A Family of Riordan Group Automorphisms

Ângela Mestre and José Agapito

Centro de Análise Funcional, Estruturas Lineares e Aplicações Grupo de Estruturas Algébricas, Lineares e Combinatórias Departamento de Matemática Faculdade de Ciências, Universidade de Lisboa 1749-016 Lisboa Portugal

                   amestre@fc.ul.pt
                    jaruiz@fc.ul.pt

Abstract

In 2006, Bacher introduced a family of Riordan group automorphisms parametrized by three complex numbers. Bacher’s family is a subgroup of the group of automorphisms of the Riordan group and so is the subfamily parametrized only by two real numbers. Here, we study some of the algebraic properties of this subfamily and use the elements to point out isomorphisms between Riordan subgroups. In this context, we prove that the set of Riordan arrays whose row sum sequence is a sequence of partial sums, forms a Riordan subgroup. Moreover, we show that the well-known recursive matrices may be constructed from sequences of images of a Riordan array under automorphisms. Our construction also discloses a correspondence between the recursive matrices and a pair of well-defined Riordan arrays.

1 Introduction

Let f(t) = ∑_{k=0}^{∞} f_{k}t^{k} be a formal power series whose coefficient f_{k} = [t^{k}]f(t) is over a field, say, the field of complex numbers ℂ. Recall that [t^{k}] is the usual coefficient extraction operator. Often, f(t) is called the generating function of the sequence (f_{k})_{k≥0}. Let d(t) and h(t) be two generating functions and consider the infinite matrix D = (D_{k,n})_{k≥n≥1} whose generic entry is given by

D_{k,n} = [t^{k−1}]d(t)h(t)^{n−1}.  (1)

If h_{0} = 0, then D is an infinite lower triangular matrix, known as a Riordan array, usually represented as the pair D = (d(t), h(t)). If, in addition, d_{0} ≠ 0 and h_{1} ≠ 0, then D is said to be a proper Riordan array. The matrix multiplication rule for Riordan arrays in terms of their generating functions is

(d_{1}(t), h_{1}(t))(d_{2}(t), h_{2}(t)) = (d_{1}(t)d_{2}(h_{1}(t)), h_{2}(h_{1}(t))).  (2)

Also, if f(t) and g(t) are the generating functions of the sequences (f_{k})_{k≥0} and (g_{k})_{k≥0}, then the following identity is often referred to as the Fundamental Theorem of Riordan Arrays (FTRA)

g(t) = (d(t), h(t))f(t) = d(t)f(h(t)).  (3)

The first instances where Riordan arrays were considered in the literature are probably Jabotinsky’s work on Bell-type arrays [12, 13, 14], Shapiro’s Catalan triangle [23], and Rogers’ renewal arrays [22]. The formal definition of (proper) Riordan arrays was introduced years later by Shapiro, Getu, Woan, and Woodson [25]. The extra conditions, namely, d_{0} ≠ 0 and h_{1} ≠ 0 allow for the set of proper Riordan arrays to form a group. Here, the Riordan group, denoted by ℜ𝔦𝔬, is the set ℜ𝔦𝔬 = {(d(t), h(t)) | d_{0} = 1, h_{0} = 0, h_{1} = 1}, where the group multiplication is given by formula (2). The group identity is id_{ℜ𝔦𝔬} = (1, t) and the inverse of any element D = (d(t), h(t)) is D^{−1} = (1/d(h̄(t)), h̄(t)), where h̄(t) denotes the compositional inverse of the generating function h(t), which satisfies h(h̄(t)) = h̄(h(t)) = t. Throughtout the paper, unless otherwise stated, by Riordan array we actually mean Riordan group element.

This paper aims at giving further insight into the Riordan group ℜ𝔦𝔬 through some of its automorphisms. A Riordan group automorphism ϕ is a 1-1 mapping from ℜ𝔦𝔬 onto itself such that ϕ(D_{1}D_{2}) = ϕ(D_{1})ϕ(D_{2}) for all D_{1}, D_{2} in ℜ𝔦𝔬. We denote the set of Riordan group automorphisms by Aut(ℜ𝔦𝔬). Here, we consider a family of Riordan group automorphisms, denoted by ϕ_{r,s}, which depend on two real numbers, r and s with s ≠ 0. The definition is ϕ_{r,s}(d(t), h(t)) = ((h(t)/t)^{r}d(t)^{s}, h(t)). They are a special case of Bacher’s automorphisms on three complex numbers introduced in the context of the Lie algebra structure of the Riordan group [5].

The paper is organized as follows. Let ℝ denote the real numbers and ℝ^{∗} = ℝ\{0}. Section 2 shows that the set {ϕ_{r,s}}_{r∈ℝ,s∈ℝ^{∗}}, has a very rich algebraic structure. We give the involutions, the commutator subgroup of {ϕ_{r,s}}_{r∈ℝ,s∈ℝ^{∗}}, and show that the set {ϕ_{r,±1}}_{r∈ℝ} is a normal subgroup. Moreover, let AAut(ℜ𝔦𝔬) denote the group of automorphisms and antiautomorphisms of the Riordan group ℜ𝔦𝔬. Let ψ_{r,s} denote the anti-automorphism induced by ϕ_{r,s}. We give the multiplication and inverse rules for the subgroup of AAut(ℜ𝔦𝔬) consisting of {ϕ_{r,s}, ψ_{r,s}}_{r∈ℝ,s∈ℝ^{∗}}. We end up this section by posing two questions on the general characterization of the automorphisms of ℜ𝔦𝔬.

Section 3 determines the images of most of the well-known Riordan subgroups under the automorphisms ϕ_{r,s}. This points out an infinity of isomorphisms between Riordan subgroups. We use the row sum generating function [24, 11] to show that the image of the stochastic subgroup under the automorphism ϕ_{1,1} is the subgroup of Riordan arrays whose row sum sequence is the partial sum of the coefficients of a generating function h(t). We also give a generic closed form for the row-sum sequence of the arrays in the Riordan subgroup obtained by applying ϕ_{1,n} to the stochastic subgroup, for any n ≥ 1.

Furthermore, Section 4 shows that for a given a Riordan array D (base array), for very large m, the Riordan array ϕ_{−m,1}(D) is an approximation to a bi-infinite triangle which coincides with the recursive matrix of Luzón, Merlini, Morón, and Sprugnoli [17] associated with D. The fact that the automorphism setting does not involve Laurent series is an advantage over the recursive matrices. Moreover, we show that both the east and west side of the bi-infinite triangles can be described by well-defined Riordan arrays. In this context, a bi-infinite triangle turns out to be equivalent to a pair of Riordan arrays. Finally, we observe that the [−m]-complementary of a Riordan array D introduced in the former paper (see also [27, 8, 18]), is the image of the inverse matrix D^{−1} under a Bacher automorphism with appropriate parameters.

Throughtout the paper, the integer sequences which are well-known are referred to by their number from the On-Line Encyclopedia of Integer Sequences [26].

2 A family of automorphisms of ℜ𝔦𝔬

We study some of the algebraic structure of an infinite subgroup of Aut(ℜ𝔦𝔬) parametrized by two real numbers. We obtain the involutions, the commutator subgroup, and the normal subgroups of the considered subgroup of Aut(ℜ𝔦𝔬). We also raise the question of the general characterization of the automorphisms of the Riordan group.

We consider the following transformation:

Φ : ℝ × ℝ^{∗} → Aut(ℜ𝔦𝔬)
(r, s) ↦ ϕ_{r,s},

where for any r ∈ ℝ and s ∈ ℝ^{∗} the mappings ϕ_{r,s} are defined by:

ϕ_{r,s} : ℜ𝔦𝔬 → ℜ𝔦𝔬
(d(t), h(t)) ↦ ((\frac{h(t)}{t})^{r}d(t)^{s}, h(t)).  (4)

Note that, ϕ_{0,1} = id_{Aut(ℜ𝔦𝔬)}. It is easy to verify that the mappings ϕ_{r,s} are automorphisms of the Riordan group. Moreover, ϕ_{r,s} = φ_{s,r,0}, where for any λ, μ ∈ ℂ and κ ∈ ℂ^{∗}, with ℂ^{∗} = ℂ\{0}, the automorphisms φ_{λ,κ,μ} are defined by Bacher [5] as follows:

φ_{λ,κ,μ} : ℜ𝔦𝔬 → ℜ𝔦𝔬
D = (d(t), h(t)) ↦ ((\frac{h(t)}{t})^{λ}d(t)^{κ}h′(t)^{μ}, h(t)) ,  (5)

where h′(t) denotes the derivative of h(t). Note that here the order of the parameters λ and κ is reversed in φ_{λ,κ,μ} in comparison with [5].

The automorphism ϕ_{1,1} (as well as its iterations) has recently been studied in the literature. Actually, ϕ_{1,1} is the diagonal translation operator used by Luzón, Merlini, Morón, and Sprugnoli [17] (see also [5]). The automorphism ϕ_{1,1} is also the mapping T : ℜ𝔦𝔬 → ℜ𝔦𝔬 of He [10], defined by T (D) = UDU^{T}, where (U_{i,j})_{i,j≥1} = (δ_{i+1,j})_{i,j≥1}.

For any real numbers r and s ≠ 0, the array ϕ_{r,s}(d(t), h(t)) has the following semidirect product factorization:

ϕ_{r,s}(d(t), h(t)) = (d(t)^{s}, t)((\frac{h(t)}{t})^{r}, h(t)).  (6)

Clearly, the set Φ(ℝ × ℝ^{∗}) = {ϕ_{r,s}}_{r∈ℝ,s∈ℝ^{∗}} forms a group under composition.

Proposition 1.

Φ(ℝ × ℝ^{∗}) ≤ Aut(ℜ𝔦𝔬).

Proof. Any mapping ϕ_{r,s} is invertible for all r ∈ ℝ and s ∈ ℝ^{∗}, the inverse being given by

ϕ_{r,s}^{−1} = ϕ_{−r/s,1/s}.  (7)

The set is clearly closed under composition:

ϕ_{r′,s′} ∘ ϕ_{r,s} = ϕ_{r′+rs′,ss′}.  (8)

Now, let ℤ and ℚ denote the integer and rational numbers, respectively. Let ℤ^{∗} = ℤ\{0} and ℚ^{∗} = ℚ\{0}. Observe that whereas Φ(ℚ × ℚ^{∗}) remains a subgroup of Aut(ℜ𝔦𝔬), the same does not hold for Φ(ℤ × ℤ^{∗}), except for Φ(ℤ × {±1}). Next, we explore some of the algebraic properties of the sets Φ(ℤ × {±1}).

Proposition 2. Any element in Φ(ℝ × {−1}) is an involution.

Proof. By equation (8), we have ϕ_{r,−1}^{2} = ϕ_{r,−1} ∘ ϕ_{r,−1} = ϕ_{0,1}.

Moreover, it follows from equation (8) that, for all integers n ≥ 0, the nth iterate of ϕ_{r,s} is

ϕ_{r,s}^{n} = ϕ_{r(1+s+s^{2}+···+s^{n−1}),s^{n}}.

Hence, there are no elements in Φ(ℝ × ℝ^{∗}) of order higher than 2, which implies that the set Φ(ℝ × {−1}) contains all the involutions of Φ(ℝ × ℝ^{∗}).

Proposition 3. The set Φ(ℝ × {1}) is the commutator subgroup of Φ(ℝ × ℝ^{∗}).

Proof. It is clear that Φ(ℝ × {1}) is a subgroup of Φ(ℝ × ℝ^{∗}). Also, by equations (7) and (8), we have

[ϕ_{r′,s′}, ϕ_{r,s}] = ϕ_{r′,s′}^{−1} ∘ ϕ_{r,s}^{−1} ∘ ϕ_{r′,s′} ∘ ϕ_{r,s}
= ϕ_{−r′/s′,1/s′} ∘ ϕ_{−r/s,1/s} ∘ ϕ_{r′,s′} ∘ ϕ_{r,s}
= ϕ_{−r′/s′−r/ss′,1/ss′} ∘ ϕ_{r′+rs′,ss′}
= ϕ_{(r′−r+rs′−r′s)/ss′,1}.

Therefore, the commutator of any two elements of the group Φ(ℝ × ℝ^{∗}) is in Φ(ℝ × {1}) and so is any finite product of commutators of Φ(ℝ × ℝ^{∗}).

Proposition 4. The set Φ(ℝ × {−1, 1}) is a normal subgroup of Φ(ℝ × ℝ^{∗}).

Proof. It is straightforward to check that Φ(ℝ × {−1, 1}) is a subgroup of Φ(ℝ × ℝ^{∗}). The normality property follows from Proposition 3 and from equations (7) and (8):

ϕ_{r,s} ∘ ϕ_{r′,−1} ∘ ϕ_{r,s}^{−1} = ϕ_{r+r′s,−s} ∘ ϕ_{−r/s,1/s} = ϕ_{2r+r′s,−1},
ϕ_{r,s} ∘ ϕ_{r′,1} ∘ ϕ_{r,s}^{−1} = ϕ_{r+r′s,s} ∘ ϕ_{−r/s,1/s} = ϕ_{r′s,1}.

We define as well a family of anti-automorphisms of ℜ𝔦𝔬 induced by Φ as follows:

Ψ : ℝ × ℝ^{∗} → AAut(ℜ𝔦𝔬)
(r, s) ↦ ψ_{r,s},

where for any r ∈ ℝ and s ∈ ℝ^{∗}, the mapping ψ_{r,s} is defined by:

ψ_{r,s} : ℜ𝔦𝔬 → ℜ𝔦𝔬
D ↦ ϕ_{r,s}(D^{−1}) = ((\frac{h̄(t)}{t})^{r} \frac{1}{d(h̄(t))^{s}} , h̄(t)).

It is easy to see that Φ(ℝ × ℝ^{∗}) ∪ Ψ(ℝ × ℝ^{∗}) ≤ AAut(ℜ𝔦𝔬). Moreover, the automorphisms ϕ_{r,s} commute with the mapping that sends a Riordan group element to its inverse.

Lemma 5. Let Inv : ℜ𝔦𝔬 → ℜ𝔦𝔬; D ↦ D^{−1} be the mapping that sends a Riordan group element D to its inverse D^{−1}. Then, we have

ϕ_{r,s} ∘ Inv = Inv ∘ ϕ_{r,s}.

The following four equations are immediately derived from the lemma above:

ψ_{r,s}^{−1} = ψ_{−r/s,1/s},
ψ_{r′,s′} ∘ ψ_{r,s} = ϕ_{r′+rs′,ss′},
ϕ_{r′,s′} ∘ ψ_{r,s} = ψ_{r′+rs′,ss′},
ψ_{r,s} ∘ ϕ_{r′,s′} = ψ_{r+r′s,ss′}.

Now, let D_{1} = (d_{1}(t), h_{1}(t)) and D_{2} = (d_{2}(t), h_{2}(t)) denote two Riordan matrices. One can partition the set of Riordan arrays into equivalence classes by setting D_{1} ∼ D_{2} if and only if h_{1}(t) = h_{2}(t). Clearly, the family {ϕ_{r,s}}_{r∈ℝ,s∈ℝ^{∗}} (as well as Bacher’s larger family) preserves these equivalence classes, something which does not happen, for instance, with the inner automorphisms. Recall that for the Riordan group, the inner automorphism ϕ_{ℑ} associated with the element ℑ = (g(t), f(t)) is

ϕ_{ℑ} = (g(t), f(t))(d(t), h(t))(g(t), f(t))^{−1}
= (g(t)d(f(t))\frac{1}{g(f̄(h(f(t))))}, f̄(h(f(t)))) .

We did not find any automorphism, other than the inner automorphisms (or a composition with them), that modifies the second element of the pair (d(t), h(t)). Thus, we pose the following two questions: Question 6. Is there any automorphism in Aut(ℜ𝔦𝔬) up to composition with inner automorphisms, that sends a Riordan array (d_{1}(t), h_{1}(t)) to (d_{2}(t), h_{2}(t)) with h_{1}(t) ≠ h_{2}(t)?

Due to Bacher’s definition of the Riordan group as the semidirect product of the group of formal power series with constant term 1 by the group of formal power series under substitution [5]. We believe that the results of Muckenhoupt [20] and Johnson [16] are likely to be relevant to answer the question above in the negative. Moreover, a more general question in this context is the following: Question 7. Which is the general characterization of the group of automorphisms of the Riordan group? That is, in terms of the generating functions d(t) and h(t) such that d_{0} = 1, h_{0} = 0, and h_{1} = 1, which properties must a mapping (d_{1}(t), h_{1}(t)) ↦ (d_{2}(t), h_{2}(t)) satisfy in order to be a Riordan group automorphism?

3 Isomorphisms between Riordan subgroups via ϕ_{r,s}

Since the mappings ϕ_{r,s} are automorphisms of the Riordan group ℜ𝔦𝔬, for any subgroup 𝔰𝔲𝔟ℜ𝔦𝔬 ≤ ℜ𝔦𝔬, the sets ϕ_{r,s}(𝔰𝔲𝔟ℜ𝔦𝔬) are also subgroups of ℜ𝔦𝔬. In this section, we characterize the images under ϕ_{r,s} of several well-known subgroups of the Riordan group. For brevity, we shall write 𝔰𝔲𝔟ℜ𝔦𝔬_{r,s} = ϕ_{r,s}(𝔰𝔲𝔟ℜ𝔦𝔬). Observe that the groups 𝔰𝔲𝔟ℜ𝔦𝔬_{r,s} and 𝔰𝔲𝔟ℜ𝔦𝔬_{r′,s′} are isomorphic via ϕ_{r′−rs′/s,s′/s} for all pairs (r, s) and (r′, s′).

3.1 The c-Appell subgroup {(d(t), ct) | c ≠ 0}

Proposition 8. Let c-𝔄𝔭𝔭_{r,s} = {(c^{r}d^{s}(t), ct) | c ≠ 0}. Then, for all r ∈ ℝ and s ∈ ℝ^{∗}:

c-𝔄𝔭𝔭_{r,s} ≤ ℜ𝔦𝔬.

The usual Appell subgroup 𝔄𝔭𝔭 is obtained for c = 1 and s = 1; that is 1-𝔄𝔭𝔭_{r,s} = 𝔄𝔭𝔭_{s}.

3.2 The Lagrange subgroup {(1, h(t))}

Proposition 9. Let 𝔏𝔞𝔤_{r} = {((h(t)/t)^{r}, h(t))}. Then, for all r ∈ ℝ:

𝔏𝔞𝔤_{r} ≤ ℜ𝔦𝔬.

Note that when r ≠ 0 the group 𝔏𝔞𝔤_{1/r} coincides with the r-power-Bell subgroup of Jean-Louis and Nkwanta [15], whose arrays are of the form ((h(t)/t)^{1/r}, h(t)). When r = 1 we recover the well-known isomorphism between the Lagrange and the Bell subgroups given by Jean-Louis and Nkwanta [15] and by He [10]. Furthermore, observe that by equation (6), the group {ϕ_{r,s}(ℜ𝔦𝔬)}_{r∈ℝ,s∈ℝ^{∗}} can be written as the semidirect product of the groups 𝔄𝔭𝔭_{s} and 𝔏𝔞𝔤_{r}.

3.3 The c-Bell subgroup {(h(t)/t, c h(t)) | c ≠ 0}

Proposition 10. Let c-𝔅𝔢𝔩𝔩_{r,s} = {(c^{r}(h(t)/t)^{r+s}, c h(t)) | c ≠ 0}. Then, for all r ∈ ℝ and s ∈ ℝ^{∗}:

c-𝔅𝔢𝔩𝔩_{r,s} ≤ ℜ𝔦𝔬.

Here, the Bell subgroup for c = 1 is simply denoted by 𝔅𝔢𝔩𝔩. We have 𝔅𝔢𝔩𝔩_{r,s} = 𝔏𝔞𝔤_{r+s}.

Example 11. Recall that c(t) = (1 − \sqrt{1 − 4t})/2t is the generating function of the Catalan numbers C_{n} = 1/(n + 1)\binom{2n}{n}, n ≥ 0 \underline{A000108}. The function c(t) satisfies the functional equation c(t) = 1 + tc(t)^{2}. There are several well-known Bell matrices in the literature generated by c(t) and accordingly referred to as Catalan triangles, namely,

• Aigner’s array C = (c(t), tc(t)) of the ballot numbers [1, 2];

• Shapiro’s Catalan triangle B = (c(t)^{2}, tc(t)^{2}) [23];

• Radoux’s triangle of numbers R = (c(t), tc(t)^{2}) [21].

It is easy to see that ϕ_{r,s}(B) = R when r + s = 1/2. Also, ϕ_{r,s}(R) = B when 2r + s = 2. We thus have a system of two linear equations with two unknowns r and s. The solution gives the involution ϕ_{3/2,−1} such that ϕ_{3/2,−1}(B) = R and ϕ_{3/2,−1}(R) = B.

Moreover, recall that Shapiro’s triangle satisfies the following factorization property B = CP, where P is the classical Pascal triangle (see, e.g., [3]). Obviously, ϕ_{r,s}(B) = ϕ_{r,s}(C)ϕ_{r,s}(P). A similar factorization holds for Radoux’s triangle. Indeed, for every pair (r, s) such that r + s = 1/2 it follows that R = ϕ_{r,s}(C)ϕ_{r,s}(P). For the involution ϕ_{3/2,−1}, we have R = (c(t)^{1/2}, tc(t))(p(t)^{1/2}, tp(t)) and ϕ_{r,s}(R) = ϕ_{r+3s/2,−s}(C)ϕ_{r+3s/2,−1}(P).

Furthermore, the authors, Petrullo, and Torres [4] showed that the matrices above are particular cases of a wide family of Catalan triangles given by

C^{a,b}(q, z) = (c(q, z; t)^{a}, t c(q, z; t)^{b}), a, b ≥ 1,

where

c(q, z; t) = \frac{1 − zt − \sqrt{(1 − zt)^{2} − 4qt}}{2qt} with q, z ≥ 0.

The family C^{a,b}(q, z) includes:

• He’s parametric Catalan triangles C^{1,1}(q, z − q) [9];

• Yang’s generalized Catalan triangles C^{a,b}(q, 0) [28].

It is easy to check that ϕ_{r,s}(C^{a,b}(q, z)) = C^{br+as,b}(q, z). Also, note that the action of the automorphisms ϕ_{r,s} on the generalized Catalan triangles of He yields ϕ_{r,s}(C^{1,1}(q, z − q)) = C^{r+s,1}(q, z − q). Hence, the image under ϕ_{r,s} of any He’s parametric Catalan triangle is not a He’s parametric Catalan triangle unless r + s = 1.

3.4 The checkerboard subgroup {(d(t), h(t)) | d(t) = d(−t) , h(t) =

−h(−t)}

Proposition 12. Let ℭ𝔥𝔢𝔠𝔨_{r,s} = { ((h(t)/t)^{r}d(t)^{s}, h(t)) |d(t) = d(−t) , h(t) = −h(−t)}. Then, for all r ∈ ℝ and s ∈ ℝ^{∗}:

ℭ𝔥𝔢𝔠𝔨_{r,s} ≤ ℜ𝔦𝔬.

Jean-Louis and Nkwanta [15] proved that the checkerboard subgroup is the centralizer of the element (1, −t). We note that the checkerboard subgroup ℭ𝔥𝔢𝔠𝔨 is also the centralizer of ϕ_{r,s}(1, −t) for every pair (r, s).

3.5 The derivative subgroup {(h′(t), h(t))}

Proposition 13. Let 𝔇𝔢𝔯_{r,s} = { ((h(t)/t)^{r}h′(t)^{s}, h(t)) }. Then, for all r ∈ ℝ and s ∈ ℝ^{∗}:

𝔇𝔢𝔯_{r,s} ≤ ℜ𝔦𝔬.

The derivative subgroup is isomorphic to the Lagrange subgroup via Bacher’s automorphism φ_{0,1,−1} : (h′(t), h(t)) ↦ (1, h(t)). The analogous result for the subgroups 𝔇𝔢𝔯_{r,s} and 𝔏𝔞𝔤_{r} follows via φ_{0,1,−s} : ((h(t)/t)^{r}h′(t)^{s}, h(t)) ↦ ((h(t)/t)^{r}, h(t)). Note that the subgroup 𝔇𝔢𝔯_{r,s} was previously introduced in [17].

3.6 The hitting time subgroup {(t h′(t)/h(t), h(t))}

Proposition 14. Let ℌ𝔦𝔱_{r,s} = { ((h(t)/t)^{r−s}h′(t)^{s}, h(t)) }. Then, for all r ∈ ℝ and s ∈ ℝ^{∗}:

ℌ𝔦𝔱_{r,s} ≤ ℜ𝔦𝔬.

The traditional hitting time subgroup corresponds to the case r = −1 and s = 1. As a direct consequence of the above definition, we have ℌ𝔦𝔱_{r,s} = 𝔇𝔢𝔯_{r−s,s}. Also, the groups ℌ𝔦𝔱_{r,s} and 𝔏𝔞𝔤_{r−s} are isomorphic via Bacher’s automorphism φ_{1,1,−s}.

3.7 The Jean-Louis–Nkwanta subgroup {(f(t)/f(h(t)), h(t))}

Recall that the stochastic Riordan subgroup is the stabilizer of the column vector (1, 1, 1, …). More generally, let f(t) = f_{0} + f_{1}t + f_{2}t^{2} + ⋯ be the generating function of the sequence (f_{0}, f_{1}, f_{2}, …). By using the fundamental theorem of Riordan arrays, Jean-Louis and Nkwanta [15] showed that the set of Riordan matrices of the form (f(t)/f(h(t)), h(t)) is the stabilizer of the column vector (f_{0}, f_{1}, f_{2}, …). In particular, the elements of the stochastic Riordan subgroup are of the form ((1 − h(t))/(1 − t), h(t)).

Proposition 15. Let 𝔍𝔏𝔑_{r,s} = { ((h(t)/t)^{r}(f(t)/f(h(t)))^{s}, h(t)) }. Then, for all r ∈ ℝ and s ∈ ℝ^{∗}:

𝔍𝔏𝔑_{r,s} ≤ ℜ𝔦𝔬.

3.7.1 The stochastic subgroup

Let 𝔇^{h} = ϕ_{1,1}((1 − h(t))/(1 − t), h(t)). Note that the row sum sequence of 𝔇^{h}, here denoted 𝔯^{𝔇^{h}}, is the partial sum of the coefficients of the generating function h(t). That is, 𝔯_{n}^{𝔇^{h}} = ∑_{i=1}^{n} h_{i}. This follows from observing that the generating function for the row sum of a Riordan array is given by d(t)/(1−h(t)) [24, 11]. Therefore, the Riordan subgroup 𝔖𝔱𝔬𝔠𝔥_{1,1} = {𝔇^{h}} is the set of Riordan arrays whose row sum sequence is the partial sum of the coefficients of the generating function h(t).

Example 16. Let h(t)/t = p(t), c(t), c^{2}(t), where p(t) = 1/(1 − t) and c(t) the generating function of the Catalan numbers. The elements of the group 𝔖𝔱𝔬𝔠𝔥_{1,1} related to these functions are:

𝔇^{tp} = (\frac{p(t) − tp^{2}(t)}{1 − t}, tp(t)) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ −2 & 2 & 3 & 1 & 0 & 0 & ⋯ \\ −5 & 0 & 5 & 4 & 1 & 0 & ⋯ \\ −9 & −5 & 5 & 9 & 5 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

𝔇^{tc} = (\frac{c(t) − tc^{2}(t)}{1 − t}, tc(t)) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ 1 & 4 & 3 & 1 & 0 & 0 & ⋯ \\ 1 & 9 & 8 & 4 & 1 & 0 & ⋯ \\ 1 & 23 & 22 & 13 & 5 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

𝔇^{tc^{2}} = (\frac{c^{2}(t) − tc^{4}(t)}{1 − t}, tc^{2}(t)) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 3 & 4 & 1 & 0 & 0 & 0 & ⋯ \\ 3 & 12 & 6 & 1 & 0 & 0 & ⋯ \\ −3 & 33 & 25 & 8 & 1 & 0 & ⋯ \\ −36 & 88 & 91 & 42 & 10 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Their row sum sequence numbers are \underline{A000027}, \underline{A014137}, and \underline{A014138}. Obviously, we have 𝔇^{tc}𝔇^{tp} = 𝔇^{tc^{2}}.

The above result fixes s = 1. Restricting s to integers greater than 1 gives the next result, which is a closed form for the row-sum sequences of the the arrays in the image of the Stochastic subgroup under the mapping ϕ_{1,n}.

Proposition 17. Let 𝔇_{n}^{h} = ϕ_{1,n}((1−h(t))/(1 − t), h(t)), where h(t) = h_{1}t+h_{2}t^{2} +h_{3}t^{3} +⋯ with h_{1} ≠ 0 is any power series over the complex numbers. Then, for all n ≥ 1 the Riordan subgroup 𝔖𝔱𝔬𝔠𝔥_{1,n} = {𝔇_{n}^{h}} is the set of Riordan arrays whose row sum sequence is given by:

𝔯^{𝔇_{n}^{h}} = ∑_{j=0}^{∞} ∑_{i=0}^{j} ∑_{k=0}^{n−1} ∑_{i_{1}+i_{2}+···+i_{k+1}=i} h_{i_{1}+1} … h_{i_{k+1}+1} ×

(−1)^{k}\binom{n + j − i − 1}{n − 1}\binom{n − 1}{k}t^{j+k},

where the third sum runs over all sequences of non-negative integers (i_{1}, …, i_{k+1}) satisfying i_{1} + i_{2} + ⋯ + i_{k+1} = i.

Proof. The proof is a straightforward computation of the product of the power series h(1 − h(t))^{n−1}/t and 1/(1 − t)^{n} = ∑_{i=0}^{∞} \binom{n+i−1}{n−1}t^{i}.

Example 18. We consider again h(t)/t = p(t), c(t), c^{2}(t). The row sum sequences associated with the elements 𝔇_{2}^{tp}, 𝔇_{2}^{tc}, and 𝔇_{2}^{tc^{2}} of the group 𝔖𝔱𝔬𝔠𝔥_{1,2} are:

(𝔯^{𝔇_{2}^{tp}}) = 1, 2, 2, 0, −5, −14, −28, −48, −75, −110, −154, −208, −273, −350, −440, …
(\underline{A005586} from the fourth term on),
(𝔯^{𝔇_{2}^{tc}}) = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, … (\underline{A000027}),
(𝔯^{𝔇_{2}^{tc^{2}}}) = 1, 3, 6, 9, 6, −30, −209, −960, −3921, −15280, −58293, −220170, −827787, ….

4 Bi-infinite triangles defined via automorphisms

This section uses automorphisms to introduce bi-infinite triangles. These bi-infinite triangles coincide with the recursive matrices of Luzón, Merlini, Morón, and Sprugnoli [17]. Recall that a recursive matrix (d_{k′,n′})_{k′≥n′∈ℤ} = χ(d(t), h(t)) is defined by

d_{k′,n′} = [t^{k′}]d(t)h(t)^{n′} ∀k′, n′ ∈ ℤ, k′ ≥ n′,

where d(t) and h(t) are formal power series as in the definition of Riordan arrays. Also, d(t)h(t)^{n′} = ∑_{i=n′}^{∞} l_{i}t^{i} is a formal Laurent series (which yields a formal power series if n′ ≥ 0).

Let m be a non-negative integer. Let D be any Riordan array. We examine the matrix ϕ_{−m,1}(D) for large m. Though, for simplicity, we fix the second parameter of the automorphism equal to 1, our results generalize straightforwardly for ϕ_{−m,s}(D), with s ∈ ℝ^{∗}, by replacing d(t) by d(t)^{s}.

For any non-negative integer m, the submatrix of ϕ_{−m,1}(D) that results from deleting the first m columns and the first m rows is the base Riordan array D. Moreover, any array ϕ_{−m,1}(D) may be obtained from ϕ_{−(m+1),1}(D) by removing the first row and the first column. That is to say that the array sequence (ϕ_{−m,1}(D))_{m≥1} is an ascending chain of matrices, in the north west direction. Thus, every array in the sequence (ϕ_{−m,1}(D))_{m≥1} is a submatrix of a bi-infinite triangle, denoted by ϕ_{−∞,1}(D), constructed as follows. Take the matrix D and place on the right of D, the first column of each of the matrices ϕ_{−m,1}(D) (from right to left). Then, add infinite rows of zeros on the top of each column (including the columns of D) to yield bi-infinite columns of a bi-infinite triangle. Then, for any pair of integers (k′, n′) the entries of ϕ_{−∞,1}(D) are given by

(ϕ_{−∞,1}(D))_{k′≥n′} = \begin{cases}D_{k′,n′}, & if n′ ≥ 1; \\ (ϕ_{n′−1,1}(D))_{k′−n′+1,1}, & otherwise, \end{cases}

where (ϕ_{n′−1,1}(D))_{k′−n′+1,1} = [t^{k′−1}]d(t)h(t)^{n′−1}. Therefore, our construction of the bi-infinite triangle ϕ_{−∞,1}(D) leads to the definition of the recursive array χ(d(t), h(t)) of Luzón, Merlini, Morón, and Sprugnoli [17], up to a column phase shift of one unit. Observe that the phase shift is due to the fact that in Equation (1) the rows and columns, indexed by k and n, respectively, are numbered from 1 on, while in [17] they go from 0 on.

The array ϕ_{−∞,1}(D) can be visualized as follows:

ϕ_{−∞,1}(D) = [ 𝔏_{D} ℜ_{D} ] ,

where ℜ_{D} = [ D ]^{T} and the matrix 𝔏_{D} is a lower-triangular array which grows to the west region of ϕ_{−∞,1}(D), while the matrix D grows to the south east region.

Although the bi-infinite triangle ϕ_{−∞,1}(D) is not a Riordan array, this can be approximated by a Riordan array ϕ_{−m,1}(D) when m is large. For practical calculations, one can thus work with a Riordan array ϕ_{−m,1}(D) after choosing m large enough to fit the purpose.

Indeed, the matching between the columns of both bi-infinite triangles can also be seen from the large m approximation of ϕ_{−∞,1}(D). We have

ϕ_{−m,1}(D)_{k,n} = [t^{k−m−1}]d(t)h(t)^{n−m−1}, ∀k ≥ n ≥ 1.

Let k′ = k−m−1 and n′ = n−m−1. Then, when m is large, say m approaches infinity, both k′ and n′ run through all integers in ℤ. Hence, the array ϕ_{−∞,1}(D) has the same columns as the recursive matrix χ(d(t), h(t)). However, since the image of a Riordan array under an automorphism is a Riordan array, the expansion of the columns of ϕ_{−m,1}(D) never involves formal Laurent series, only formal power series whose coefficient sequences coincide with the coefficient sequences of the formal Laurent series in the corresponding recursive matrices. Note that in this case, there is a phase shift of m + 1 units between the index of a column in ϕ_{−m,1}(D) and the index of the same column in χ(d(t), h(t)). A recursive matrix χ(D) can thus be studied through a Riordan array ϕ_{−m,1}(D) after choosing an appropriate value for m.

Furthermore, the matrix 𝔏_{D} (read from right to left) mirrors a well-defined Riordan array.

Definition 19. Let D be any Riordan array. Consider the matrix sequence (ϕ_{−m,1}(D))_{m≥1}. We define the lower triangular matrix 𝔩_{D} as one whose mth column is the first column of the matrix ϕ_{−m,1}(D) for each m ≥ 1, with m rows of zeros added to the top of the column. Hence, the entries of 𝔩_{D} read as follows:

(𝔩_{D})_{k,n} = [t^{k−2n}]d(t)h(t)^{−n}, k ≥ n ≥ 1.

Thus, by definition, 𝔩_{D} is the following Riordan array:

𝔩_{D} = (\frac{t}{h(t)}d(t), \frac{t^{2}}{h(t)}).  (9)

Note that a similar idea to that of defining the matrices ϕ_{−∞,1}(D) and 𝔩_{D} from a sequence of images of the Riordan array D under some automorphisms, was recently given by the authors for the construction of some square matrices related to Riordan arrays [19].

Remark 20. Let D be any Riordan array. The bi-infinite triangle ϕ_{−∞,1}(D) corresponds to the following pair of Riordan arrays:

(𝔩_{D}, D) = ((\frac{t}{h(t)}d(t), \frac{t^{2}}{h(t)}), (d(t), h(t))) ,

in the sense of yielding exactly the same columns.

Now, recall briefly that for m ∈ ℤ, the [m]-complementary array χ(D)^{[m]} of a recursive matrix χ(D), is defined by χ(D)_{k,n}^{[m]} = χ(D)_{m−n,m−k} [17]. Next, for fixed m ≥ 1, we introduce the following lower triangular array with m columns 𝔏_{D}^{(m)}:

(𝔏_{D}^{(m)})_{k,n} = [t^{k−m−1}]d(t)h(t)^{n−m−1}, 1 ≤ n ≤ k, m.

That is,

(𝔏_{D}^{(m)})_{k,n} = ϕ_{−∞,1}(D)_{k−m,n−m} = χ(D)_{−n+1,−k+1}^{[−m]}.

We also remark that the [m]-complementary array of a Riordan array D reads as

D^{[m]} = ((\frac{t}{h̄(t)})^{m+1} d(h̄(t))h̄′(t), h̄(t))

yields φ_{−(m+1),−1,1}(D^{−1}), where Bacher’s automorphism φ_{−(m+1),−1,1} is defined by formula (5).

Additionally, the first l ≥ 1 rows of the matrix 𝔏_{D}^{(m)} yield a finite matrix, say, 𝔏_{D}^{(l,m)}. Moreover, consider the submatrix, say D^{(l′,m′)}, of the base array D consisting of the first l′ ≥ 1 rows and m′ ≥ 1 columns of D. In this context, for fixed q ≥ 0 we consider the following finite (2q + 1) × (2q + 1) and (2q + 2) × (2q + 2) matrices:

A_{q} = [ 𝔏_{D}^{(2q+1,q)} ℜ_{D}^{(q+1,q+1)} ] , where ℜ_{D}^{(q+1,q+1)} = [ \begin{matrix}↕ q rows of zeros \\ D^{(q+1,q+1)} \end{matrix} ] ,

B_{q} = [ 𝔏_{D}^{(2q+2,q)} ℜ_{D}^{(q+2,q+2)} ] , where ℜ_{D}^{(q+2,q+2)} = [ \begin{matrix}↕ q rows of zeros \\ D^{(q+2,q+2)} \end{matrix} ] .

Note that the west side matrices do not occur in A_{0} and B_{0}. We observe that (A_{q})_{q≥0} and (B_{q})_{q≥0} are sequences of finite matrices such that A_{q} = γ_{q}(D), B_{q} = δ_{q}(D), where (γ_{q})_{q≥0} and (δ_{q})_{q≥0} are sequences of projections introduced by Luzón, Merlini, Morón, Prieto-Martinez, and Sprugnoli [18].

Example 21. Consider Pascal’s array P = (p(t), tp(t)), where p(t) = 1/(1 − t), sequence \underline{A007318}:

P = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 3 & 3 & 1 & 0 & 0 & 0 & ⋯ \\ 1 & 4 & 6 & 4 & 1 & 0 & 0 & ⋯ \\ 1 & 5 & 10 & 10 & 5 & 1 & 0 & ⋯ \\ 1 & 6 & 15 & 20 & 15 & 6 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .  (10)

Hence, P_{k,n} = \binom{k−1}{n−1}, n ≥ k ≥ 1. Let m ≥ 1. We now examine the matrix ϕ_{−m,1}(P) = ((1 − t)^{m−1}, t/(1 − t)) for large m. For instance, we have for m = 10:

ϕ_{−10,1}(P) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −9 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 36 & −8 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −84 & 28 & −7 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 126 & −56 & 21 & −6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −126 & 70 & −35 & 15 & −5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 84 & −56 & 35 & −20 & 10 & −4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −36 & 28 & −21 & 15 & −10 & 6 & −3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 9 & −8 & 7 & −6 & 5 & −4 & 3 & −2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −1 & 1 & −1 & 1 & −1 & 1 & −1 & 1 & −1 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 & 3 & 1 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 & 6 & 4 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

We observe that the matrix 𝔏_{P}^{(10)} is zero below the equator. Note that is not true in general for the matrix 𝔏_{D}^{(m)}, with m ≥ 1 associated with an arbitrary Riordan array D. Indeed, this happens for all m ≥ 1 with the Pascal array by chance, for only the first two coefficients of p^{−1}(t) are different from zero.

When m is large, in the north west region of the matrix ϕ_{−m,1}(P), the transpose of the reflection of the matrix 𝔏_{P} (read from right to left) over the equator yields the matrix P^{−1} = (1/(1 + t), t/(1 + t)), the inverse of Pascal’s array. The columns of 𝔏_{P} (read from right to left) also coincide with the sequence \underline{A110555} when seen as an upper right triangle. Besides, the lower triangular matrix 𝔩_{P} of formula (9) is

𝔩_{P} = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & −1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & −2 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 1 & −3 & 1 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 3 & −4 & 1 & 0 & ⋯ \\ 0 & 0 & 0 & −1 & 6 & −5 & 1 & ⋯ \\ 0 & 0 & 0 & 0 & −4 & 10 & −6 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & −10 & 15 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 5 & −20 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & −1 & 15 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & −6 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Then, (𝔩_{P})_{k,n} = (−1)^{k−n}\binom{n−1}{k−n}. The diagonals of 𝔩_{P} yield the columns in the north east region of the bi-infinite matrix of Barnabei [6, p. 1132]. Note that a similar correspondence also holds for the matrix of p. 1139 of the same paper.

Example 22. We compute the array 𝔩_{T} (x, y) of formula (9), where T (x, y) is a Pascal-like array of polynomials in the variables x and y, introduced by Barry [7]:

T (x, y) = (\frac{1}{1 − yt}, \frac{yt(1 + xt)}{1 − yt})

= [ \begin{matrix}1 & 0 & 0 & 0 & 0 & ⋯ \\ y & y & 0 & 0 & 0 & ⋯ \\ y^{2} & xy + 2y^{2} & y^{2} & 0 & 0 & ⋯ \\ y^{3} & 2xy^{2} + 3y^{3} & 2xy^{2} + 3y^{3} & y^{3} & 0 & ⋯ \\ y^{4} & 3xy^{3} + 4y^{4} & x^{2}y^{2} + 6xy^{3} + 6y^{4} & 3xy^{3} + 4y^{4} & y^{4} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The general expression for the entries is:

T_{k,n}(x, y) = ∑_{j=0}^{n−1} \binom{n − 1}{j}\binom{k − 1 − j}{n − 1}x^{j}y^{k−1−j}, k, n ≥ 1.

Then, the matrix 𝔩_{T} is

𝔩_{T} (x, y) = (\frac{1}{y(1 + xt)}, \frac{t(1 − yt)}{y(1 + xt)})

= [ \begin{matrix}\frac{1}{y} & 0 & 0 & 0 & 0 & ⋯ \\ −\frac{x}{y} & \frac{1}{y^{2}} & 0 & 0 & 0 & ⋯ \\ \frac{x^{2}}{y} & −\frac{2x+y}{y^{2}} & \frac{1}{y^{3}} & 0 & 0 & ⋯ \\ −\frac{x^{3}}{y} & \frac{3x^{2}+2xy}{y^{2}} & −\frac{3x+2y}{y^{3}} & \frac{1}{y^{4}} & 0 & ⋯ \\ \frac{x^{4}}{y} & −\frac{4x^{3}+3x^{2}y}{y^{2}} & \frac{6x^{2}+6xy+y^{2}}{y^{3}} & −\frac{4x+3y}{y^{4}} & \frac{1}{y^{5}} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Hence, the complete central coefficients sequence introduced by Barry [7] of the matrix T (x, y) is the sequence:

T_{k,⌈ \frac{k}{2} ⌉}(x, y) = ∑_{j=0}^{⌈ \frac{k}{2} ⌉−1} \binom{⌈\frac{k}{2}⌉ − 1}{j}\binom{k − 1 − j}{⌈\frac{k}{2}⌉ − 1}x^{j}y^{k−1−j}, k, n ≥ 1.

Note that Barry considers ⌊k/2⌋ instead of ⌈k/2⌉ for in [7] k, n ≥ 0, while here k, n ≥ 1. The general term of the Riordan array 𝔩_{T} (x, y) is given by:

(𝔩_{T})_{k,n}(x, y) = \frac{(−1)^{k+1}}{y^{n}} ∑_{j=0}^{n−1} \binom{n − 1}{j}\binom{k − 1 − j − 1}{n − 1}x^{k−j−n}y^{j}, k, n ≥ 1.

Therefore, the complete central coefficients sequence of Barry of the matrix 𝔩_{T} (x, y) is the sequence:

(𝔩_{T})_{k,⌈ \frac{k}{2} ⌉}(x, y) = \frac{(−1)^{k+1}}{y^{⌈ \frac{k}{2} ⌉}} ∑_{j=0}^{⌈ \frac{k}{2} ⌉−1} \binom{⌈\frac{k}{2}⌉ − 1}{j}\binom{k − 1 − j − 1}{⌈\frac{k}{2}⌉ − 1}x^{k−j−⌈ \frac{k}{2} ⌉}y^{j}, k, n ≥ 1.

Example 23. We compute the Riordan array 𝔩_{F C}, where

F C = (\frac{1}{1 − t − t^{2}} , tc(t))
= (\frac{1}{1 − t − t^{2}} , \frac{1 − \sqrt{1 − 4t}}{2})

= [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ 3 & 5 & 3 & 1 & 0 & 0 & ⋯ \\ 5 & 12 & 9 & 4 & 1 & 0 & ⋯ \\ 8 & 31 & 26 & 14 & 5 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ]

is the Fibonacci-Catalan triangle whose first column yields the Fibonacci numbers \underline{A000045} (without the first element which is zero). Moreover, the second column and the row sum sequence number both yield the convolution of the Fibonacci and the Catalan numbers \underline{A090826}. The sequence numbers for the first three descending diagonals are \underline{A000012}, \underline{A000027}, and \underline{A000096}. The next three descending diagonals yield:

F C_{3+n,n} = \frac{n^{3} + 9n^{2} + 20n − 12}{3!}, n ≥ 1,
F C_{4+n,n} = \frac{n^{4} + 18n^{3} + 107n^{2} + 162n − 168}{4!}, n ≥ 1,
F C_{5+n,n} = \frac{n^{5} + 30n^{4} + 335n^{3} + 1530n^{2} + 1824n − 2760}{5!}, n ≥ 1.

The array 𝔩_{F C} of formula (9) is

𝔩_{F C} = (\frac{2t}{(1 − t − t^{2})(1 − \sqrt{1 − 4t})}, \frac{2t^{2}}{1 − \sqrt{1 − 4t}})

= [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & −1 & 1 & 0 & 0 & 0 & ⋯ \\ −2 & −1 & −2 & 1 & 0 & 0 & ⋯ \\ −7 & −4 & −1 & −3 & 1 & 0 & ⋯ \\ −23 & −10 & −4 & 0 & −4 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The coefficients of the generating function 2t/(1 − \sqrt{1 − 4t}) yield the sequence \underline{A115140}. Then, the first column of 𝔩_{F C} is the convolution of \underline{A115140} and \underline{A000045} (after removing the first element of the latter). The first descending diagonal is of course \underline{A000012}. The second is \underline{A001478}. The third is \underline{A000096} from the fourth element on. We now give recursion formulas for the fourth and fifth descending diagonals from the sixth and seventh element on, respectively. The following holds:

(𝔩_{F C})_{3+n,n} = −\frac{n^{3} − 12n^{2} + 41n − 18}{3!}, n ≥ 6,
(𝔩_{F C})_{4+n,n} = \frac{n^{4} − 22n^{3} + 167n^{2} − 434n + 120}{4!}, n ≥ 7.

Alternatively, we have the following recursion formulas:

(𝔩_{F C})_{9,6} = −2, (𝔩_{F C})_{10,7} = −4,
(𝔩_{F C})_{3+n,n} = −((𝔩_{F C})_{2+n,n−2} − 2(𝔩_{F C})_{3+n,n−1} + n − 5), n ≥ 8,
(𝔩_{F C})_{11,7} = 5, (𝔩_{F C})_{12,8} = 7, (𝔩_{F C})_{13,9} = 11,
(𝔩_{F C})_{4+n,n} = 3(𝔩_{F C})_{4+n,n−1} − 3(𝔩_{F C})_{3+n,n−2} + (𝔩_{F C})_{2+n,n−3} + n − 7, n ≥ 10.

5 Acknowledgments

We would like to thank the anonymous referee for his valuable suggestions. This research was made within the activities of the Group for Linear, Algebraic and Combinatorial Structures of the Center for Functional Analysis, Linear Structures and Applications (University of Lisbon, Portugal). The first and the second authors were supported by the research fellowships SFRH/BPD/100567/2014 and SFRH/BPD/111317/2015, respectively, provided by the Portuguese Foundation for Science and Technology (FCT).

References

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2010 Mathematics Subject Classification: Primary 05B20; Secondary 18A30, 20E18, 20H20. Keywords: Riordan group, Riordan subgroup, automorphism, isomorphism, bi-infinite triangle.

(Concerned with sequences \underline{A000012}, \underline{A000027}, \underline{A000045}, \underline{A000096}, \underline{A000108}, \underline{A001478}, \underline{A005586}, \underline{A007318}, \underline{A014137}, \underline{A014138}, \underline{A090826}, \underline{A110555}, and \underline{A115140}.)

Received July 10 2019; revised versions received December 23 2019; December 24 2019. Published in Journal of Integer Sequences, December 26 2019.

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