Riordan Arrays Research Articles

Notes on the total positivity of Riordan arrays

Let R=(d(t),h(t)) be a Riordan array. We show that if both d(t) and h(t) are Pólya frequency formal power series, then R is totally positive.

Functions and Jordan canonical forms of Riordan matrices

This paper collects results about Riordan arrays in the framework of matrix functions; actually, the following methodology applies to any square matrix m×m with exactly one eigenvalue λ of algebraic multiplicity m∈N. Generalized Lagrange bases are used to construct Hermite polynomials that interpolate a family of functions; moreover, we show a parallel application of such functions via Jordan canonical forms and case studies are given.

Bijections of Motzkin Paths Using Shifted Riordan Decompositions

This paper concerns bijections between Motzkin and Łukasiewicz paths arising from Riordan array decompositions. Bijections have been shown between Motzkin paths and Łukasiewicz paths with constant weights (Hennessy in A study of riordan srrays with applications to continued fractions, orthogonal polynomials and lattice paths. Ph.D. thesis, Waterford Institute of Technology, 2011). We introduce a shifting matrix technique to induce a bijection between Motzkin paths and Łukasiewicz paths with non constant weighted steps. We will show a generating function proof and a construction of these bijections.

A unified approach for the Catalan matrices by using Riordan arrays

We study generalized Catalan matrices based on the Riordan array and Fuss–Catalan numbers. A unified combinatorial interpretation for the entries of the generalized Catalan matrices is presented by means of m-Dyck paths. Some properties and examples of the generalized Catalan matrices are given including a new convolution formula for the generalized Catalan numbers. Finally, we present applications of generalized Catalan matrices to the problems in counting the hill-free and lower peak-free m-Dyck paths.

A new aspect of Riordan arrays via Krylov matrices

In this paper, we give a new angle to interpret Riordan arrays by showing that every Riordan array can be expressed as a Krylov matrix. We then use this idea to obtain some groups containing the Riordan group as a subgroup. Moreover, we study Lie algebras for the extended Riordan groups as Lie groups.

Hankel Determinants of the Generalized Factorials

Denote 〈x|d〉 n = x(x + d)(x + 2d) · · · (x + (n - 1)d) for n = 1, 2, · · ·, and 〈x|d〉0 = 1, where 〈x|d〉 n is called the generalized factorial of x with increment d. In this paper, we present the evaluation of Hankel determinants of sequence of generalized factorials. The main tool used for the evaluation is the method based on exponential Riordan arrays. Furthermore, we provide Hankel determinant evaluations of the Eulerian polynomials and exponential polynomials.

Documenting and treating a dysregulated gut-immune-brain axis in schizophrenia

We study generalized Catalan matrices based on the Riordan array and Fuss–Catalan numbers. A unified combinatorial interpretation for the entries of the generalized Catalan matrices is presented by means of m-Dyck paths. Some properties and examples of the generalized Catalan matrices are given including a new convolution formula for the generalized Catalan numbers. Finally, we present applications of generalized Catalan matrices to the problems in counting the hill-free and lower peak-free m-Dyck paths.

Sequence characterizations of double Riordan arrays and their compressions

Inspired by the Fibonacci tree shown in the recent book, Catalan Numbers, by Richard Stanley, we present a combinatorial way to construct double Riordan arrays by using ECO technique. The sequence characterizations of double Riordan arrays and subgroups of the double Riordan group are found. As an extension of Pascal–Fibonacci triangle, the compression forms of double Riordan arrays called double quasi-Riordan arrays are defined. The connections of lower triangular matrices and double Riordan arrays as well as their compressions are given. Those results are also extended to the case of high order Riordan arrays, which are defined in the paper. The pairs of Sheffer polynomials and pairs of summation formulas associated with double Riordan arrays are defined and discussed.

Some Identities Involving the Higher-Order Changhee Numbers and Polynomials

In this paper, by the classical method of Riordan arrays, establish several general involving higher-order Changhee numbers and polynomials, which are related to special polynomials and numbers. From those numbers, we derive some interesting and new identities.

English

For integers m > r ≥ 0, Brietzke (2008) defined the (m, r)-central coefficients of an infinite lower triangular matrix G = (d, h) = (dn,k)n,k∈N as dmn+r,(m−1)n+r, with n = 0, 1, 2,..., and the (m, r)-central coefficient triangle of G as $${G^{\left( {m,r} \right)}} = {\left( {{d_{mn + r,\left( {m - 1} \right)n + k + r}}} \right)_{n,k \in \mathbb{N}}}.$$ It is known that the (m, r)-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = (d, h) with h(0) = 0 and d(0), h′(0) ≠ 0, we obtain the generating function of its (m, r)-central coefficients and give an explicit representation for the (m, r)-central Riordan array G(m,r) in terms of the Riordan array G. Meanwhile, the algebraic structures of the (m, r)-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of m and r. As applications, we determine the (m, r)-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.

Half of a Riordan array and restricted lattice paths

For an infinite lower triangular matrix G=(gn,k)n,k≥0, we define the half of G to be the infinite lower triangular matrix H=(hn,k)n,k≥0 such that hn,k=g2n−k,n for all n≥k≥0. In this paper, we will show that if G=(gn,k)n,k≥0 is a Riordan array, then its half H=(hn,k)n,k≥0 is also a Riordan array, and we obtain new combinatorial interpretations for some Riordan arrays in terms of weighted lattice paths.

Log-concavity and strong q-log-convexity for Riordan arrays and recursive matrices

Let [An,k]n,k⩾0 be an infinite lower triangular array satisfying the recurrencefor n ⩾ 1 and k ⩾ 0, where A0,0 = 1, A0,k = Ak,–1 = 0 for k > 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schröder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not only preserves the strong q-log-convexity property, but also preserves the strong q-log-concavity property. Finally, we demonstrate that the strong q-log-convexity property is preserved by the Stirling transformation and Whitney transformation of the second kind, which extends some known results for the strong q-log-convexity property.

Some matrix identities on colored Motzkin paths

Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form (z+b)n.

Fuss–Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group

In this paper, we present the Riordan arrays called Fuss–Catalan matrices which are constructed by the convolutions of the generating functions of the Fuss–Catalan numbers. We also discuss weighted sums of the Fuss–Catalan matrices, using such matrices as transformations of recursive sequences, and their connection with stabilizer subgroups of the Riordan group.

The group of multi-dimensional Riordan arrays

We first consider two groups, F0={g∈C[[z]]|g(0)≠0} under multiplication and F1=zF0 under composition, where C[[z]] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F0⋊F1. It may be viewed as a group extension of F1 by F0. In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F0. This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums.

Row polynomial matrices of Riordan arrays

Let R=[rn,k]n,k≥0 be a Riordan array. Define the row polynomials Rn(q)=∑k=0nrn,kqk and the row polynomial matrix R(q)=[rn,k(q)]n,k≥0 by rn,k(q)=∑j=knrn,jqj−k. Then R(q) is also a Riordan array with the Rn(q) located on the leftmost column of R(q). In this paper we investigate combinatorial properties of the matrix R(q) and the sequence (Rn(q))n≥0, including their characterizations, the q-total positivity of R(q) and the q-log-convexity of (Rn(q))n≥0.

On the restricted Chebyshev–Boubaker polynomials

ABSTRACTUsing the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev–Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they satisfy, and we study certain central sequences defined by their coefficient arrays. We give an integral representation for their moments, and we show that the Hankel transforms of these moments have a simple form. We show that the (sequence) Hankel transform of the row sums of the corresponding moment matrix is defined by a family of polynomials closely related to the Chebyshev polynomials of the second kind, and that these row sums are in fact the moments of another family of orthogonal polynomials.

Riordan arrays and d-orthogonality

Using the theory of the Riordan group and d-orthogonal polynomials, we shall show that sequences of d-orthogonal polynomials, can be also generated by the Riordan group. We interpret some families of d-orthogonal polynomials within the framework of the Riordan group.

Multivariate Riordan groups and their representations

Riordan bases in a power series ring of several variables are introduced as a special type of Schauder bases. As a part of the method of generating differentials, interplay of Riordan bases generalizes Lagrange inversion formulas. The set of Riordan bases has a group law with a specified set of variables as the unit element. Riordan arrays appear in a representation of the group.

Production matrices for geometric graphs

We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.