On Heisenberg--Weyl algebra and subsets of Riordan subgroups

(The present paper is an extended and corrected second version of [25].) In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with dif-ferential operators and the associated one-parameter group. Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e. substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the"striped"Ri-ordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.


Introduction
Quantum physics has revealed many interesting formal properties associated to an associative and unitary algebra of operators a and a † meeting the partial commutation relation aa † −a † a = 1. The approach of considering an algebraic normal-ordering problem and have it transformed into a combinatorial (enumeration) problem is the source of a number of recent research works on the borders of quantum physics and algebraic or analytic combinatorics.
The present work is mainly motivated by the algebraic introductory survey of Duchamp, Penson and Tollu [18], by the article of Blaziak and Flajolet [7] and Blasiak, Dattoli, Duchamp, Penson and Solomon [6,8,9,17,19] as well as by the whole bunch of recent papers on Riordan arrays and the Riordan group [4,16,29,41], which extensively investigate the topic. Blaziak and Flajolet's paper provides a notably comprehensive and insighful synthetic presentation, especially with respect to the correspondence with combinatorial objects and structures, such as models that involve special numbers such as generalized Stirling numbers, set partitions, permutations, increasing trees, as well as weighted Dyck and Motzkin lattice paths, rook placement, extensions to q-analogues, multivariate frameworks, urn models, etc. In the first four sections, the paper is concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg-Weyl, the Bargmann-Fock representation of operators of creation-annihilation and the transformations arising from the one-parameter group as Riordan matrices which are substitutions with prefunctions. Upon this basis, the goal of the second part of the paper is to adapt the former parts to the theory of Riordan arrays. The next four sections are thus devoted to the associated groups of Riordan matrices and, thereby, collect the various relations and properties arising in Riordan arrays and in the Riordan group. More specifically, "striped" Riordan subgroups, a quasigroup and a semigroup for two appropriate operations are respectively defined and studied in Section 8. A few applications to combinatorial structures are also briefly mentioned in Appendix D.

The algebra of Heisenberg-Weyl
As customary in quantum physics, the Lie bracket in the associative and unitary Heisenberg-Weyl algebra of operators is the relation [a, a + ] := aa + − a + a = 1, (2.1) where a + stands for the usual a † , 1 is the identity operator of the algebra and aa + − a + a is the commutator. This partial commutation relation is referred to as the bosonic commutation rule or the creation-annihilation condition. As a matter of fact, it is satisfied in quantum physics by the creation and annihilation operators a and a + , which are adjoint to each other and serve to decrease or increase the number or the energy level of bosons by 1. where C A, B is the free associative algebra over C of the free monoid {A, B} * with identity 1.

Namely, C A, B stands for the algebra of polynomials in noncommuting indeterminates A, B
and I HW is the two-sided ideal generated by the polynomial AB − BA − 1.
Def. 2.1 is given by the mapping s : C A, B → HW C defined as a = s(A) and a + = s(B) (see [6,9,18]).
Since HW C is generated by a + and a, any element ω ∈ HW C writes as a linear combination of finite products of such generators in the form ω = r,s α r,s a + r1 a s1 · · · a + rj a sj , (2.3) where r = (r 1 , r 2 , . . . , r j ) and s = (s 1 , s 2 , . . . , s j ) are multi-indices of non-negative integers (denoted by Z ≥0 or N) with the convention a 0 = a + 0 = 1.
Observe that the representation given by formula (2.3) is ambiguous in so far as the rewriting rule of the commutation relation in eq. (2.1) allows different representations of a same element, e.g., aa + or equally a + a + 1. To remedy this situation, a preferred order of the generators is fixed by conventionally choosing the normally ordered form in which all annihilators stand to the right of creators (see Wick's Theorem, e.g. in [7]).

Definition 2.2. (Normal form, normal order.)
The commutation relation (2.1) may be regarded as a directed rewriting rule aa + −→ a + a + 1 (the normalization), which makes a systematic use of the reduction of aa + − a + a − 1 to 0. Any general expression F(a + , a) in C a + , a is thus completely reduced to a unique and equivalent normal form N F(a + , a) ≡ F(a + , a), such that, in each monomial, all the occurrences of a + precede all the occurrences of a.
Every element of HW C is written i,j β i,j b i,j with b i,j = (a + ) i a j in its normal form. By using [11,Chap. 2], one can show that (b i,j ) i,j∈N is a natural linear basis of HW C (the basis of normal forms). Remark 2. 1. Formula (2.4) can be obtained either from an algebraic approach (similar to Wick's Theorem or Rook numbers) [18], or (without loss of generality) by simply setting k = s = 0 in (2.4) and by using Leibniz's rule on the appropriate operators introduced further in §3. 1. (The beginning of this section consists of fundamental notions available in several articles [6,20,7,8,18], books [27,Chap. 2], [47,Chap. 6], among (many) others; see also §3.3 and Appendix A.)

The Lie bracket in HW ≤1 \HW 0
HW ≤1 denotes the subalgebra of elements in HW C for the usual product, which consist in polynomial operators of degree at most one in the variable a, i.e. with only one or zero annihilation. By Def. 2.1, such polynomials can be regarded as words in general form ω = (a + ) k a δ (a + ) , where k, are non-negative integers and the degree δ of operator a equals 0 or 1. By the commutation relation (2.1), the Lie bracket is a binary operation for HW C induced for the subalgebra HW ≤1 .
After simplification and normalization of all the above products (such as a + k+1 a a + +1 a , a + k+1 a a + , etc.), the final expression of the Lie bracket makes all terms of degree two in the variable a disappear, and we obtain

Remark 2.2.
When |ω 1 | = |ω 2 | (k = ) in HW ≤1 \HW 0 , the Lie bracket makes the vector part disappear and there remains the scalar part only in eq. (2.6). As regards the algebra HW C , it is unitary and associative and hence the Lie bracket in HW C fulfils the Jacobi identity. As the Lie bracket also satisfies the axioms of bilinearity and alternating on HW C , HW C is a (graded) Lie algebra (see §A in Appendix A).

The Lie algebra HW ≤1 \HW 0
Consider an equivalence relation in HW ≤1 between words of equal lengths for which the annihilation a lays at different places. In that case, for any n, the words a + k1 aa + k2 and a + 1 aa + 2 with k 1 + k 2 = 1 + 2 = n belong to the same equivalence class in HW ≤1 \HW 0 . Let a + n a be the representative of each class in that set and let N (HW ≤1 \HW 0 ) denote the set of the normalized elements of the subalgebra HW ≤1 \HW 0 . Then, N (HW ≤1 \HW 0 ) ∼ = HW ≤1 \HW 0 , since there exists a one-to-one correspondence between the sets of class representatives a + n a and a + n a . More precisely, given any operation for a + n a , there exists also a unique operation for a + n a . Thus, these two structures are isomorphic.
Theorem 2. 1. The Lie bracket on the set HW ≤1 \HW 0 defines a Lie algebra which is isomorphic to the set of vector fields on the real line. Proof. The bases of HW ≤1 \ HW 0 (as a vector space) are a + n a and a + n a up to an equivalence, for all n ∈ N. From eq. (2.6), the properties of the Lie bracket make HW ≤1 \HW 0 into a Lie algebra (i.e., bilinearity, alternating on HW ≤1 \HW 0 and the Jacobi identity).

Operators in the creation-annihilation model
The Heisenberg-Weyl algebra comes with the natural representation HW C → End K (Φ) (K = R or C), where Φ is some general class of smooth functions. Thus, Φ may be the class C ∞ of infinitely differentiable functions over some suitable domain, the space of analytic functions, or the vector space C[x] of polynomials in indeterminate x, etc. For example, End C[x] is the algebra of endomorphisms of C[x], which explains why it appears so often in many branches of mathematics and physics. The map HW C → End C (Φ) is given on the generators of HW C through the Bargmann-Fock representation of HW C .

The Bargmann-Fock representation
A particular realization of the commutation relation is obtained by choosing some sufficiently general space {f (x)} of smooth functions (typically C ∞ (0, 1) or C[x], see Comment 1), on which two operators X and D are defined as and Then, the creation-annihilation principle is obviously satisfied by a = D and a + = X. One recovers the Weyl relations [D, X] = 1 of abstract differential algebra [36,Ch. 1]. The interest of such a differential model of creation-annihilation is that it is faithful, meaning that any identity (without any additional assumption regarding the space of functions) is true in all generality under the commutation relation. This differential view will prove central to the following developments.
Definition 3. 1. The Bargmann-Fock representation of HW C is given by the application φ : where Ω(C ∞ ) = End C ∞ (0, 1), R is the set of such differential operators on a class of C ∞ functions over an appropriate domain.
The transformation of a term in HW C is given by a word whose letters are operators a + and a, where the multiplication is represented by x and the derivation by d dx , respectively. In this For example, any normalized operator x k d dx in Ω(C ∞ ) on a smooth real function x f (x) defined on the open set (0, 1) with k, ∈ N, is written where x (k+ ) d dx is the vector field part of the operator and v(x) := x (k+ −1) is its scalar field part on the real line. For = 0, v ≡ 0, the operator reduces to a pure tangent vector field.
From now on, we will denote ∂ x := d dx and ϑ x := x∂ x or also ∂ x := D and ϑ x := XD, as the case may be.
There follows the computation of the Lie bracket of two differential operators in HW C .

The Lie Bracket of two differential operators
Let q 1 (x)∂ x + v 1 (x) and q 2 (x)∂ x + v 2 (x) be two differential operators (the sum of one vector field and one scalar field on the line). By making use of Poisson's braces notation q 1 (x), q 2 (x) = q 1 (x)q 2 (x) − q 2 (x)q 1 (x), the Lie bracket expresses as Taking When there exists a scalar part (v(x) ≡ 0) and under the assumption that, either vector parts meet the condition v(x) = q(x)/x (x = 0), or all operators have the form q(x)ϑ x + tq(x) (t ∈ Q, t > 1). Then, by eq. (3.1) (Poisson's notation), the Lie bracket is written as Specifically, when q 1 (x) = x k and q 2 (x) = x , where k, , r, s ∈ Z >0 , one gets Eq. (2.6) is recovered in the first equality. Note also that, in either case, if k = , the Lie bracket applies to two operators of equal degrees and therefore reduces to its scalar parts only.

Normally ordered powers of strings in HW C
As noticed in Section 2, the normally ordered form of a general expression F(X, D) of C X, D is obtained by making use of the commutation relation (2.1) by moving all the annihilation operators a to the right. Whence a variety of properties provided by the normal ordering. All coefficients of the normal ordering of a boson string (or word) ω ∈ {X, D} * (more precisely the coefficients of the decomposition of ω on the basis (X i D j ) i,j∈N ) are positive integers, which suggests that such integers count combinatorial objects. For example, the Bell and Stirling numbers have a combinatorial origin, nevertheless one may consider them (and their generalizations) also as coefficients of the normal ordering [7, §4.2-4.3] and [6,9,18]. (Generalized ω-Stirling numbers are (re)introduced in this way in eq. (3.2) below.) Before the representations (or realizations) of the one-parameter group e λω λ∈R are defined formally, one must undertake the problem of ordering the powers of ω ∈ HW C in normal form N (ω n ) = n,i,j α(n, i, j)X i D j . Though, in general, such is a three parameters problem, it can be reduced to two parameters for homogeneous operators by help of the gradation property introduced in Appendix A. In this respect, any normalized polynomial operator ω ∈ HW C can be expressed as This is the definition of the generalized ω-Stirling numbers of the second kind, as recently introduced and used for polynomials and generalized to homogeneous operators, e.g. in [6,7,9].
(ii) The algebraic reduction of operators (X r D s ) n with r, s, n ∈ N has the form The above formula obviously holds in the case when e ≥ 0. The case e < 0 also gives rise to similar coefficients (see Lemma 2.1), as results from the "duality argument" shown in [7, Note 2, p. 11]. n k r,s denotes an operator formulation of generalized Stirling numbers of the second kind, which appears, for example, in Comtet's book [15,Ex. 2,p. 220], in Katriel [32] and, recently, in [7,18,22]). When r = s = 1, n k 1,1 := n k denotes the usual Stirling numbers of the second kind as defined after Stirling in 1730 and (re)considered later from a combinatorial viewpoint, e.g. by Carlitz [13], Comtet [15], [26, §6.1], Riordan [35, §6.6], etc.
Remark 3. 1. The words ω ∈ HW ≤1 \ HW 0 (with one annihilation only) have the general form w = (a + ) n−p a(a + ) p . If p = 0, n k ω is the matrix of a unipotent substitution, while if p > 0, n k ω is the matrix of a unipotent substitutions with prefunctions (see Lemma 4.1). This will prove of prime importance in the integration of the one-parameter group in next Section 4.

One-parameter groups
In general, a lot of meaningful operators can be associated to elements in HW C . In particular, for any given polynomial ω ∈ C X, D , the one-parameter group e λω λ∈R with sufficienly small parameter λ is of particular interest in quantum physics and for related identities on combinatorial numbers. (Full details on one-parameter groups may be found e.g. in [18,19,24].) Now, introduce the exponential as an operator on C X, D is a power series in z whose coefficients are in the polynomial ring C a, a + with the normal form The sum on the right-hand side is no other than the EGF of the total number of specific combinatorial objects, for example the total weight of the corresponding diagrams in the combinatorial context of [7, §2.1] and [22, p. 531].

Comment 1.
It is a well-known property that the exponential of a derivative plays the role of a shift (or translation) operator. As an example, the operator e λD (λ ∈ R, D := ∂) may be interpreted symbolically through its Taylor's expansion in λ.
The action of the shift operator on the monomial x n is evident by the binomial theorem, and thus on all polynomials f ∈ C[x]. This is also the case on all (formally convergent) series f ∈ C[[x]], due to the purely symbolic nature of the calculation (see Appendix C). Taylor formula also applies to any complex (or analytic) polynomial, thus preserving the shift operator property.
Recall that a function f is said to be of class C ∞ , or smooth, if it has derivatives of all orders. Hence, in the context of Section 4, we must first consider operators of the form e λD (for a sufficiently small parameter λ ∈ R) on a typically general space of holomorphic functions on a suitable domain, such as (0, 1). Besides, one has also to assume that |λ| < R, where R is the radius of convergence of f in order to ensure at least the necessary conditions of formal displacements for series in zC [[z]] [12, Ch. IV. 4.3.]. Fortunately, this is verified in the situation of such one-parameter groups.
Thereby, holomorphic functions on (0, 1) appear a suitable candidate in being the actual maximal class of functions satisfying the shift operator requirements.

Integration of the one-parameter group
The normal ordering of elements in HW ≤1 \ HW 0 (with one annihilation only), i.e. differential operators of the first order exactly, is shown (e.g. in [10,19]) to be of a special type. More precisely, the integration of the one-parameter groups generated by such operators involves transformations going on the name of substitutions with prefunctions in combinatorial physics, that is actually Riordan arrays. In the following, we are concerned with monomials in the general form the sum of one vector field and one scalar field on the line. The object of this subsection is the integration of the one-parameter groups associated to (4.3). Taking a geometric viewpoint, one uses the fact that any such one-parameter group is conjugate to the pure vector field q(x)∂ x on the line (tangent vector field paradigm). This will result further in the one-parameter group where g λ and s λ are both analytic in a neighbourhood of the origin; and under the assumptions that q and v are at least continuous, λ ∈ R is sufficiently small and f is a function in an appropriate space (see Comment 1). The calculations of s λ and g λ are deferred until next §4. 2.1 in Appendix B and what follows.

Substitutions with prefunctions in HW
Following [19,18], the integration of the one-parameter group e λq(x)∂x+v(x) [f (x)] (x) can be considered in the first place when the problem involves a pure vector field only (i.e. v ≡ 0).

Evaluation of the substitution in a pure vector field
In this case, the transformation of f (x) = 0 is given by a substitution factor s λ only, which means that eq. (4.4) admits the form The computation of that substitution function is given in Appendix B. Now, whenever an expression takes the (general) form a + n a with integer n ≥ 2, the operator is e (λx n ∂x) , which gives rise to the substitution function absolutely convergent for |x| < 1 n−1 (n − 1)λ. Formulas (4.5)-(4.6) translate into For any n ∈ Z >0 , the related geometric transformation turns out to be the conjugate of a homography and a transformation π n : x → x n , whose reversal is π n : x → x 1/n , where π n and π n are both C ∞ in an appropriate space depending on f . Let h n denote the homography relative to the substitution function s λ (x) = x 1−λnx . Since h n and π n are conjugates with respect to the composition, s λ writes as Actually, for any n > 0 the substitution simplifies to s λ (x) = which is the homography h n . Finally, for any n > 0, y 1−nλy = y 1−λy n and h n = h n 1 . Hence, by plugging h n into eq. (4.8), one can rewrite s λ for any integer n ≥ 1 in the simpler form The cases of n = 0, 1 and 2 are treated in the example of Appendix B, where the corresponding substitution functions s λ evaluate respectively to a translation (n = 0), to an homothety (n = 1) and to a homography (n = 2).

Examples 4.1.
(The Lie bracket and the substitution factor.) Given two positive integers k and , take q 1 (x) = x k+1 and q 2 (x) = x +1 . From the previous §4.2.1, the substitution functions associated to q 1 (x) and q 2 (x) are, respectively, Hence, the Lie bracket of the two operators is , standing for the substitution function, admits the general form (according to the sign of − k) This formula is at the basis of the three cases discussed in §4.2.3.

Substitutions with prefunction: the general case of a vector field
In the general case of a vector field with scalar part v ≡ 0, the integration of the associated one-parameter group e λ(q(x)∂x+v(x)) results in the basic form (4.4), under the assumptions on q(x), v(x), parameter λ and f (x) already drawn in §4.1 (from Comment 1 and Appendix B). A few transformations in eq. (4.5) allow to integrate the one-parameter group e λ(q(x)∂x+v(x)) for a general scalar field v(x).
Taking again a geometric viewpoint, we make use here of the fact that a general field of type q(x)∂ x + v(x) is conjugate to the tangent vector field q(x)∂ x on the line (with respect to composition). So, on the same assumptions as in §4. 2 , which gives the conjugate to the tangent vector field q(x)∂ x in the neighbourhood of the origin λ = 0 (see Comment 2). Due to the fact that the exponential commutes with the conjugacy, we thus have Then, by the calculations performed in §4. 2.1 and in Appendix B, one obtains the integration of the general one-parameter group (at least locally) under the transformation f → g λ (f • s λ ): where s λ is the substitution factor with prefunction g λ = (u • s λ )/u.

Comment 2.
The "conjugacy trick" and the "tangent paradigm" (so called in [19]) which lie behind the result in 4.10 may explain themselves as follows.
Regarding vector fields as infinitesimal generators of one-parameter groups leads to conjugacy since, if U λ is a one-parameter group of transformation, so too is V U λ V −1 (V being a continuous invertible operator). In the context, we can formally consider (a + ) n−p a(a + ) p with p > 0 as conjugate to (a+) n a.

Examples 4.2. (Conjugacy trick)
More generally, supposing all the terms well-defined let Eq. (4.12) is conjugate to a vector field and integrates as a substitution with prefunction factor. Finally, by straightening the vector field on the line by the technique described in Remark 1, Appendix B, the required formula in (4.10) holds. (This method also amounts to use the "ad" operator (conjugacy) of derivation in the Lie algebra.) Now, the "tangent paradigm" works in the following manner: if the tangent vector is adjusted so as to coincide with x n−p ∂ x x p , then we get the right one-parameter group. By virtue of the "conjugacy trick" and the "tangent paradigm", the integration of the one-parameter group yields It can be checked that, if s λ (x) is a substitution factor, in other words (at least locally) are conducted near x = 0 and the result must also stay in that neighbourhood. Thus, whenever the composition of two such transformations, say U λ1 and U λ2 , is realized, the values λ 1 and λ 2 of parameter λ have to be chosen small enough to keep (U λ1 • U λ2 ) [f ](x) stay also close to zero. Then, the correctness of the computational procedure can also be stated a posteriori by making use of a technique of tangent vector as follows (see [18]). Check that, for small values λ 1 and λ 2 of the parameter λ, U λ1 • U λ2 = U λ1+λ2 (local one-parameter group) and check that

The Lie bracket and the prefunction
As examined in Ex. 3.1 in §3.2, given k, and r, s in Z >0 , consider the words ω 1 = a + (k+1) aa + r and ω 2 = a + ( +1) aa + s in HW ≤1 \ HW 0 , whose normal forms are N (ω 1 ) = a + (k+1) + ra + k and N (ω 2 ) = a + ( +1) + sa + . They are represented here as x k ϑ x + rx k and x ϑ x + sx , respectively (the case of x k ϑ x − rx k and x ϑ x − sx , as given in Ex. 3.1, is discussed in Remark 4.2). The Lie Bracket and the prefunction follow: . (4.14) To draw conclusions as to the characteristics of the prefunction g λ (x) in eq. (4.14), the problem at stake is to determine the sign of s −rk ( −k) , according to the respective values of integers k, , r, s ≥ 1. First, without loss of generality, we can assume k = , otherwise g λ (x) ≡ 1 (see §4.1). Next, the case of > k gives rise to three subcases which make the prefunctions g λ , up to the sign of the numerator, denoted by θ := s − rk.
Finally, in the symmetric case of < k, the behaviour of θ = s − rk is similar to the one examined above, except (up to a sign) for the expressions of g λ (x) within the previous last two subcases. More precisely, when < k and according to the sign of θ.
This completes the discussion regarding the values of the prefunction factor.

Remark 4.2.
Turning now to the features of the Lie bracket of the two differential operators x k ϑ x − rx k and x ϑ x − sx as summarized in Ex. 3.1, the discussion runs along the same lines as in the above one for eq. (4.14). Still assuming k = and θ := s − rk, we have (i) In the case when the scalar part is θ x k+ , the expressions of the prefunction g λ are similar to the ones in the above three cases up to the sign of θ.
(ii) In the case when the scalar part is −(rk + s ) < 0, the sign of rk+s k− depends only on whether k > or not. Then, the prefunction turns out to take the general forms The goal of next §4.3 and following Sections is to adapt the aforegoing parts to the theory of Riordan arrays by regarding the framework of the transformations involved in the integration of the one-parameter group in HW ≤1 \HW 0 (i.e. substitutions with prefunctions) as transformation matrices.

Transformation of matrices and Riordan arrays
Due to the emphasis placed on generating functions in what follows, we first recall some basic notations (see also Section 5 and Appendix 5). Usual generating functions f (z) = n≥0 f n w n z n are specialized to w n = 1/c n , where (c n ) (n ∈ N) is a fixed sequence of non-zero constants with c 0 = 1, given once and for all (see Appendix C). In particular, if c n = 1, f (z) = n f n z n /c n is an ordinary generating function (OGF) and, if c n = n!, f (z) is an exponential generating function (EGF). The notation [z n ] (or z n cn , f (z) ) stands for the coefficient extractor operator.
denotes for the coefficient of z n in the OGF f (z) and similarly, the coefficient of z n /n! in the EGF f (z) is n![z n ]f (z) := f n ; that is the respective coefficients of z n and z n /n! in the expansion of f (z) into powers of z [22, p. 19] Consider, as examples, the upper-left corner of the (doubly infinite) matrices M ω , each representing a word ω ∈ HW C , such as the Pascal matrices or the Stirling matrix (as exemplified in Ex. 3.2, §3.3, and Ex. 5.1, §5.2). More precisely, for ω = a + a (ω → XD) one gets the array of the usual Stirling numbers of the second kind n k n,k∈N from their classical recurrence relation: once the first line is fixed, the Stirling array can be constructed iteratively, whose bivariate EGF is k≥0 [18,19]) For any homogeneous operator ω ∈ HW C , all lines M (n, k) k∈N of the Stirling type matrix are finitely supported. Such matrices are said row finite with set denoted as RFM N (C), and their composition makes RFM N (C) into an algebra. Proof. In each case, the matrix has the form of a staircase where each "step" depends on the number δ = |ω| a . One can prove precisely that each row ends with a "one" in the cell (n, nδ), and we number the entries from (0, 0). Thus, all matrices are row finite and unitriangular if, and only if, δ = 1 (the matrices of coefficients for words in HW ≤1 \HW 0 ). Moreover, the first column is (1, 0, . . . , 0, . . . , 0, . . .) if, and only if, ω ends with an "a" (this means that N(ω n ) has no constant term for all n > 0).
Note that such matrices belong to the group of unipotent matrices. As emphasized in §4.1 and in Remark 3.1 of §3.3, unipotent matrices are required for words in HW ≤1 \HW 0 , i.e. which have been proved of special type: the matrices of substitutions with prefunctions. Note also that the matrices in RFM N (C) are coding certain classes of operators, such as the continuous operators in the Fréchet space C[[z]] endowed with the (semi-norms) Treves topology.

The algebra L(C N ) of sequence transformations
Let C N be the vector space of all complex sequences, equipped with the Treve product topology. It is easy to check that the algebra L(C N ) of all continuous operators C N −→ C N is the space RFM N (C). For a sequence A = (a n ) n≥0 , the transformed sequence B = M A is given by B = (b n ) n≥0 with b n = k≥0 M (n, k)a k . We may associate a series with a given sequence (a n ) n∈N and a sequence of prescribed (non-zero) denominators (c n ) n∈N to a GF n≥0 a n z n /c n . Thus, once the c n 's have been chosen, to every (linear continuous) transformation of generating functions, one can associate a corresponding matrix.
The algebra L(C N ) possesses many interesting subalgebras and subgroups, such as the algebra of lower triangular transformations T N (C), the group T N (C) of invertible elements of the latter (which is the set of infinite lower triangular matrices with non-zero elements on the diagonal), the subgroup of unipotent transformations UT N (C) (i.e. the set of infinite lower triangular matrices with elements on the diagonal all equal to 1) and its Lie algebra N T N (C), the algebra of locally nilpotent transformations (with zeroes on the diagonal), etc. (see [19]).
To each matrix M (n, k) n,k∈N ∈ RFM N (C), one may associate an operator Note that if C[[x]] is endowed with the structure of Fréchet space of simple convergence of the coefficients (also called the "Treves topology") [18, p. 43], each Φ M is continuous. Then, the next proposition states that there exists no other case. The proposition applies immediately to the one-parameter groups e λω generated by homogeneous operators ω through the Bargmann-Fock representation (a → D and a + → X). Indeed, the matrix ϕ −1 (ω) is (e denotes the excess of ω) • Strictly lower triangular when e < 0, • Diagonal when e = 0, • Strictly upper triangular when e > 0.
(These matrices are different from the "generalized Stirling matrices" defined by eq.(3.2): their non-zero elements are supported by a line parallel to th diagonal.) Consequently, e λω always admits a representation as a group of operators in an appropriate space. For e < 0, one gets each of the polynomial in the spaces C ≤n [x] (i.e. the sets of polynomials whose degree is less than n). For e ≥ 0, one gets C[[x]] endowed with the Treves topology.

Substitutions with prefunctions
Let (c n ) n≥0 bet a fixed set of denominators. For a generating function f , we consider the The matrix of this transformation M g,φ is given by the transforms of the monomials with a , α m = 0 and then, by (4.15,4.16), Therefore, the equivalence M g,φ ∈ RFM N (C) ⇐⇒ φ has no constant term holds true (and, in this case, M g,φ is always lower triangular). The converse is true in the following sense. Let T ∈ L C N be a matrix with non-zero first two columns and suppose that the first index n such that T (n, k) = 0 is less for k = 0 than for k = 1 (which is the case, by (4.15), when T = M g,φ ). Set  . f analytic on (0, 1)).
Proof. From eq. (3.2) in §3. 3, one first has the following equality between continuous operators Assuming condition i), let us check ii) for f a monomial, namely choose the test functions f = x p , for p = 0, 1, . . .
Now, as the two members of ii) are continuous and linear in f and the set of monomials is total [11] in the space of formal power series endowed with the Treves topology (the usual ultrametric topology would not be enough for e = 0), condition ii) holds for any suitable function f . Conversely, assume condition ii), then U λ (e yx ) = g(λx e )e yx(1+φ(λx e )) and, from eq. (4.17), one gets n,k≥0 Finally, the change of variables (λx e → x and xy → y) yields condition i). This proves the required Prop. 4 [38,46,49]. From now on, we will suppose that φ has no constant term (α 0 = 0). Moreover, M g,φ ∈ T iff a 0 , α 1 = 0, which implies that (on the diagonal) M g,φ (n, n) = a 0 /c 0 (α 1 /c 1 ) n . Hence, which, for EGF and OGF, reduces to the equivalence M g,φ ∈ UT ⇐⇒ a 0 = α 1 = 1. In this setting, g and φ meet the conditions that g ∈ C[[x]] with g 0 = g(0) = 1 and φ ∈ xC [[x]]. In classical combinatorics (using OGF and EGF), the matrices M g,φ (n, k) are known as Riordan matrices (see Def. 5.1 and Thm. 5.2 in § §5.1-5.2 and, for example, [38,41]).

Riordan arrays and Riordan group
Since power series (OGF and EGF) play a prominent role in the present paper, some basic useful definitions are recalled below, in the beginning of § 4. 3

Riordan arrays
Riordan arrays, named after John Riordan, were introduced by Shapiro et al. in [41] and Roman [38] to generalize the properties of the Pascal Triangle. Along which, the group structure of the set of Riordan arrays was shown, and called the Riordan group. In [45], Sprugnoli made use of Riordan arrays for proving combinatorial identities, and then he showed how Riordan arrays can be used to perform combinatorial sum inversions. These seminal articles were followed by a number of papers, e.g. [4,28,29,33,48] exploring classical or generalized Riordan arrays and the theory of the Riordan group and subgroups.
are referred to as the column GFs (or the GF of the kth column) of the Riordan array. Furthermore, if f ∈ xS, i.e. f is such that f 1 = f (0) = 0 (or ord(f ) = 1), the Riordan array is said to be proper.
By definition, any proper Riordan array is invertible; that is, the infinite lower triangular array T = d n,k (0 ≤ k ≤ n) is such that d n,n = 0 for all n. Furthermore, for any proper Riordan array T = g(x), f (x) , its diagonal sums are just the row sums of the vertically stretched array g(x), xf (x) and hence have OGF g(x) 1−xf (x) . Ordinary Riordan arrays correspond to the case of c n = 1, while exponential Riordan arrays corresponds to the case of c n = n!.  2) or equivalently, the so-called "fundamental theorem of Riordan arrays", where * denotes the usual array product.
The fundamental theorem of Riordan Arrays is a somewhat practical statement of Thm. 5. 1. Given a Riordan array T = (g, f ) and an integer sequence (λ n ) with OGF Λ, the OGF of the sequence T λ n is g(x)Λ(x)f (x), where a sequence is regarded as an (infinite) column vector. The (infinite) array T can thus be considered to act on the ring of integer sequences Z N by multiplication. This action can be extended to the ring Z

Remark 5.1. A useful (combinatorial) alternative definition to the (algebraic) Def. 5.1 of proper
Riordan arrays is in terms of the so-called A-sequence and Z-sequence. It was found by Rogers in [37] and extensively developed after Shapiro et al. [41] by Sprugnoli, Merlini et al. [33,45], etc. Let A(x) in S (a 0 = 0) and Z(x) be the two GFs of A-sequences (a i ) and Z-sequences (z i ) (i ∈ N) that satisfy the relations f (x) = xA f (x) and g(x) =

The Riordan group
The Riordan group introduced by Shapiro et al. in [41] is algebraically defined by Bacher in [2] as the interpolation group, which has a faithful representation into the infinite lower triangular matrices and carries thus the natural structure of a Lie group. With Thm. 5.1, we can further compute the product (g, f ) * (h, ) of two Riordan arrays. In fact, the column GFs of h(x), (x) are h(x) (x) k /c k . Thus, by matrix multiplication, the kth column GF of the product f (x) k /c k , which means that the product is also a Riordan array, i.e., Proof. According to eq. (5.4), the Riordan group R is closed under the Riordan product and the product is associative. The matrix (1, x) is an element of R and, for each matrix g(x), f (x) ∈ R, there exists a matrix 1 Hence, the inverse of a matrix L = g(x), f (x) under the matrix product in R is which proves the assertion of the theorem.
The group R introduced in Thm. 5.2 is called the Riordan group with respect to (c n ). For any reference sequence (c n ) (with c 1 = 1 and c n = 0), the identity I = (1, x) in R is the usual infinite identity matrix and each matrix has an inverse in R, defined by eq. (5.5). Actually, by eq. (5.1), the coefficients d n,k of the matrix (1, x)

In addition, for any series f ∈ C[[x]
] such that f 1 = 1, 1, f = B f is also an iteration matrix. Hence, the set of iteration matrices with respect to (c n ) is a nonempty subset of the Riordan group R (with respect to (c n ); it is also closed under multiplication and inversion in R, and as such it is a subgroup of R).
In Shapiro et al. [41], the Riordan group is a unipotent group of infinite lower-triangular integer matrices that are defined by a pair of power series (g, f ) such that g ∈ 1 + xC [[x]] and f ∈ xS with f 1 = 1. Such is a stronger definition than usually required for the Riordan group. However, the paper makes only use of proper Riordan arrays whose diagonal elements equal 1. The series f = n≥1 f n x n /c n in xS will henceforth meet also the additional property that f 1 = 1, which naturally suppose infinite lower-triangular integer matrices with diagonals d n,n = 1 (i.e. the unipotent Riordan group).
Examples 5. 1. Let the matrix P of the Riordan group be defined by the pair and P is the well-known Pascal (or binomial) matrix. The inverse of P in R is P −1 = 1 1+x , x 1+x , with coefficients (−1) n−k n k . In the Riordan group, for m integer, P m = 1 1−mx , x 1−mx , the general term of which is m n−k n k , and the inverse P −m is given by The row OGFs of P are (x+1) n , and the row OGFs of P −1 are (x−1) n . In addition, P * (1, −x) is the diagonal matrix with alternating 1's and −1's on the diagonal; so, P * (1, −x) is said to have "order two" and P to have "pseudo-order two", which also means that P is a pseudo-involution. Similarly, let the Pascal matrixP in R defined by the exponential Riordan array (e x , x).
More generally,P m is the element (e mx , x) of R, and the inverseP −m ofP m is (e −mx , x). The row EGFs ofP are e x x n /n! and the row EGFs ofP −1 are (−1) n e −x x n /n!. In addition, since each Riordan array g(x), f (x) has the bivariate OGF g(x) 1−yf (x) , P has the bivariate OGF 1 1−x(1+y) andP has the bivariate EGF g(x)e yf (x) .
As another example, let us consider the exponential Riordan array 1, ln(1 + z) . The general term of the columns EGF is n! [z n ] ln(1 + z) k /k! = n k (or also s(n, k)): the usual Stirling number of the first kind. The row EGFs are n k=0 n m z k = z n = z(z − 1)(z − 2) · · · (z − n + 1) (the "falling factorial" polynomials) and form the sequence associated to e z − 1 [15, p. 206 & p. 212].
The inverse of the array 1, ln(1 + z) is (1, e z − 1), whose general term of the columns EGF is n! [z n ] (e z − 1) k /k! = n k (or also S(n, k)): the usual Stirling number of the second kind. The polynomials n (x) = n k=0 n m x k are referred to as the exponential polynomials. The sequence { n (x)} is associated to ln(1 + z).

Riordan subgroups
A number of basic subgroups of the Riordan group R can be found e.g. in [16]. Among others, the set of Riordan arrays (g(x), x) , where g ∈ S, is also a subgroup of R, * . It is called the Appell subgroup of R and denoted by A. Still assuming g ∈ S and f ∈ xS, the Bell subgroup of R is defined by B = (g, xg) and the Lagrange (or associated) subgroup of R is given by L = (1, f ) . Furthermore, for all (g, f ) ∈ R, Thm. 5

.2 implies
and Since A is normal, R A B and R = A L: the Riordan group is a semidirect product of the Appell normal subgroup and the Lagrange and Bell subgroups, respectively.
In what follows, these two classic subgroups of R, along with another subgroup which generalizes L and B, that is the set of proper Riordan arrays of the form g(x) ρ , xg(x) with ρ rational (and, by definition, g ∈ S) are under consideration. The latter group, is defined in next Def. 6.1 and plays a key role in Sections. 7 and 8. (See e.g., Def. 7.2 in §7. 1.) Definition 6.1. By Thm. 5.2, the set g ρ , xg with ρ ∈ Q of proper Riordan arrays is a subgroup of R with respect to the Riordan product; it is denoted by G(ρ), * .
More precisely, the identity in G(ρ) is I = (1, x). Now, let h(x) ∈ xS be the reverse series of xg(x), that is h(x) = x (g•h)(x) . Since g ∈ S, any matrix L = (g ρ , xg) has a unique inverse L −1 = 1 g ρ •h , h in G(ρ). When ρ is specialized to 0 and 1, respectively, the Lagrange and the Bell subgroups of R are obtained as subgroups of G(ρ), * .

Automorphic Riordan subgroups and Puiseux series
As defined in Appendix C, if K is a field then the field of Puiseux series with coefficients in K is defined informally as the field K{{x}} of formal Laurent series K((x)) of the form n≥k a n x n/N , where N ∈ Z >0 and k ∈ Z. In other words, K{{x}} with coefficients in K is where each element of the union is a field of formal Laurent series over x 1/N (considered as an indeterminate). In the following, we set K = C and C{{x}} denotes the field (and also the vector space over C) of Puiseux series with complex coefficients.

Definition 6.2.
Let M = {µ ρ , ρ ∈ Q} be the subgroup of the group GL C{{x}} of linear transformations of C{{x}} such that for any U ∈ C{{x}}, µ ρ U (x) = x ρ U (x) and, similarly, Further, M is the group of transformations of the Riordan group R, and so it is for the Riordan subgroups of R such as G(ρ).
The usual Riordan product * is thus extended from R×C [[x]] to the product · for R×C{{x}}.
More precisely, f (x), g(x) · U (x) = f (x)U g(x) for any matrix (f, g) in R and any power series U ∈ C{{x}}. Now, since U (x) = n≥k a n x n/N , we have (f, g) · U = f (x) n≥k a n g(x) n/N , where k ∈ Z, N ∈ Z >0 and a n ∈ C.
By combining the actions of groups in Def. 6.2, we can construct the diagram in Fig. 6. 1. In this diagram, for any L = g(x) ρ , xg(x) ∈ G(ρ) (ρ ∈ Q), ϕ and ψ stand respectively for where the product · is the extension of the product * for R × C{{x}}). From the two actions of the groups G(ρ) on C{{x}} for all ρ ∈ Q, the subgroups of G(ρ) are determined by the outer automorphisms Out Lemma 6.1. For any ρ 1 , ρ 2 ∈ Q, there exists a unique µ ρ1 such that the action on the set C{{x}} . This completes the proof.

Striped Riordan subgroups
Again at this point, we make use of the subgroup G(ρ) of R defined in Def. 6.1, Section 6. In the context of striped Riordan subgroups of G(ρ), we will extensively refer to power series (or GFs) restricted to the form g n (x) = 1 n √ 1−nλx n with integer n > 0. The families of function g n ∈ S and xg n (x) ∈ xS are actually replacing the usual prefunctions g λ with substitutions factor s λ arising in the context of the Bargmann-Fock representation in §3.1 (Section 3), and of one-parameter groups in Section 4.

Striped Riordan matrices and subgroups of R
The next two definitions are at the basis of the notions required in the present Section 8. Definition 7.1. Any Bell matrix g(x), xg(x) with g ∈ S and g 0 = 1, such that there exists an integer ν ≥ 1 with h(x) = n g n x νn = 1 + g 1 x ν + g 2 x 2ν + g 3 x 3ν + · · · will be referred to as a ν-striped Riordan matrix. The number ν represents the size of blocks made of ν − 1 consecutive zeros in each column of the matrix (whence the names "ν-striped" or "(ν − 1)-aerated"): in other words, the matrix coefficients meet the condition d n,k = 0 when n − k ≡ 0 (mod ν) for all k ≤ n.
The set of ν-striped Riordan arrays is similar to B, with stronger properties however: on each array diagonal, d n,n = 1 and stripes of ν − 1 consecutive zeros may appear along the columns. In addition, this set is a unipotent subgroup of B under the Riordan product, since g 0 = 1, (1, x) is the identity and (g, xg) −1 = 1 g•(xg) , xg , is the unique inverse of (g, xg).

Definition 7.2.
Any n-striped Riordan array g ρ n , xg n with ρ ∈ Q such that the EGF g n is of the form g n (x) = 1 n √ 1−nλx n for n positive integer will be referred to as a ρ-prefunction n-striped Riordan array.
We let G(n, ρ) stand for the set of the ρ-prefunction n-striped Riordan arrays. Further, according to the definition 6.1, G(n, ρ), * denotes the particular subgroup g ρ n (x), xg n (x) of G(ρ), * for any integer n > 0 and rational ρ. From Def. 6.1, the identity in G(n, ρ) is I = (1, x) and the inverse of any matrix g ρ n , G n is 1 g ρ n •Gn , G n , where G n ∈ xS is the reverse series of G n . Furthermore, the group (G(n, ρ), * ) (n ∈ Z >0 and ρ ∈ Q) is generated by the Riordan matrices Extending eq. (7.1) to L mn,ρ and L −1 mn,ρ , we can fix the values of n ∈ Z >0 and ρ ∈ Q; then for all m ∈ Z, the functions g ρ mn 's are the germs of the infinite group G(mn, ρ), each for one given m.
Remark 7. 1. The designation of G(n, 1) as the "prefunction n-striped Riordan subgroup" of B simply follows the definition of an n-striped Riordan matrix, restricted to the usual prefunction g n designed in Section 4. Indeed, let γ ∈ Q and p ∈ N, and consider for example the real linear combination of operators in HW ≤1 \HW 0 of the form Then pγ ∈ Q and plays the role of the rational ρ in the subgroup G(ρ) (see Def. 6.1) In the cases of ρ = 0 and ρ = 1, the subgroup G(n, ρ) of G(ρ) reduces respectively to the subgroup G(n, 0) of L and the subgroup G(n, 1) of B defined in Def. 7.1. Moreover, the group G(n, 1) generates the group which contains every G(n, m) with m ∈ Z as subgroup. Notice also that, in the formal topology defined in Appendix C, lim n→0+ g n (x) = e λ and lim n→0+ g n (x) = e −λ for all |x| < 1/(nλ) 1/n . Thus, both matrices e ±λ , e ±λ x = e ±λ I are distinct from I (unless λ = 0), but also belong to R. Although e ±λ I are not in G(n, ρ), they are the two generators of the subgroup e ±λ I of R for any real |λ| < 1.

The Lie bracket and the Riordan subgroups of G(n, ρ)
Under the same assumptions as in the above eq. (7.1), the following generalization of the statements in §4. 2.3 to the group G(n, ρ) is now under consideration. Let k, and r, s in Z >0 represent the weights of the polynomials of the two differential operators x k ϑ x + rx k and x ϑ x + sx . By Eq (4.14) in §4. 2.3, the Lie Bracket of the above operators is Note that the evaluation of the Lie bracket of the differential operators x k ϑ x ± rx k is discussed in where ρ = θ/m ∈ Q is expressed here in terms of the integer θ := s − rk.

Remark 7.2.
For all k, ∈ Z >0 and ρ ∈ Q, the subgroup G(mn, ρ) of R under the Riordan product * is generated by the two prefunction mn-striped Riordan matrices L mn,ρ = g ρ mn , s mn and L −1 mn,ρ = 1 g ρ mn •smn , s mn , where mn = ( − k)(k + ) ∈ Z and ρ = θ/m ∈ Q. The case when ρ = 1 leads to the three subcases investigated in §4. 1. Similarly, when > k and r = s, θ = rm and the subgroup G(mn, ρ) reduces to a particular case which is treated in next Ex. 7.1. Under the current assumptions on k, , r, s, let θ := s − rk ∈ Z and mn = ( − k)(k + ), the present discussion about the expression of g ρ mn follows the same lines as in §4. 2.3. More precisely, to draw conclusions as to g ρ mn in eq. (7.3), the problem at stake is to determine the sign of the rational ρ = θ/m according to the respective values of integers k, , r, s ≥ 1. The general form of each striped Riordan matrix in each case can then be deduced from each associated pair g ρ mn , s mn in eq. (7.3). First, without loss of generality, we can assume k = , otherwise (m = 0), g ρ mn (x) = 1 and s mn (x) = x (see eq. (4.9) in §4.1), with the identity (1, x) as associated striped Riordan matrix. Next, the case of > k (m > 0) gives rise to three subcases where, as the case may be, each one yields the expression of g ρ mn (x), where mn = ( − k)(k + ) > 0. So, according to the sign of ρ only, 1. If s/r = k/ , then ρ = θ = 0 and g ρ mn (x) = 1. As a consequence, The corresponding striped Riordan matrices take the general form L mn,0 = (1, s mn ). 2. If s/r > k/ , then ρ > 0 and the prefunction simplifies to g ρ mn (x) = 1 1−mnλx n ρ/n . As a consequence, The corresponding striped Riordan matrices take the general form L mn,ρ = g ρ mn , s mn . 3. If s/r < k/ , then ρ < 0 and the prefunction simplifies to g ρ mn (x) = (1 − mnλx n ) |ρ|/n . As a consequence, (1−mnλx n ) 1/n , which is an element of the form L mn,ρ = g r mn , G mn in G(mn, r) (m > 0). • If k > , then g ρ mn (x), s mn (x) = 1 (1+|m|nλx n ) r/n , x (1+|m|nλx n ) 1/n , which is an element of the form L −1 mn,r = g r −|m|n , G −|m|n in G(mn, r) (m < 0).
One of the simplest applications is when r = 1 and m = − k = 1, which correspond to the Pascal matrix P = 1 1−x , x 1−x and its inverse P −1 = 1 1+x , x 1+x (see Ex. 5.1). To summarize, whatever , k ∈ Z and ρ ∈ Q, the discussion leads to the two matrices L mn,ρ and L −1 mn,ρ (where m := −k, n := k + and ρ := θ/m), which are the two generators of G(mn, ρ). Remark 7. 3. Turning back to Remark 4.2 of §4.2.3 regarding differential operators of the form x k ϑ x − rx k or x ϑ x − sx , we obtain a general expression of the prefunction g ρ mn (x) as follows (still on the assumption that k = , otherwise m = 0 so g ρ mn (x) = 1 for any ρ ∈ Q). (i) In the case when the scalar part of the Lie bracket writes ±|θ|x n , where again θ := s − rk and ρ = θ/m, the expressions of the prefunction g ρ mn are similar to the above cases in eqs. (7.4),(7.5),(7.6),(7.7), according to the sign of ρm.
(ii) In the case when the scalar part of the Lie bracket is −(rk+s ) < 0, the sign ofρ := − rk+s m depends only on whether m := − k > 0 or not, which implies the expression of the prefunction,

Striped quasigroup and semigroup operations
The subgroup G(n, ρ) of G(ρ) is introduced in Def. 7.2 and eq. (7.1) in the latter Section 7. We consider now the set G(n, ρ; µ) with µ ∈ Q of all pairs L µn,ρ := L (µ) n,ρ = g ρ µn , xg µn , where g µn ∈ S is of the form 1−µnλx n −1/n . G(n, ρ; µ) is constructed from the group G(n, ρ), * and endowed with a new binary operation defined from the results obtained in §7.2.
Note that taking µ ∈ Z (denoted m in that case: see Notations 7.1.1), the substitution function G n (x) ∈ xS corresponding to ω verifies by definition, with G mn := G (m) n , That is, for all m ∈ Z, G ±mn (x) = x 1±mnλx n −1/n , as the case may be. Since G mn •G −mn = x, each G mn and G −mn is the reverse series of the other in xS. Moreover, this enables to interpolate between g mn ∈ S and G mn ∈ xS for any m ∈ Z. From the Lie group structure of the interpolation unipotent subgroup of G(n, ρ), • , m can be taken in Q, R or C (see Bacher in [2]). So, in order to avoid any confusion, the parameter µ ∈ Q will henceforth replace the parameter m previously used in Section 7.
The next Def. 8.2 of weak (or non strict) associativity appears primarily important for characterizing the structure of the set G(n, ρ; µ) and the operations on its elements in this section.

Definition 8.2.
The operation * will be said weakly associative iff it is not associative, unless it is closed under belonging to the conjugacy classes in G(n, ρ; µ).
In other words, although the operation * is never associative (in the strict sense), a weak form of associativity holds however in G(n, ρ; µ), according to the cases. On the one hand (in the general case), the product * makes the set of all the conjugacy classes in the Riordan subgroup G(n, ρ) closed under weak associativity. On the other hand, whenever associativity is proved valid within at least one conjugacy class in the group G(n, ρ) for every pair (n, ρ) ∈ Z >0 ×Q (e.g., for say ρ = ϕ r, ϕ(s, t) , one unique conjugacy class is closed under the product * ). Theorem 8. 1. With respect to the operation * , G(n, ρ; µ) has a quasigroup structure. Proof. The set G(n, ρ; µ) in Def. 8.1 is closed under the binary operation * . However, G(n, ρ; µ) has no identity element (except for ρ = µ = 0) and neither exists an inverse for every element L µn,ρ indexed by n ∈ Z >0 and ρ, µ ∈ Q. Therefore, by Def. 8.1 and Def. 8.2, the product * is not associative. The latter properties make G(n, ρ; µ), * into a quasigroup 1 .
Proof. Given n = k + ≥ 2 and m = − k ∈ Z, we can generate all the elements G(n, ρ; µ) in H(n, ρ) by composition of G(n, r; σ) with G( , s; τ ) for all σ, τ ∈ Q. Indeed, we can always find two rationals σ and τ such that there exists µ = στ m in Q with L µn,ρ = L σk,r * L τ ,s for any matrix L µn,ρ . Hence, is an involution within H(n, ρ) if, and only if, µ = στ m = 0, that is either if m = 0 (i.e. k = and n = 2k), or m = 0 and σ = 0 or τ = 0. This completes the proof of the lemma.

Theorem 8.2.
With respect to the operation , H(n, ρ) has a semigroup structure. Proof. The set H(n, ρ) is closed under the binary operation . However, by lemma 8.2, H(n, ρ) has no identity element for all values of µ ∈ Q, except when µ = 0. Furthermore, there exists no inverse for every element G(n, ρ; µ) indexed by n ∈ Z >0 and ρ, µ ∈ Q. Yet, H(n, ρ) may be embedded into a monoid simply by adjoining the set {(1, x)}, which is not a subset of the set H(n, ρ). Therefore, the identity may be defined in H(n, ρ) ∪ {(1, x)}.
Hence, H(n, ρ), is a semigroup and the theorem is established.
Remark 8. 4. A monoid is a semigroup with an identity element. Any semigroup S may be embedded into a monoid (generally denoted as S 1 ) simply by adjoining an element e not in S and defining es = s = se for all s ∈ S ∪ {e}. In the present case, the set {(1, x)} plays the role of the identity in the monoid H(n, ρ) ∪ {(1, x)} denoted here as H(n, ρ) 1 .
Laurent series n≥k a n z n/N , where N ∈ Z >0 is a positive integer, k ∈ Z, and each a n belongs to K. K{{z}} sometimes denotes the field of Puiseux series (with respect to the usual sum and Cauchy product). If K is algebraically closed and has characteristic 0, then the field of Puiseux series over K is the algebraic closure of the field of Laurent series over K. ( N) is a sequence of real or complex numbers, the power series f (x) = n≥0 f n ω n x n is then called the generating function (GF) of the sequence (f n ) with respect to the given reference sequence (ω n ) [15].

D Application to combinatorial structures
Exponential generating functions (EGFs) of the form enumerates several sorts of various combinatorial structures, e.g. varieties of increasing ordered rooted trees on d or d + 1 vertices, weights of Dyck d-paths, multiple factorials, etc. (see the examples below). Note that there exists a one-to-one correspondence between G(z) and the EGFs g d (z) for all d ∈ Z >0 defined by substituting z d for z in G(z).
Consider for instance the (d + 1)-ary increasing trees, the d-plane recursive trees and the d-Stirling permutations defined in [23]. Recall that, for d ≥ 1, the degree-weight generating function of (d + 1)-ary increasing trees is given by ϕ(t) = (1 + t) d+1 , i.e. ϕ 0 = 1. Consequently, the generating function G(x) and the numbers G n of (d + 1)-ary trees of order n are obtained by (jd + 1) for n ≥ 1 and G 0 = 0.
In addition, G n = Q n , the number of d-Stirling permutation, which helps have a combinatorial interpretation of Gessel's following theorem (see e.g. [5,22] and [30, p. 7] for a detailed proof).
Theorem (Gessel) Let d ≥ 1. The family T n (d + 1) of (d + 1)-ary increasing trees of order n is in a natural bijection with d-Stirling permutations: T n (d + 1) Q n (d).
For d = 1, the well known bijection between 1-Stirling permutations (ordinary permutations) and binary increasing trees is recovered.