Paul Barry | South East Technological University (original) (raw)

Papers by Paul Barry

Journal of Integer Sequences, 2010

We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal... more We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal polynomials. In so doing, we are led to introduce a family of polynomials, which includes the Boubaker polynomials, and a scaled version of the Chebyshev polynomials, using the techniques of Riordan arrays. We classify these polynomials in terms of the Chebyshev polynomials of the first and second kinds. We also examine the Hankel transforms of sequences associated with the inverse of the polynomial coefficient arrays, including the associated moment sequences.

arXiv (Cornell University), Jan 13, 2011

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, wi... more 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.

Journal of Algebraic Combinatorics, Jun 15, 2020

We present properties of the group structure of Riordan arrays. We examine similar properties amo... more We present properties of the group structure of Riordan arrays. We examine similar properties among known Riordan subgroups, and from this, we define H [r , s, p], a family of Riordan arrays. We generalize conditions for involutions, and pseudoinvolutions of H [r , s, p], and we present stabilizers of this family. We find abelian subgroups as intersections of Riordan subgroups and show some alternative semidirect products of the Riordan group. Keywords Riordan subgroup • Involution • Pseudo-involution • Semi-direct product • Isomorphism • Stabilizer 1 Introduction The group structure of the set of Riordan arrays has been the subject of a number of papers [3,9,12,15,19], where Riordan subgroups and their group theoretical properties are presented. In this paper, we exclusively focus on the algebraic elements of Riordan arrays, by providing new findings. The motivation of our research was "Some algebraic structure of the Riordan group" by Jean-Louis et al. [12], and we hope that our work will be considered as a complementary to the results presented there. The paper is arranged as follows. In Sect. 2, we present the definitions of a Riordan array, the fundamental theorem of Riordan arrays and the Riordan group, together with the Riordan subgroups that have been found so far. We define H [r , s, p], a family of Riordan subgroups, based on a collection of isomorphic Riordan subgroups, as shown B Nikolaos Pantelidis

Journal of Integer Sequences, 2009

We examine a set of special Riordan arrays, their inverses and associated Hankel transforms.

We study a family of polynomials in two variables, identifying them as the moments of a two-param... more We study a family of polynomials in two variables, identifying them as the moments of a two-parameter family of orthogonal polynomials. The coefficient array of these orthogonal polynomials is shown to be an ordinary Riordan array. We express the generating function of the sequence of polynomials under study as a continued fraction, and determine the corresponding Hankel transform. An alternative characterization of the polynomials in terms of a related Riordan array is also given. This Riordan array is associated with Lukasiewicz paths. The special form of the production matrices is exhibited in both cases. This allows us to produce a bijection from a set of colored Lukasiewicz paths to a set of colored Motzkin paths. The polynomials studied generalize the notion of Narayana polynomial.

We study the Euler-Seidel matrix of certain integer sequences, using the binomial transform and H... more We study the Euler-Seidel matrix of certain integer sequences, using the binomial transform and Hankel matrices. For moment sequences, we give an integral representation of the Euler-Seidel matrix. Links are drawn to Riordan arrays, orthogonal polynomials, and Christoffel-Darboux expressions.

Czechoslovak Mathematical Journal, Mar 1, 2012

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Linear Algebra and its Applications, 2016

Using elements of the group of Riordan arrays we define a family of generalized Narayana triangle... more Using elements of the group of Riordan arrays we define a family of generalized Narayana triangles and their associated generalized Catalan numbers, and study their links to series reversion. In particular we use Lagrange inversion techniques to determine the generating functions for these generalized Catalan numbers.

DR COLETTE MOLONEY, MS ANGELA BUCKLEY, MS DEIRDRE MCDONALD, MS LISA MORRISSEY AND MS ALISON

72 WIT Research Day 2011 Book of Abstracts Ms Aoife Hennessy and Dr Paul Barry A MIMO calculation... more 72 WIT Research Day 2011 Book of Abstracts Ms Aoife Hennessy and Dr Paul Barry A MIMO calculation involving Narayana triangles and Riordan arrays Department of Computing, Mathematics and Physics Email: Aoife. hennessy@ gmail. com Abstract: Here we look at the mathematics of one application in the area of MIMO (multiple input, multiple output) wireless communication through the combinatorics of Riordan arrays. Due to the ever increasing popularity of wireless communications and the increased desire for efficient use of bandwidth, MIMO systems ...

Journal of Integer Sequences, 2011

By studying various ways of describing the Narayana triangle, we provide a number of generalizati... more By studying various ways of describing the Narayana triangle, we provide a number of generalizations of this triangle and study some of their properties. In some cases, the diagonal sums of these triangles provide examples of Somos-4 sequences via their Hankel transforms.

Journal of Integer Sequences, 2011

By studying various ways of describing the Narayana triangle, we provide a number of generalizati... more By studying various ways of describing the Narayana triangle, we provide a number of generalizations of this triangle and study some of their properties. In some cases, the diagonal sums of these triangles provide examples of Somos-4 sequences via their Hankel transforms.

In this paper, we study closed form evaluation for some special Hankel determinants arising in co... more In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directlyconnectedwiththetheoryofquasi-definitediscreteSobolev orthogonalpolynomials.Itopensalotofproceduraldilemmaswhich we will try to exceed. A few examples deal with Fibonacci numbers and power sequences will illustrate our considerations. We believe that our usage of Sobolev orthogonal polynomials in Hankel determinant computation is quite new.

Journal of Integer Sequences, 2011

We use the Lagrange inversion theorem to characterize the central coefficients of matrices in the... more We use the Lagrange inversion theorem to characterize the central coefficients of matrices in the Bell subgroup of the Riordan group of matrices. We give examples of how by using different means of calculating these coefficients we can deduce the generating functions of interesting sequences.

J. Integer Seq, 2011

We study the Narayana triangles and related families of polynomials. We link this study to Riorda... more We study the Narayana triangles and related families of polynomials. We link this study to Riordan arrays and Hankel transforms arising from a special case of capacity calculation related to MIMO communication systems. A link is established between a channel capacity calculation and a series reversion.

We define two notions of partial sums of a Riordan array, corresponding respectively to the parti... more We define two notions of partial sums of a Riordan array, corresponding respectively to the partial sums of the rows and the partial sums of the columns of the Riordan array in question. We characterize the matrices that arise from these operations. On the one hand, we obtain a new Riordan array, while on the other hand, we obtain a rectangular array which has an inverse that is a lower Hessenberg matrix. We examine the structure of these Hessenberg matrices. We end with a generalization linked to the Fibonacci numbers and phyllotaxis.

We study the Hankel transforms of sequences related to the central coefficients of a family of Pa... more We study the Hankel transforms of sequences related to the central coefficients of a family of Pascal-like triangles. The mechanism of Riordan arrays is used to elucidate the structure of these transforms.

We study a sequence transformation pipeline that maps certain sequences with rational generating ... more We study a sequence transformation pipeline that maps certain sequences with rational generating functions to permutation-based sequence families of combinatorial significance. Many of the number triangles we encounter can be related to simplicial objects such as the associahedron and the permutahedron. The linkages between these objects is facilitated by the use of the previously introduced T transform.

We characterize a family of number triangles whose production matrices are closely related to the... more We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these triangles relate to sequences that have interesting convolution recurrences and continued fraction generating functions.

The Hankel transform of an integer sequence is a much studied and much applied mathematical opera... more The Hankel transform of an integer sequence is a much studied and much applied mathematical operation. In this note, we extend the notion in a natural way to sequences of d integer sequences. We explore links to generalized continued fractions in the context of d-orthogonal sequences.

We give an explicit formula for the Hankel transform of a regular sequence in terms of the coeffi... more We give an explicit formula for the Hankel transform of a regular sequence in terms of the coefficients of the associated orthogonal polynomials and the sequence itself. We apply this formula to some sequences of combinatorial interest, deriving interesting combinatorial identities by this means. Further insight is also gained into the structure of the Hankel transform of a sequence.

Journal of Integer Sequences, 2010

We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal... more We determine which (ordinary) Riordan arrays are the coefficient arrays of a family of orthogonal polynomials. In so doing, we are led to introduce a family of polynomials, which includes the Boubaker polynomials, and a scaled version of the Chebyshev polynomials, using the techniques of Riordan arrays. We classify these polynomials in terms of the Chebyshev polynomials of the first and second kinds. We also examine the Hankel transforms of sequences associated with the inverse of the polynomial coefficient arrays, including the associated moment sequences.

arXiv (Cornell University), Jan 13, 2011

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, wi... more 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.

Journal of Algebraic Combinatorics, Jun 15, 2020

We present properties of the group structure of Riordan arrays. We examine similar properties amo... more We present properties of the group structure of Riordan arrays. We examine similar properties among known Riordan subgroups, and from this, we define H [r , s, p], a family of Riordan arrays. We generalize conditions for involutions, and pseudoinvolutions of H [r , s, p], and we present stabilizers of this family. We find abelian subgroups as intersections of Riordan subgroups and show some alternative semidirect products of the Riordan group. Keywords Riordan subgroup • Involution • Pseudo-involution • Semi-direct product • Isomorphism • Stabilizer 1 Introduction The group structure of the set of Riordan arrays has been the subject of a number of papers [3,9,12,15,19], where Riordan subgroups and their group theoretical properties are presented. In this paper, we exclusively focus on the algebraic elements of Riordan arrays, by providing new findings. The motivation of our research was "Some algebraic structure of the Riordan group" by Jean-Louis et al. [12], and we hope that our work will be considered as a complementary to the results presented there. The paper is arranged as follows. In Sect. 2, we present the definitions of a Riordan array, the fundamental theorem of Riordan arrays and the Riordan group, together with the Riordan subgroups that have been found so far. We define H [r , s, p], a family of Riordan subgroups, based on a collection of isomorphic Riordan subgroups, as shown B Nikolaos Pantelidis

Journal of Integer Sequences, 2009

We examine a set of special Riordan arrays, their inverses and associated Hankel transforms.

We study a family of polynomials in two variables, identifying them as the moments of a two-param... more We study a family of polynomials in two variables, identifying them as the moments of a two-parameter family of orthogonal polynomials. The coefficient array of these orthogonal polynomials is shown to be an ordinary Riordan array. We express the generating function of the sequence of polynomials under study as a continued fraction, and determine the corresponding Hankel transform. An alternative characterization of the polynomials in terms of a related Riordan array is also given. This Riordan array is associated with Lukasiewicz paths. The special form of the production matrices is exhibited in both cases. This allows us to produce a bijection from a set of colored Lukasiewicz paths to a set of colored Motzkin paths. The polynomials studied generalize the notion of Narayana polynomial.

We study the Euler-Seidel matrix of certain integer sequences, using the binomial transform and H... more We study the Euler-Seidel matrix of certain integer sequences, using the binomial transform and Hankel matrices. For moment sequences, we give an integral representation of the Euler-Seidel matrix. Links are drawn to Riordan arrays, orthogonal polynomials, and Christoffel-Darboux expressions.

Czechoslovak Mathematical Journal, Mar 1, 2012

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Linear Algebra and its Applications, 2016

Using elements of the group of Riordan arrays we define a family of generalized Narayana triangle... more Using elements of the group of Riordan arrays we define a family of generalized Narayana triangles and their associated generalized Catalan numbers, and study their links to series reversion. In particular we use Lagrange inversion techniques to determine the generating functions for these generalized Catalan numbers.

DR COLETTE MOLONEY, MS ANGELA BUCKLEY, MS DEIRDRE MCDONALD, MS LISA MORRISSEY AND MS ALISON

72 WIT Research Day 2011 Book of Abstracts Ms Aoife Hennessy and Dr Paul Barry A MIMO calculation... more 72 WIT Research Day 2011 Book of Abstracts Ms Aoife Hennessy and Dr Paul Barry A MIMO calculation involving Narayana triangles and Riordan arrays Department of Computing, Mathematics and Physics Email: Aoife. hennessy@ gmail. com Abstract: Here we look at the mathematics of one application in the area of MIMO (multiple input, multiple output) wireless communication through the combinatorics of Riordan arrays. Due to the ever increasing popularity of wireless communications and the increased desire for efficient use of bandwidth, MIMO systems ...

Journal of Integer Sequences, 2011

By studying various ways of describing the Narayana triangle, we provide a number of generalizati... more By studying various ways of describing the Narayana triangle, we provide a number of generalizations of this triangle and study some of their properties. In some cases, the diagonal sums of these triangles provide examples of Somos-4 sequences via their Hankel transforms.

Journal of Integer Sequences, 2011

By studying various ways of describing the Narayana triangle, we provide a number of generalizati... more By studying various ways of describing the Narayana triangle, we provide a number of generalizations of this triangle and study some of their properties. In some cases, the diagonal sums of these triangles provide examples of Somos-4 sequences via their Hankel transforms.

In this paper, we study closed form evaluation for some special Hankel determinants arising in co... more In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directlyconnectedwiththetheoryofquasi-definitediscreteSobolev orthogonalpolynomials.Itopensalotofproceduraldilemmaswhich we will try to exceed. A few examples deal with Fibonacci numbers and power sequences will illustrate our considerations. We believe that our usage of Sobolev orthogonal polynomials in Hankel determinant computation is quite new.

Journal of Integer Sequences, 2011

We use the Lagrange inversion theorem to characterize the central coefficients of matrices in the... more We use the Lagrange inversion theorem to characterize the central coefficients of matrices in the Bell subgroup of the Riordan group of matrices. We give examples of how by using different means of calculating these coefficients we can deduce the generating functions of interesting sequences.

J. Integer Seq, 2011

We study the Narayana triangles and related families of polynomials. We link this study to Riorda... more We study the Narayana triangles and related families of polynomials. We link this study to Riordan arrays and Hankel transforms arising from a special case of capacity calculation related to MIMO communication systems. A link is established between a channel capacity calculation and a series reversion.

We define two notions of partial sums of a Riordan array, corresponding respectively to the parti... more We define two notions of partial sums of a Riordan array, corresponding respectively to the partial sums of the rows and the partial sums of the columns of the Riordan array in question. We characterize the matrices that arise from these operations. On the one hand, we obtain a new Riordan array, while on the other hand, we obtain a rectangular array which has an inverse that is a lower Hessenberg matrix. We examine the structure of these Hessenberg matrices. We end with a generalization linked to the Fibonacci numbers and phyllotaxis.

We study the Hankel transforms of sequences related to the central coefficients of a family of Pa... more We study the Hankel transforms of sequences related to the central coefficients of a family of Pascal-like triangles. The mechanism of Riordan arrays is used to elucidate the structure of these transforms.

We study a sequence transformation pipeline that maps certain sequences with rational generating ... more We study a sequence transformation pipeline that maps certain sequences with rational generating functions to permutation-based sequence families of combinatorial significance. Many of the number triangles we encounter can be related to simplicial objects such as the associahedron and the permutahedron. The linkages between these objects is facilitated by the use of the previously introduced T transform.

We characterize a family of number triangles whose production matrices are closely related to the... more We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these triangles relate to sequences that have interesting convolution recurrences and continued fraction generating functions.

The Hankel transform of an integer sequence is a much studied and much applied mathematical opera... more The Hankel transform of an integer sequence is a much studied and much applied mathematical operation. In this note, we extend the notion in a natural way to sequences of d integer sequences. We explore links to generalized continued fractions in the context of d-orthogonal sequences.

We give an explicit formula for the Hankel transform of a regular sequence in terms of the coeffi... more We give an explicit formula for the Hankel transform of a regular sequence in terms of the coefficients of the associated orthogonal polynomials and the sequence itself. We apply this formula to some sequences of combinatorial interest, deriving interesting combinatorial identities by this means. Further insight is also gained into the structure of the Hankel transform of a sequence.