arXiv subjects
Publications and source records attributed to Paul Barry.
The group of almost Riordan arrays contains the group of Riordan arrays as a subgroup. In this note, we exhibit examples of pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays.
We define a three parameter family of Bell pseudo-involutions in the Riordan group. The defining sequences have generating functions that are expressible as continued fractions. We indicate that the Hankel transforms of the defining sequences, and of the $A$ sequences of the corresponding Riordan arrays, can be associated with Somos $4$ sequence. We give examples where these sequences can be associated with elliptic curves, and we exhibit instances where elliptic curves can give rise to associated Riordan pseudo-involutions. In the case of a particular one parameter family of elliptic curves, we show how we can associate to each such curve a unique Bell pseudo-involution.
We define and characterize the $f$-matrices associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. These generalize the face matrices of simplices and hypercubes. Their generating functions can be expressed simply in terms of continued fractions, which are shown to be transformations of the generating functions of the corresponding $γ$- and $h$-matrices.
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.
We define and characterize the $γ$-matrix associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. We also define and characterize the $γ$-matrix of the reversions of these triangles, in the case of ordinary Riordan arrays. We are led to the $γ$-matrices of a one-parameter family of generalized Narayana triangles. Thus these matrices generalize the matrix of $γ$-vectors of the associahedron. The principal tools used are the bivariate generating functions of the triangles and Jacobi continued fractions.
We show how some special production matrices may be used to define families of generalized Eulerian triangles. We furthermore show that these generalized Eulerian triangles are the coefficient arrays of polynomials which are the moments of families of orthogonal polynomials. Using the previously defined $\mathcal{T}$ transform, we associate these generalized Eulerian triangles to triangles defined by Catalan generating functions. Again, these new triangles are the coefficient arrays of polynomial moment sequences. The main tools used are the Sumudu transform, Jacobi continued fractions and Riordan arrays.
Using the theory of exponential Riordan arrays, we show that the $1/k$-Eulerian polynomials are moments for a paramaterized family of orthogonal polynomials. In addition, we show that the related Savage-Viswanathan polynomials are also moments for appropriate families of orthogonal polynomials. We provide continued fraction ordinary generating functions and Hankel transforms for these moments, as well as the three-term recurrences for the corresponding orthogonal polynomials. We provide formulas for the $1/k$-Eulerian polynomials and the Savage-Viswanathan polynomials involving the Stirling numbers of the first and the second kind. Finally we show that the once-shifted polynomials are again moment sequences.
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 $\mathcal{T}$ transform.
We define a transformation that associates certain exponential moment sequences with ordinary moment sequences in a natural way. The ingredients of this transformation are series reversion, the Sumudu transform (a variant of the Laplace transform), and the inverting of generating functions. This transformation also has a simple interpretation in terms of continued fractions. It associates lattice path objects with permutation objects, and in particular it associates the Narayana triangle with the Eulerian triangle.
The Riordan group, along with its constituent elements, Riordan arrays, has been a tool for combinatorial exploration since its inception in 1991. More recently, this group has made an appearance in the area of mathematical physics, where it can be used as a toy model in the theory of the renormalization of scalar fields. In this context, its Hopf algebra nature is of importance. In this note, we explain these notions. Power series play a fundamental role in this discussion.
Using 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.
Using the theory of exponential Riordan arrays, we show that the Eulerian-Dowling polynomials are moments for a paramaterized family of orthogonal polynomials. In addition, we show that the related Dowling and the Tanny-Dowling polynomials are also moments for appropriate families of orthogonal polynomials. We provide continued fraction generating functions and Hankel transforms for these polynomials.
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.
Sigmoid functions play an important role in many areas of applied mathematics, including machine learning, population dynamics and probability. We place the study of sigmoid functions in the context of the derivative sub-group of the group of exponential Riordan arrays. Links to families of polynomials are drawn, and it is shown that in some cases these polynomials are orthogonal. In the non-orthogonal case, transformations are given that produce orthgonal systems. Alternative means of characterisation are given, based on the production (Stieltjes) matrix associated to the relevant Riordan array.
We study a super group of the group of Riordan arrays, where the elements of the group are given by a triple of power series. We show that certain subsets are subgroups, and we identify a normal subgroup whose cosets correspond to Riordan arrays. We give an example of an almost-Riordan array that has been studied in the context of Hankel and Hankel plus Toepliz matrices, and we show that suitably chosen almost-Riordan arrays can lead to transformations that have interesting Hankel transform properties.
Every ordinary Riordan array contains two naturally embedded Riordan arrays. We explore this phenomenon, and we compare it to the situation for certain moment matrices of families of orthogonal polynomials.
We study two matrices $N$ and $M$ defined by the parameters of equivalent $S$- and $J$-continued fraction expansions, and compare them by examining the product $N^{-1}M$. Using examples based on the Catalan numbers, the little Schröder numbers and powers of $q$, we indicate that this matrix product is an object worthy of study. In the case of the little Schröder numbers, we find that the matrix $N$ has an interleaved structure based on two Riordan arrays.
We show that Laurent biorthogonal polynomials whose defining three-term recurrence have constant coefficients have coefficient arrays that are Riordan arrays. For each such family of Laurent biorthogonal polynomials we associate in a natural way a family of orthogonal polynomials. We also extend these results to a notion of generalized orthogonal polynomials whose recurrence depends on three parameters.