of increasing $(1-2)$-trees (see [28]). Recalling that $Z_{E_2}(z_1, z_2, \cdots) = \frac{z_1^2+z_2^2}{2}$, the generating series is the solution of : $y'(t) = \frac{y(t)^2+1}{2}$, $y(0) = 1$, and is given by : $y(t) = \tan(2t+\frac{\pi}{4}) = \tan(t)+\sec(t)$. The coefficients of this series are the Euler numbers.

More generally, increasing $(1-n)$-trees are the solutions of the equation

$$Y' = E_n(Y), Y(0) = 1.$$

Each vertex in such a tree has at most $n$ children, and the generating series is the solution of the differential equation $y' = Z_{E_n}(y, 1, 1, \cdots)$, $y(0) = 1$. Knowing that $Z_{E_k}(z, z, z, \cdots) = \frac{z(z+1)\cdots(z+k-1)}{k!} = \frac{z^{(k)}}{k!}$ is the generating polynomial of permutations $\sigma \in S_k$ by number of cycles, we obtain $Z_{E_n}(z+t, z, z, \cdots) = \sum_{k=0}^{n} \frac{t^{n-k}}{(n-k)!k!}z(z+1)\cdots(z+k-1)$. By setting $z=1$, we can deduce that the generating series of increasing $(1-n)$-trees is the solution of the differential equation $y' = \sum_{k=0}^{n} \frac{(y-1)^k}{k!}$, $y(0) = 1$, or, setting $u = y-1$, we obtain $u' = \sum_{k=0}^{n} \frac{u^k}{k!} = E_{\le n}(u)$, $u(0) = 0$.

**Remark 5.8.** For any $\mathbb{B}$-species, we have $Z_G(y, \alpha^2, \alpha^3, \cdots) = \sum_{k \ge 0} \frac{(y-\alpha)^k}{k!} \tilde{G}^{[k]}(\alpha)$. Therefore, the equation $y' = Z_G(y, \alpha^2, \alpha^3, \cdots)$ is equivalent to $y' = \sum_{k \ge 0} \frac{(y-\alpha)^k}{k!} \tilde{G}^{[k]}(\alpha)$.

**5.3. System of combinatorial differential equations.** Let $(F_i)_{1 \le i \le k}$ be a family of species, each of which is either an $\mathbb{L}$-species or a $\mathbb{B}$-species, and $(Z_i)_{1 \le i \le k}$ a family of sorts of points representing initial conditions. Consider the combinatorial system of differential equations :

$$Y'_{i} = F_{i}(Y_{1}, Y_{2}, \cdots, Y_{k}), Y_{i}(0) = Z_{i}, 1 \le i \le k \qquad (16)$$

**Definition 5.9.** A solution to system (16) is a pair $(A, \Phi)$, where $A = (A_1, A_2, \cdots, A_k)$ is a $k$-tuple of mixed species of type $\mathbb{L} \times \mathbb{B}^k$ if all $F_i$ are $\mathbb{B}$-species, or $(k+1)$-sorted $\mathbb{L}$-species if all $F_i$ are $\mathbb{L}$-species, and $\Phi = (\phi_1, \phi_2, \cdots, \phi_k)$ is a $k$-tuple of isomorphisms $\phi_i : \frac{\partial A_i}{\partial T} \to F_i(A)$.

Let $\vec{F} = (F_1, F_2, \cdots, F_k)$, $\vec{Y} = (Y_1, Y_2, \cdots, Y_k)$, and $Z = (Z_1, Z_2, \cdots, Z_k)$. The system (16) can be written in integral form :

$$Y_i(T, \vec{Z}) = Z_i + \int_0^T F_i(\vec{Y}(X, \vec{Z}))dX, 1 \le i \le k. \qquad (17)$$

An interpretation of these integral equations is given in Figure 8(A), where the minimal

<figure><img src="image_9.png" /><figcaption>FIGURE 8 – (A) Integral equation; (B) increasing $\vec{F}$-enriched trees of type 1</figcaption></figure>

element is of color $i$ and is followed by an $F_i(Y_1, Y_2, \cdots, Y_k)$-structure. Iterating this process for each $1 \le i \le k$ yields the species of increasing $\vec{F}$-enriched trees of type $i$, which we denote
by $A_{i,F} = A_{i,F}(T, Z_1, \dots, Z_k)$ (Figure 8(B)). This species is characterized by the fact that each vertex is colored by a label $j \in [k]$. The fiber of an internal vertex of color $j$, viewed as a $k$-set, is then endowed with an $F_j$-enrichment (and the root with an $F_i$-enrichment). We have the following theorem (see [1] or [2] or [17]).

**Theorem 5.10.** The system (16) admits as canonical solution the pair $(A_F, \phi)$, where $A_F = (A_{1,F}, A_{2,F}, \dots, A_{k,F})$, and $\phi = (\phi_i)_{1 \le i \le k}$ with $\phi_i : A_{i,F} \to F(A_{i,F})$ the canonical isomorphism that forgets the root.

Moreover, denoting $y(t) = (y_1(t), y_2(t), \dots, y_k(t))$, the generating series of the $A_{i,F}$ are solutions to the differential system :

$$y'_{i} = F_{i}(y(t)), y_{i}(0) = z_{i}, 1 \le i \le k. \qquad (18)$$

Now assume that all $F_i$ are $\mathbb{B}$-species. For each $i$, let $\alpha_i \in \mathbb{A}$. Theorem 5.5 extends as follows :

**Theorem 5.11.** If the $F_i$ are $\mathbb{B}$-species, then the differential system : $\{Y'_i = F_i(Y), Y_i(0) = 1_{\alpha_i}, 1 \le i \le k\}$ admits as solution the $k$-tuple of $\mathbb{L}$-species

$$T_{\vec{Z}:\vec{\alpha}} A_F = (T_{\vec{Z}:\vec{\alpha}} A_{1,F}, T_{\vec{Z}:\vec{\alpha}} A_{2,F}, \dots, T_{\vec{Z}:\vec{\alpha}} A_{k,F})$$

where $\vec{Z}: \vec{\alpha}$ means that for each $i$, $Z_i : \alpha_i$. Moreover, the generating series of the species $T_{\vec{Z}:\vec{\alpha}} A_{i,F}$ are solutions of the system :

$$\{y'_{i} = Z_{F_{i}}(y_{1}, \alpha_{1}^{2}, \alpha_{1}^{3}, \dots; y_{2}, \alpha_{2}^{2}, \alpha_{2}^{3}, \dots; \dots; y_{k}, \alpha_{k}^{2}, \alpha_{k}^{3}, \dots), y_{i}(0) = \alpha_{i}, 1 \le i \le k\}$$

where $Z_{F_i}(x_{11}, x_{12}, \dots; x_{21}, x_{22}, \dots; \dots; x_{k1}, x_{k2}, \dots)$ is the cycle index series of the $\mathbb{B}$-species $F_i$.

**Example 5.12.** Complete increasing $m$-ary trees.

Consider the pair $(U, V) = (\mathcal{A}_{m,c}^{\uparrow}, \mathcal{B}_{m,c}^{\uparrow})$, solution of the system :

$$\left\{ \begin{array}{ll} U' = 1 + U^m & U(0) = 0 \\ V' = U^{m-1}V & V(0) = 1 \end{array} \right. \qquad (19)$$

The species $U = \mathcal{A}_{m,c}^{\uparrow}$ is called the species of complete increasing $m$-ary trees of the first

<figure><img src="image_10.png" /><figcaption>FIGURE 9 – (A) An $\mathcal{A}_{3,c}^{\uparrow}$-structure; (B) A $\mathcal{B}_{3,c}^{\uparrow}$-structure</figcaption></figure>

type. It lives on sets of cardinality $mk + 1$, $k \in \mathbb{N}$. On the other hand, the species $V = \mathcal{B}_{m,c}^{\uparrow}$ is the species of complete $m$-ary trees of the second type, where the unique white point is a point of sort $Z$. It lives on sets of cardinality $mk$, $k \in \mathbb{N}$. Figure 9 represent an $\mathcal{A}_{3,c}^{\uparrow}$-structure on [10], and $\mathcal{B}_{3,c}^{\uparrow}$-structure on [12].

Moreover, setting $\mathcal{A}_{m,c}^{\uparrow}(t) = \sum_{n \ge 0} e_{m,n} \frac{t^n}{n!}$ and $\mathcal{B}_{m,c}^{\uparrow}(t) = \sum_{n \ge 0} f_{m,n} \frac{t^n}{n!}$, we have the relations :

$$e_{m,n+1} = \sum_{n_1+n_2+\dots+n_m=n} e_{m,n_1} e_{m,n_2} \cdots e_{m,n_m},$$
$$f_{m,n+1} = \sum_{n_0+n_1+\cdots+n_{m-1}=n} f_{m,n_0} e_{m,n_1} \cdots e_{m,n_{m-1}}$$

with $e_{m,n} = 0$, if $n \notin m\mathbb{Z} + 1$ and $f_{m,n} = 0$ if $n \notin m\mathbb{Z}$. Note that in the particular case $m = 2$, the solutions of system (19) correspond to complete and semi-complete binary trees. The sequences $e_{m,n}$ and $f_{m,n}$ therefore the Euler numbers.

**Proposition 5.13.** The species $U = A_{m,c}^{\uparrow}$ and $V = B_{m,c}^{\uparrow}$ satisfy the equations $(B_{m,c}^{\uparrow})^m \simeq 1 + (A_{m,c}^{\uparrow})^m$ and $B_{m,c}^{\uparrow} \simeq E(f(A_{m,c}^{\uparrow})^{m-1})$.

Proof. For $m = 3$, the isomorphisms are illustrated in Figures 10 and 11.

<figure><img src="image_11.png" /><figcaption>FIGURE 10 – Isomorphism between $(B_{m,c}^{\uparrow})^m$ and $1 + (A_{m,c}^{\uparrow})^m$</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIGURE 11 – Isomorphism between $B_{m,c}^{\uparrow}$ and $E(f(A_{m,c}^{\uparrow})^{m-1})$</figcaption></figure>

**Corollary 5.14.** The solutions of the analytic system

$$y'(t) = 1 + y^m(t), z'(t) = y^{m-1}(t), y(0) = 0, z(0) = 1$$

satisfy the identities :

$$z^m(t) = 1 + y^m(t) \text{ and } Z(t) = e^{\int_0^t y^{m-1}(u)du} \qquad (20)$$

## 6. A BOUND FOR THE COEFFICIENTS.

Let us now consider the differential system

$$Y_i' = F_i(Y_1, Y_2, \cdots, Y_k), Y_i(0) = 0, i = 1, \cdots, k. \qquad (21)$$

Suppose that the generating series of the solutions $Y_i = A_{i,\bar{F}}$ is $Y_i(t) = A_{i,\bar{F}}(t) = \sum_{n \ge 0} a_{i,n} \frac{t^n}{n!}$, and that

$$F_i(y_1, \cdots, y_k) = \sum_{n_1, \cdots, n_k} c_{n_1, \cdots, n_k}^{(i)} \frac{y_1^{n_1} \cdots y_k^{n_k}}{n_1! \cdots n_k!}$$

with $c_{n_1, \cdots, n_k}^{(i)} \in \mathbb{C}$.

Then we have the following proposition, which provides an upper bound for the coefficients $a_{i,n}$ and generalizes Theorem 3(i) from [4], where the series $F_i(y_1, \cdots, y_k)$ are assumed to be polynomials.
**Theorem 6.1.** If there exist constants $\alpha_1, \dots, \alpha_k \in \mathbb{R}^+$ such that, for all $i$ and all $(n_1, \dots, n_k)$, $|c_{n_1, \dots, n_k}^{(i)}| \le \alpha_1^{n_1} \cdots \alpha_k^{n_k} n_1! \cdots n_k!$ then there exists a constant $\alpha \in \mathbb{R}^+$ such that, for all $n$ and all $i$,

$$|a_{i,n}| \le (k+1)^{n-1} \alpha^{n-1}(n-1)!.$$

*Proof:* Consider the auxiliary system (22), whose combinatorial solutions are trees obtained by iterating the process described in Figure 12 :

$$Z_i' = \prod_{j=1}^{k} \mathbb{L}(\alpha_j Z_j), Z_j(0) = 0, 1 \le i \le k \quad (22)$$

where $\mathbb{L}$ is the species of lists and $\alpha_j \in \mathbb{R}^+$. Let $Z_i(t) = \sum_{n \ge 0} b_n \frac{t^n}{n!}$ be the generating function of the $i$-th component of the solution.

<figure><img src="image_13.png" /><figcaption>FIGURE 12 – Combinatorial solution of auxiliary system</figcaption></figure>

**Lemma 6.2.** Let $\alpha_i \in \mathbb{R}^+$. If for all $i$ and all $(n_1, \dots, nk)$ we have $|c_{n_1, \dots, n_k}^{(i)}| \le \alpha_1^{n_1} \cdots \alpha_k^{n_k} n_1! \cdots n_k!$, then for all $i$ and all $n$, $|a_{i,n}| \le b_{i,n}$.

*Proof:* Let $F$ be an $\mathbb{L}$-species with $k$ sorts, and $G_1, \dots, G_k$ are one-sort species. An $F(G_1, \dots, G_k)$-structure on a finite set $l$ consists of a quadruple $(\pi, \epsilon, f, (g_c)_{c \in \pi})$, where $:\pi$ is a partition of $l$, $\epsilon: \pi \to [k]$ a function, $g_c \in G_{\epsilon(c)}$ and $f \in F[\epsilon^{-1}(1), \epsilon^{-1}(2), \dots, \epsilon^{-1}(k)]$ (see [22]). Equations (21) and (22) show that :

$$a_{i,n+1} = \sum_{\pi \in \mathrm{Par}[n]} \sum_{\epsilon; \pi \to [k]} \alpha_{|\epsilon^{-1}(1)|, \dots, |\epsilon^{-1}(1)|}^{(i)} \prod_{j=1}^{k} \prod_{c \in \epsilon^{-1}(j)} a_{j,|c|} \quad \text{and}$$

$$b_{i,n+1} = \sum_{\pi \in \mathrm{Par}[n]} \sum_{\epsilon; \pi \to [k]} |\epsilon^{-1}(1)|! \cdots |\epsilon^{-1}(k)|! \alpha_1^{\epsilon^{-1}(1)|!} \cdots \alpha_k^{\epsilon^{-1}(k)|!} \prod_{j=1}^{k} \prod_{c \in \epsilon^{-1}(j)} b_{j,|c|}$$

An easy induction shows that $|a_{i,n}| \le b_{i,n}$ for all $i$ and $n$. $\diamond$

*Continuation of the proof of 6.1.* It is clear that if $Z$ is the solution of equation $Z' = \prod_{j=1}^{k} L(\alpha_j Z), Z(0) = 0$, then the solutions of system (22) satisfy $Z_i = Z_j = Z$ for all $i, j$. Let $\alpha = \max\{\alpha_i, 1 \le i \le k\}$ and consider the equation :

$$U' = \prod_{i=1}^{k} \mathbb{L}(\alpha U) = \mathbb{L}^{k}(\alpha U), U(0) = 0.$$

If we write $U(t) = \sum_{n \ge 0} u_n \frac{t^n}{n!}$, then Lemma 6.2 implies that for all $n$, $b_{i,n} \le u_n$. Since $U(t)$ satisfies the analytic equation $U' = \frac{1}{(1-\alpha U)^k}$, with $U(0) = 0$. It follows that $U(t) = \frac{1}{\alpha} [1 - (1 - (k+1)t)^{\frac{1}{k+1}}]$. Using the binomial expansion, we have $u_n = k(2k+1)(3k+2) \cdots ((n-1)k + n - 2)\alpha^{n-1}$, and it suffices to observe that $k(2k+1)(3k+2) \cdots ((n-1)k + n - 2) \le (k+1)(2k+2) \cdots ((n-1)k + n - 1) = (k+1)^{n-1}(n-1)!$. $\diamond$
## 7. DIFFERENTIAL OPERATORS

**Definition 7.1.** A combinatorial differential operator is an operator $\mathcal{D}$ defined on the class of species with $k$ sorts ($k \ge 1$) such that:

(1) To every species $FF$, it assigns a species $\mathcal{D}(F)$ satisfying, for all species $F$ and $G$:

$$ \mathcal{D}(F + G) = \mathcal{D}(F) + \mathcal{D}(G), \text{ and } \mathcal{D}(F \cdot G) = \mathcal{D}(F) \cdot G + F \cdot \mathcal{D}(G) $$

(2) To every morphism $\varphi: F \to G$, it assigns a morphism $\mathcal{D}(\varphi): \mathcal{D}(F) \to \mathcal{D}(G)$ satisfying, for all morphism $\varphi: F \to G$ and $\psi: H \to K$:

$$ \mathcal{D}(\varphi + \psi) = \mathcal{D}(\varphi) + \mathcal{D}(\psi), \text{ and } \mathcal{D}(\varphi \cdot \psi) = \mathcal{D}(\varphi) \cdot \psi + \varphi \cdot \mathcal{D}(\psi) $$

For example, to the differential system $\{\frac{\partial Y_i}{\partial T} = F_i(Y_1, \dots, Y_k), Y_0(Z) = Z_i, 1 \le i \le k\}$ is canonically associated the differential operator

$$ \mathcal{D} = \sum_{i=1}^{k} F_i(Z_1, \dots, Z_k) \frac{\partial}{\partial Z_i} $$

which assigns to any $k$-sort species $G$ the species $\mathcal{D}(G) = \sum_{i=1}^{k} F_i(Z_1, \dots, Z_k) \frac{\partial G}{\partial Z_i}$, and to any morphism $\phi$ the morphism $\mathcal{D}(\phi) = \sum_{i=1}^{k} F_i(Z_1, \dots, Z_k) \frac{\partial \phi}{\partial Z_i}$.

To the differential operator $\mathcal{D} = \sum_{i=1}^{k} F_i(Z_1, \dots, Z_k) \frac{\partial}{\partial Z_i}$, is associated the operator $e^{\mathcal{D}} = \sum_{n \ge 0} \frac{T^n}{n!} \mathcal{D}^n$, called by Gilbert Labelle the multiple blossoming operator associated with $\mathcal{D}$ in [15] (see [1] or [2] on [17]).

We then have the following proposition:

**Proposition 7.2.** (Leroux-Viennot [17]). The solution $Y = A_{\vec{F}} = (A_{\vec{F},1}, \dots, A_{\vec{F},k})$ of the system $\{\frac{dY_i}{dT} = F_i(Y_1, \dots, Y_k), Y_i(0) = Z_i, 1 \le i \le k\}$, satisfy

$$ A_{\vec{F}} = (A_{\vec{F},1}, \dots, A_{\vec{F},k}) = (e^{\mathcal{D}}(Z_1), \dots, e^{\mathcal{D}}(Z_k)). $$

More generally, for any species $G = G(Z_1, \dots, Z_k)$, we have

$$ e^{\mathcal{D}}(G(Z_1, \dots, Z_k)) = G(A_{\vec{F}}). $$

This proposition states that the solutions of a system of combinatorial differential equations can be obtained by computing the successive powers of the associated differential operator. The following theorem by Ginocchio (see [12]), whose case $n = 1$ was proved by François Bergeron and Christophe Reutenauer in [3], provides a method for computing them.

**Notation.** Let $\mu = (i_1, i_2, \dots, i_r)$, where $i_j \in [k]$ and $0 \le r \le n$. Define $\partial\mu = \frac{\partial^r}{\partial Z_{i_1} \dots \partial Z_{i_r}}$, and set $F_i^{(\mu)} := \partial_i F_i$. If $r = 0$, $\mu = \emptyset$ is the empty sequence, and $\partial\mu = \text{Id}$.

Let $D = \sum_{i=1}^{k} F_i(Z_1, \dots, Z_k) \frac{\partial}{\partial Z_i}$ be a differential operator. Consider the set $\mathcal{F}_n$ of forests of increasing trees on $[n]$, and let $\varphi \in \mathcal{F}_n$. To each vertex $m$ of $\varphi$, we associate an integer variable $i_m$ such that $1 \le i_m \le k$, and define the sequence of integers $\mu_m = (i_{m1}, i_{m2}, \dots, i_{mr})$ where $m_1, m_2, \dots, m_r$ are the children of $m$ (if $m$ is a leaf, set $\mu_m = \emptyset$). The vertex $m$ is weighted by $F_{im}$ ($\mu_m$). On the other hand, we define $\mu_0 = (i_1, i_2, \dots, i_{j_0})$, where $\{1, j_2, \dots, j_s\}$ is the set of roots of the trees in the forest $\varphi$. The weight of $\varphi$ is, by definition,

$$ \omega(\varphi) = \sum_{1 \le i_1, i_2, \dots, i_n \le k} \Pi_{s=1}^{n} F_{is}^{(\mu_{is})} \partial_{\mu_0}. $$

Thus, for the forest $\varphi$ in Figure 13, we set $\mu_{i_1} = (i_2, i_4, i_{11}, \mu_{i_2} = \emptyset, \mu_{i_3} = (i_5, i_7), \mu_{i_4} =$
<figure><img src="image_14.png" /><figcaption>FIGURE 13 – Solution combinatoire du système auxiliaire</figcaption></figure>

$(i_8, i_{10}), \mu_{i_7} = (i_9), \mu_{i_0} = (i_1, i_3, i_6), \mu_{i_2} = \mu_{i_5} = \mu_{i_6} = \mu_{i_8} = \mu_{i_9} = \mu_{i_{10}} = \mu_{i_{11}} = \emptyset$, and

$$ \omega(\varphi) = \sum_{1 \le i_1, i_2, \dots, i_k \le 11} F_{i_1}^{(\mu_{i_1})} F_{i_2}^{(\mu_{i_2})} \dots F_{i_{11}}^{(\mu_{i_{11}})}. $$

**Proposition 7.3.** (Ginocchio [12]). Let $\mathcal{F}_n$ be the set of all forests of trees on $[n]$. Then $\mathcal{D}^n = \sum_{\varphi \in \mathcal{F}_n} \omega(\varphi)$. In particular, if $k = 1$ and $\mathcal{D} = F(Z) \frac{d}{dz}$, then $\mathcal{D}^n = \sum_{\varphi \in \mathcal{F}_n} \omega(\varphi)$, where $\omega(\varphi) = \Pi_{i \in [n]} F^{(\alpha(i))} d^m$, with $d = \frac{d}{dz}$, $\alpha(i)$ the number of children of vertex $i$, and $m$ is the number of trees constituting the forest $\varphi$.

*Proof:* It suffices to observe that for any species $G$, a $\mathcal{D}^n(G)$-structure (Figure 14(A)) is canonically identified with a pair consisting of a collection of trees where each vertex is weighted by a $F^{\mu_i}$-structure, and a $G^{(\mu_0)}$-structure (Figure 14(B)). ♦

<figure><img src="image_15.png" /><figcaption>FIGURE 14 – A $\mathcal{D}(G)$-structure</figcaption></figure>

## 8. CONNECTION WITH WILLIAM CHEN’S GRAMMARS.

In [6], William Y. C. Chen uses the concept of substitution rules from the algebraic grammars theories to study enumeration problems. This idea was later taken up by Dumont (see [10]). In this section, we will establish the connection between William Chen’s grammars and System of combinatorial differential equations.

A grammar in the sense of William Chen is a set $G = \{x \to G(x) | x \in V\}$ of substitution rules replacing a letter $x$ from the alphabet $V$ by its image $G(x)$, who is a formal function. In the context of this work, we assume that $G(x)$ is an element of the ring $\mathbb{A}[[V]]$ of formal power series, where $\mathbb{A}$ is an integral domain of characteristic 0. In other words, a grammar over $V$ is a map $G: V \to \mathbb{A}[[V]]$, where $V$ is an alphabet (possibly infinite) of formal variables commuting pairwise.

To the grammar $G$, we can associate a differential operator $\mathcal{G}$, defined by

$$ \mathcal{G} = \sum_{x \in V} G(x) \frac{\partial}{\partial x} $$
, satisfying $\mathcal{G}(x) = G(x)$, if $x \in V$, and for all $u, v \in A[[V]]$ $\mathcal{G}(u + v) = \mathcal{G}(u) + \mathcal{G}(v)$.
Let $t$ be a formal variable not belonging to $V$, and let $u \in A[[V]]$. We define

$$ \mathrm{Gen}(u, t) = \sum_{n \ge 0} \mathcal{G}^n(u) \frac{t^n}{n!} $$

**Proposition 8.1.** (cf. Chen [6], Props. 3.2 and 3.4) For every pair $(u, v) \in A[[V]]$, we have : $\mathrm{Gen}(u + v, t) = \mathrm{Gen}(u, t) + \mathrm{Gen}(v, t)$, $\mathrm{Gen}(uv, t) = \mathrm{Gen}(u, t)\mathrm{Gen}(v, t)$, and $\mathrm{Gen}(\mathcal{G}(u), t) = \frac{\partial}{\partial t}\mathrm{Gen}(u, t)$.

**Remark 8.2.** Suppose $A = \mathbb{Z}$ and $V = \{X_1, \dots, X_k\}$ is an alphabet representing types of points. Then, an element $u \in \mathbb{Z}[[V]]$, and in particular $G(X_i)$, can be identified with an asymmetric $B$-species, and the operator $\mathcal{G}$ becomes a combinatorial differential operator such that $\mathrm{Gen}(u, T) = e^{\mathcal{T}\mathcal{G}}(u)$. In this sense, a grammar corresponds to a combinatorial differential operator.

However, Chen, and later Dumont, interpret the variables as counters. The transition from level $n$ to level $n + 1$ for a given structure is interpreted in terms of grammars on these variables.

Let $\vec{x}' = \{x_1, x_2, \dots, x_n\}$ be a family of formal variables, and for each $i = 1, \dots, n$, let $G_i(\vec{x}') \in \mathbb{C}[[x]]$. Let us denote by $\mathcal{G} = \sum_{i=1}^n G_i(\vec{x}') \frac{\partial}{\partial x_i}$ the differential operator associated with the grammar defined over the alphabet $\vec{x}'$ by $G := \{x_i \to G_i(\vec{x}') \mid 1 \le i \le n\}$.

**Proposition 8.3.** (Dumont [10]) Let $\vec{y}(t) = (y_1(t), y_2(t), \dots, y_n(t))$ be the solution of the analytic differential system

$$ y'_{i}(t) = G_{i}(\vec{y}(t)), y_{i}(0) = x_{i}, i = 1, \dots, n. $$

Then, for each $i$, we have

$$ \mathrm{Gen}(x_i, t) = y_i(t). $$

*Proof:* Let $h_i(t) = \mathrm{Gen}(x_i, t) = \sum_{n \ge 0} \mathcal{G}^n(x_i) \frac{t^n}{n!}$, and set $\vec{h}(t) = (h_i(t))_{1 \le i \le n}$. We have $h_i(0) = \mathcal{G}^0(x_i) = x_i$. Moreover, by Proposition 8.1, we have : $h'_i(t) = \frac{\partial}{\partial t}(\mathrm{Gen}(x_i, t)) = \mathrm{Gen}(G_i(x), t) = G_i((\mathrm{Gen}(x_j, t))_{j \in I}) = G_i(\vec{h}(t))$. Therefore $h_i(t) = y_i(t)$. $\diamond$

Now, let $\vec{X} = \{X_i, 1 \le i \le n\}$ be an alphabet of variables representing types of points. To each $X_i$, we associate a formal variable $x_i$ that acts as a counter ot points of type $X_i$. Let $(G_i(\vec{X}))_{1 \le i \le n}$ be a family of asymmetric $\mathbb{B}$-species such that, for each $i$, the exponential generating series of $G_i$ is $G_i(\vec{x})$, where $\vec{x} = (x_i)_{1 \le i \le n}$. Moreover, we consider the combinatorial differential operator $\mathcal{D} = \sum_{i=1}^n G_i(\vec{X}) \frac{\partial}{\partial x_i}$, and the analytic differential operator $\mathcal{G} = \sum_{i=1}^n G_i(\vec{x}) \frac{\partial}{\partial x_i}$ associated with the grammar $G = \{x_i \to G_i(\vec{x}), 1 \le i \le n\}$.

**Corollary 8.4.** (1) To every family $(G_i(\vec{X}))_{1 \le i \le n}$ of asymmetric $\mathbb{B}$-species, where exponential generating series are $(G_i(\vec{x}))_{1 \le i \le n}$, one can associate :

(a) a system of combinatorial differential equations :

$$ Y'_{i} = G_{i}(Y_{1}, Y_{2}, \dots, Y_{n}), Y_{i}(0) = X_{i}, i = 1, \dots, n. \qquad (23) $$

(b) a grammar :

$$ G = \{x_i \to G_i(\vec{x}), 1 \le i \le n\} \qquad (24) $$

(c) a system of analytic differential equations

$$ y'_{i}(t) = G_{i}(\vec{y}(t)), y_{i}(0) = x_{i}, i = 1, \dots, n. \qquad (25) $$
(2) Reciprocally, with a grammar $G = \{x_j \to \sum_{\vec{i}=(i_1, \dots, i_n)} a_{j, \vec{i}} x_1^{i_1} \cdots x_n^{i_n}, 1 \le j \le n\}$, or an analytic system of differential equations $\{y_i' = \sum_{\vec{i}=(i_1, \dots, i_n)} a_{j, \vec{i}} y_1^{i_1} \cdots y_n^{i_n}, y_i(0) = x_i, 1 \le j \le n\}$ one can associate a system of combinatorial differential equations (23) where $G_j(Y_1, \dots, Y_n) = \sum_{\vec{i}=(i_1, \dots, i_n)} a_{j, \vec{i}} Y_1^{i_1} \cdots Y_n^{i_n}$.

Solutions of (23) are $\{Y_i = e^{TD}(X_i), i = 1 \cdots n\}$, where $D = \sum_{i=1}^n G_i(X_1, X_2, \cdots, X_n) \frac{\partial}{\partial X_i}$, and solutions of (25) satisfy :

$$y_i(t) = T_{\vec{X}=\vec{x}}(e^{TD}(X_i))(t) = \mathrm{Gen}(x_i, t) \quad (26)$$

and is the generating series of $T_{\vec{X}=\vec{x}}e^{TD}(X_i)$.

*Proof:* This result follows from Propositions 5.10, 7.2, 8.3 and remark 8.2. ◇

**Example 8.5.** Counting permutations by the number of fixed points. Note that one can construct $S_{n+1}$ from $S_n$ by extending each element $\sigma$ of $S_n$, placing $n+1$ either on an edge of the graph of $\sigma$ or treating it as a fixed point. Conversely, if $\sigma' \in S_{n+1}$, by removing $n+1$ from its cycle, we obtain a permutation $\sigma$ in $S_n$.

Let $x$, $y$, and $z$ be formal variables counting respectively the number of fixed points, the edges $(i, \sigma(i))$ such that $\sigma(i) \neq i$, and an additional edge $\emptyset$. If $n+1$ is placed on a non-empty edge, we obtain two edges of type $y$; and if $n+1$ is placed on the empty edge, we obtain a fixed point, that is, an edge of type $x$, along with a new empty edge of type $z$. Thus, we obtain the grammar : $G = \{x \to y^2, y \to y^2, z \to xz\}$, which is the grammar associated with the combinatorial differential system :

$$\{U' = V' = V^2, W' = UW, U(0) = X, V(0) = Y, W(0) = Z\}.$$

<figure><img src="image_16.png" /><figcaption>FIGURE 15 – Enumeration of permutations by fixed points</figcaption></figure>

Figure 15 is an example of W-structure on [9]. We have $G(z) = z \sum_{\sigma \in S_n} x^{f(\sigma)} y^{n-f(\sigma)}$, and, moreover, we obtain

$$W(x, y, z, t) = \mathrm{Gen}(z, t) = z \sum_{n \ge 0} \sum_{\sigma \in S_n} n x^{f(\sigma)} y^{n-f(\sigma)} \frac{t^n}{n!} = z \exp \frac{(x-y)t}{1-yt} \quad (27)$$

by solving the analytic system $\{u'(t) = v'(t) = v^2(t), w'(t) = u(t)w(t), u(0) = x, v(0) = y, w(0) = z\}$.

We also have the isomorphism $S_v \simeq zE_x \cdot D_y$, where $S_w$ is the species of permutation weighed by $v(\sigma) = zx^{f(\sigma)}y^{n-f(\sigma)}$, where $f(\sigma)$ is the number of fixed points of $\sigma$, $E_x$ denotes the species of sets whose elements are weighted by $x$, and $D_y$ denotes the species of derangement (fixed-point-free permutations), with each point is weighted by $y$.
## 9. APPLICATIONS TO HYPERELLIPTIC SPECIES

**9.1. Generalities.** Let $k \ge 2$ be a fixed integer. We consider the system of differential equations:

$$Y'_{i} = a_{i}Y_{1} \cdots Y_{i-1}Y_{i+1} \cdots Y_{k}, Y_{i}(0) = Z_{i}, 1 \le i \le k. \quad (28)$$

The canonical solution to system (28) is the family $(\mathcal{A}_{k,i})_{1 \le i \le k}$, where $\mathcal{A}_{k,i} = \mathcal{A}_{k,i}(T, \vec{Z})$ is the species of weighted increasing $k-1$-ary trees, where the root of weighted by $a_i$, is colored with color $i$, $\vec{Z}(Z_1, \cdots, Z_k)$. An example of an $\mathcal{A}_{4,1}$-structure is illustrated in Figure 16.

<figure><img src="image_17.png" /><figcaption>FIGURE 16 – An $\mathcal{A}_{4,1}$-structure on [10]</figcaption></figure>

**Definition 9.1.** Given an integer $k \ge 2$, the $k$-tuple $\mathcal{A}^{(k)} = (\mathcal{A}_{k,i})_{1 \le i \le k}$, solution of system (28), is called a family of hyperelliptic species of order $k$. Hyperelliptic species of order 2 are called trigonometric species, and hyperelliptic species of order 3 is Jacobi elliptic species. The integral

$$\mathrm{Am}_k(T, \vec{Z}) = \int_0^T \mathcal{A}_{k,k}(X; \vec{Z}) dX$$

is called the amplitude of $\mathcal{A}^{(k)}$ or the amplitude of order $k$.

**Proposition 9.2.** For $1 \le i \le k$, we have

$$\mathcal{A}_{k,i}(T, Z) = \mathcal{A}_{k-1,i}(\mathrm{Am}_k(T, Z), Z) \quad (29)$$

*Proof:* Figure 17(A) illustrates, for $k=4$ and $i=1$, how an $\mathcal{A}_{k,i}$-tree can be decomposed, and Figure ?? (B) shows the associated $\mathcal{A}_{k-1,i}$-structure. ◇

<figure><img src="image_18.png" /><figcaption>FIGURE 17 – Decomposition of an $\mathcal{A}_{4,1}$-structure in $\mathcal{A}_{3,1}(\mathrm{Am}_k(T, Z), Z)$-structure and the associated $\mathcal{A}_{3,1}$-structure</figcaption></figure>

**Proposition 9.3.** Given $k$, then for $i \ne j$,

$$a_i Z_j^2 + a_j \mathcal{A}_{k,i}^2(T, Z) \approx a_j Z_i^2 + a_i \mathcal{A}_{k,i}^2(T, Z).$$
<figure><img src="image_19.png" /><figcaption>FIGURE 18 – The isomorphism $a_1Z_2^2 + a_2A_{4,1}^2(T, Z) \simeq a_2Z_1^2 + a_1A_{4,2}^2(T, Z)$.</figcaption></figure>

*Proof:* Figure 18, where $k = 4$, $i = 1$, and $j = 2$, describes the isomorphism. ◇

**Corollary 9.4.** For $k \ge 2$, the amplitude $\text{Am}_k$ of order $k$ is a solution of the differential equation;

$$a_k Z_1^2 + a_1 \left( \frac{\partial}{\partial T} Y \right)^2 = a_1 Z_k^2 + a_k A_{k-1,1}^2(Y), Y(0, Z) = 0.$$

*Proof:* We have the isomorphism $a_k Z_1^2 + a_1 A_{k,i}^2(T, Z) \simeq a_1 Z_k^2 + a_k A_{1,i}^2(T, Z)$. On the other hand, Proposition 9.2 show that $A_{k,1}(T, Z) = A_{k-1,1}(\text{Am}_k(T, Z), Z)$. Moreover, setting $Y = \text{Am}_k(T, Z)Y$, we have by definition $\frac{\partial Y}{\partial T}Y = A_{k,k}(T, Z)$. Hence we obtain the result. ◇

**Remark 9.5.** Let $\mathcal{G}$ be the analytic differential operator associated with the grammar $G = \{z_i \to a_i z_1 \cdots z_{i-1} z_{i+1} \cdots z_k, 1 \le i \le k\}$, and $X_n^{(k,i)}(z_1, z_2, \cdots, z_k) = \mathcal{G}^n(z_i)$. The family of polynomials $(X_n^{(k,i)}(z_1, z_2, \cdots, z_k))_{n \in \mathbb{N}}$ is called the hyperelliptic polynomials of order $k$, initialized at $z_i$. We have $X_n^{(k,i)}(z_1, z_2, \cdots, z_k) = n![t^n]A_{k,i}(t, \bar{z})$.

Note that the polynomials $X_n^{(3,i)}$ is the Schett's polynomials, studied by Alois Schett in [24], [25] and by Dumont in [9]. Note also that the case $k = 4$ was studied by Dumont in [8]. Moreover, by setting $a_1 = 1$, $a_2 = -1$, $a_3 = -k^2$, $Z_1 = 0$ and $Z_2 = Z_3 = 1$ in (28), we obtain the Jacobi trees, which provide a combinatorial interpretation of the Jacobi elliptic functions (see Viennot [27]).

Setting

$$X_n^{(k,i)}(z_1, z_2, \cdots, z_k) = \sum_{\mu_1, \dots, \mu_k, \alpha_1, \dots, \alpha_k} b_{\mu_1, \dots, \mu_k, \alpha_1, \dots, \alpha_k} a_1^{\mu_1} \cdots a_k^{\mu_k} z_1^{\alpha_1} \cdots z_k^{\alpha_k},$$

we have:

i) $\sum_{s=1}^{k} \mu_s = n,$

ii) $\sum_{s=1}^{k} \alpha_s = n(k-2) + 1,$

iii) $\mu_i + \alpha_i = 1 + \sum_{s=1, s \neq i} \mu_s,$

iv) si $j \neq i$, $\mu_s + \alpha_s = \sum_{s=1, s \neq j} \mu_s$

**9.2. Notes on addition formulas.** An open problem concerning hyperelliptic species is the existence of an addition formula. As an example, we demonstrate an addition formula for trigonometric species (Prop. 9.6). In the case of elliptic species, addition formulas for Jacobi elliptic functions (with $z_1 = 0$) are known, but no combinatorial proof is available. In [8], Dumont shows that if we set $X = A_{2,1}$, $Y = A_{2,2}$, $Z = A_{2,3}$, $a = a_1$, $b = a_2$, and $c = a_3$, then this addition formula takes the form: $a_2 z_2^2 z_3^2 X(u + v) = a^2 z_2 z_3 X(u)Y(v)Z(v) + a_2 z_2 z_3 X(v)Y(u)Z(u) + bcX^2(u)X^2(v)X(u + v)$, or equivalently:

$$X(u + v) = \frac{\frac{1}{z_2 z_3} (X(u)Y(v)Z(v) + X(v)Y(u)Z(u))}{1 - \frac{bc}{a^2 z_2^2 z_3^2} X^2(u)X^2(v)}.$$
However, in [19], Longtin provides a combinatorial proof of the addition formulas for the tangent and secant functions, which are specializations of the Jacobi's elliptic functions sn and cn.

Let us consider the trigonometric species, defined as the solution of the system :

$$ \left\{ \begin{array}{l} Y_1' = a_1 Y_2, \quad Y_1(0) = Z_1, \\ Y_2' = a_2 Y_1, \quad Y_2(0) = Z_2 \end{array} \right. \qquad (30) $$

Figure 19 shows two examples of trigonometric structures of type $Y_1$. A structure of type $Y_2$

<figure><img src="image_20.png" /><figcaption>FIGURE 19 – A trigonometric species $Y_1$</figcaption></figure>

is obtained in the same way, by swapping the roles of $a_1$ and $a_2$, and of $z_1$ and $z_2$.

**Proposition 9.6.** (Addition formula for trigonometric species). We have the following isomorphism :

$$ a_2 z_1^2 Y_1(T_1 + T_2) + a_1 z_2 [Y_1(T_1) Y_2(T_2) + Y_2(T_1) Y_1(T_2)] \approx a_1 z_2^2 Y_1(T_1 + T_2) + z_1 [a_2 Y_1(T_1) Y_1(T_2) + a_1 Y_2(T_1) Y_2(T_2)]; \quad (31) $$

$$ a_2 z_1^2 Y_2(T_1 + T_2) + z_2 [a_2 Y_1(T_1) Y_1(T_2) + a_1 Y_2(T_1) Y_2(T_2)] \approx a_1 z_2^2 Y_2(T_1 + T_2) + a_2 z_1 [Y_1(T_1) Y_2(T_2) + Y_2(T_1) Y_1(T_2)]; \quad (32) $$

*Proof:* Figure 20 illustrates the first isomorphism. ♦

<figure><img src="image_21.png" /><figcaption>FIGURE 20 – Isomorphism between $a_2 z_1^2 Y_1(T_1 + T_2) + a_1 z_2 [Y_1(T_1) Y_2(T_2) + Y_2(T_1) Y_1(T_2)]$ (1+2+3) and $a_1 z_2^2 Y_1(T_1 + T_2) + z_1 [a_2 Y_1(T_1) Y_1(T_2) + a_1 Y_2(T_1) Y_2(T_2)]$ (4+5+6)</figcaption></figure>

**Remark 9.7.** Setting $\omega = a_2 z_1^2 - a_1 z_2^2$, equations (31) and (32) show that :

$\omega Y_2(t_1 + t_2) = a_2 z_1 [Y_1(t_1) Y_2(t_2) + Y_2(t_1) Y_1(t_2)] - z_2 [a_2 Y_1(t_1) Y_1(t_2) + a_1 Y_2(t_1) Y_2(t_2)]$, and $\omega Y_1(t_1 + t_2) = z_1 [a_2 Y_1(t_1) Y_2(t_2) + a_1 Y_2(t_1) Y_2(t_2)] - a_1 z_2 [Y_1(t_1) Y_2(t_2) + Y_2(t_1) Y_1(t_2)]$. Therefore :

$$ \begin{aligned} z_2 Y_1(t_1 + t_2) + z_1 Y_2(t_1 + t_2) &= Y_1(t_1) Y_2(t_2) + Y_2(t_1) Y_1(t_2), \\ a_2 z_1 Y_1(t_1 + t_2) + a_1 z_2 Y_2(t_1 + t_2) &= z_2 Y_1(t_1) Y_1(t_2) + z_1 Y_2(t_1) Y_2(t_2). \end{aligned} \qquad (33) $$
Setting $z_1 = 0, z_2 = 1, a_1 = a_2 = 1$, we obtain the addition formulas for $Y_1(t) = \sin t$ and $Y_2(t) = \cos t$. Setting $z_1 = 0, z_2 = 1, a_1 = -1$ and $a_2 = 1$, we obtain the addition formulas for $Y_1(t) = \sinh t$ and $Y_2(t) = \cosh t$.

**Acknowledgements.** My sincere thanks go to my two advisors, Gilbert Labelle and Pierre Leroux of the Université du Québec à Montréal, for the countless suggestions they provided during the preparation of this work, in particular, the proof of theorem 4.7.

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LABORATOIRE DE MATHÉMATIQUES ET APPLICATIONS DE L'UNIVERSITÉ DE FIANARANTSOA (LAMAF),
UNIVERSITY OF FIANARANTSOA, MADAGASCAR