Generalized Riordan arrays

In this paper, we generalize the concept of Riordan array. A generalized Riordan array with respect to c_n is an infinite, lower triangular array determined by the pair (g(t),f(t)) and has the generic element d_n,k=[tⁿ/c_n]g(t)(f(t))^k/c_k, where c_n is a fixed sequence of non-zero constants with c₀=1.We demonstrate that the generalized Riordan arrays have similar properties to those of the classical Riordan arrays. Based on the definition, the iteration matrices related to the Bell polynomials are special cases of the generalized Riordan arrays and the set of iteration matrices is a subgroup of the Riordan group. We also study the relationships between the generalized Riordan arrays and the Sheffer sequences and show that the Riordan group and the group of Sheffer sequences are isomorphic. From the Sheffer sequences, many special Riordan arrays are obtained. Additionally, we investigate the recurrence relations satisfied by the elements of the Riordan arrays. Based on one of the recurrences, some matrix factorizations satisfied by the Riordan arrays are presented. Finally, we give two applications of the Riordan arrays, including the inverse relations problem and the connection constants problem.     

Sequence characterization of Riordan arrays

In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup.     

Riordan arrays and generalized Lagrange series

The theory of Riordan arrays studies the properties of formal power series and their sequences. The notion of generalized Lagrange series proposed in the present paper is intended to fill the gap in the methodology of this theory. Generalized Lagrange series appear in it implicitly, as various equalities. No special notation is provided for these series, although particular cases of these series are generalized binomial and generalized exponential series. We give the definition of generalized Lagrange series and study their relationship with ordinary Riordan arrays and, separately, with Riordan exponential arrays.     

Combinatorial sums through Riordan arrays

The aim of this work is to show how Riordan arrays are able to generate and close combinatorial identities, by means of the method of coefficients (generating functions). We also show how the same approach can be used to deal with other combinatorial problems, for instance asymptotic approximation and combinatorial inversion. Finally, we propose a method for generating new combinatorial sums by extending the concept of Riordan arrays to bi-infinite matrices.     

Combinatorial sums and implicit Riordan arrays

In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results, both in finding closed expressions for sums and, especially, in solving classes of combinatorial sum inversions.     

Riordan arrays and r-Stirling number identities

By using the theory of Riordan arrays, we establish four pairs of general r-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling numbers of the first and second kind, the r-Stirling numbers of the first and second kind, and the r-Lah numbers. We further discuss briefly the connections between the r-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.     

A symbolic treatment of Riordan arrays

We approach Riordan arrays and their generalizations via umbral symbolic methods. This new approach allows us to derive fundamental aspects of the theory of Riordan arrays as immediate consequences of the umbral version of the classical Abel?s identity for polynomials. In particular, we obtain a novel non-recursive formula for Riordan arrays and derive, from this new formula, some known recurrences and a new recurrence relation for Riordan arrays.     

Generalized harmonic numbers with Riordan arrays

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain very simple connections between the Stirling numbers of both kinds and other generalized harmonic numbers. Further, we suggest that Riordan arrays associated with such generalized harmonic numbers allow us to find new generating functions of many combinatorial sums and many generalized harmonic number identities.     

Riordan矩阵与广义的Pell路

利用Riordan矩阵的A-矩阵得到几类广义Pell路的Riordan矩阵表达式,证明了这些矩阵的行和满足的递推关系,从而给出满足这些递推关系的序列的组合意义.最后将这些格路限制在直线x = y的上方,得出相应的Riordan矩阵表达式的一般形式.     

Products of Riordan arrays of finite orders

Journal of Algebraic Combinatorics - We prove that every Riordan array over $$mathbb {C}$$ whose main diagonal consists only of ones can be written as a product of at most five Riordan arrays of...     

Riordan arrays associated with Laurent series and generalized Sheffer-type groups

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples.     

Riordan 矩阵在广义 Motzkin 路计数中的应用

用Riordan矩阵的方法研究了具有4种步型的加权格路(广义Motzkin路)的计数问题,引入了一类新的计数矩阵,即广义Motzkin矩阵.同时给出了这类矩阵的Riordan表示,也得到了广义Motzkin路的计数公式.Catalan矩阵,Schrder矩阵和Motzkin矩阵都是广义Motzkin矩阵的特殊情形.     

Riordan arrays and the LDU decomposition of symmetric Toeplitz plus Hankel matrices

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, within the context of the Riordan group of lower-triangular matrices. This allows us to determine the LDU decomposition of certain symmetric Toeplitz plus Hankel matrices. We also determine the generating functions and Hankel transforms of associated sequences.     

Binary consecutive covering arrays

A k × n array with entries from a q-letter alphabet is called a t-covering array if each t × n submatrix contains amongst its columns each one of the q ^t different words of length t that can be produced by the q letters. In the present article we use a probabilistic approach based on an appropriate Markov chain embedding technique, to study a t-covering problem where, instead of looking at all possible t × n submatrices, we consider only submatrices of dimension t × n with its rows being consecutive rows of the original k × n array. Moreover, an exact formula is established for the probability distribution function of the random variable, which enumerates the number of deficient submatrices (i.e., submatrices with at least one missing word, amongst their columns), in the case of a k × n binary matrix (q = 2) obtained by realizing kn Bernoulli variables.     

On pseudo-involutions,involutions and quasi-involutions in the group of almost Riordan arrays

Journal of Algebraic Combinatorics - The group of almost Riordan arrays contains the group of Riordan arrays as a subgroup. In this note, we exhibit examples of pseudo-involutions, conditions under...     

Combinatoric enumeration of two-dimensional proper arrays

An n×mproper array is a two-dimensional rectangular array composed of directed cubes that obey certain constraints. Because of these constraints, the n×m proper arrays may be classified via a schema in which each n×m proper array is associated with a particular n×1 column. For a fixed n, the goal is to enumerate, modulo symmetry, all possible edge configurations associated with n×m proper arrays. By varying n, one constructs four combinatoric sequences, each of which enumerates a particular class of edge configurations. Convolution arguments and resultant calculations associate these sequences with cubic equations. These cubic equations allow one to predict M_n, the number of edge configurations, modulo symmetry, associated with n×m proper arrays.     

A combinatorial proof of the log-convexity of sequences in Riordan arrays

Journal of Algebraic Combinatorics - A Riordan array $$R=[r_{n,k}]_{n,k\ge 0}$$ can be characterized by two sequences $$A=(a_n)_{n\ge 0}$$ and $$Z=(z_n)_{n\ge 0}$$ such that $$r_{0,0}=1,...     

Iterative processes related to Riordan arrays: The reciprocation and the inversion of power series

We point out how the Banach Fixed Point Theorem, together with the Picard successive approximation methods yielded by it, allows us to treat some mathematical methods in combinatorics. In particular we get, in this way, a proof of and an iterative algorithm for deriving the Lagrange Inversion Formula.