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Paul Barry

Publications and source records attributed to Paul Barry.

64 records · Page 4Linked to original sources
arXiv2012

On the Hankel transform of C-fractions

We study the Hankel transforms of sequences whose generating function can be expressed as a C-fraction. In particular, we relate the index sequence of the non-zero terms of the Hankel transform to the powers appearing in the monomials defining the C-fraction. A closed formula for the Hankel transforms studied is given. As every power-series can be represented by a C-fraction, this gives in theory a closed form formula for the Hankel transform of any sequence. The notion of multiplicity is introduced to differentiate between Hankel transforms.

math.CA↗
arXiv2012

On sequences with {-1,0,1} Hankel transforms

We study Hankel transforms of sequences, where the transform elements are members of the set {-1,0,1}. We relate these Hankel transforms to special continued fraction expansions. In particular, we posit a conjecture relating the distribution of non-zero terms in the Hankel transform to the distribution of powers of the variable in the defining continued fractions.

math.CO↗
arXiv2011

Hankel transform of a sequence obtained by series reversion II - aerating transforms

This paper provides the connection between the Hankel transform and aerating transforms of a given integer sequence. Results obtained are used to establish a completely different Hankel transform evaluation of the series reversion of a certain rational function $Q(x)$ and shifted sequences, recently published in our paper \cite{part1}. For that purpose, we needed to evaluate the Hankel transforms of the sequences $\seqn{α^2 C_n-βC_{n+1}}$ and $\seqn{α^2 C_{n+1}-βC_{n+2}}$, where $C=\seqn{C_n}$ is the well-known sequence of Catalan numbers. This generalizes the results of Cvetkovi\' c, Rajković and Ivković \cite{CRI}. Also, we need the evaluation of Hankel-like determinants whose entries are Catalan numbers $C_n$ and which is based on the recent results of Krattenthaler \cite{krattCat}. The results obtained are general and can be applied to many other Hankel transform evaluations.

math.CO↗
arXiv2011

Invariant number triangles, eigentriangles and Somos-4 sequences

Using the language of Riordan arrays, we look at two related iterative processes on matrices and determine which matrices are invariant under these processes. In a special case, the invariant sequences that arise are conjectured to have Hankel transforms that obey Somos-4 recurrences. A notion of eigentriangle for a number triangle emerges and examples are given, including a construction of the Takeuchi numbers.

math.CO↗
arXiv2011

Eulerian polynomials as moments, via exponential Riordan arrays

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the "descending power" Eulerian polynomials, and their once shifted sequence, are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of the polynomial sequences in terms of continued fractions, and we also calculate their Hankel transforms.

math.CO↗
arXiv2011

An orthogonal polynomial coefficient formula for the Hankel transform

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.

math.CO↗
arXiv2011

Riordan arrays, orthogonal polynomials as moments, and Hankel transforms

Taking the examples of Legendre and Hermite orthogonal polynomials, we show how to interpret the fact that these orthogonal polynomials are moments of other orthogonal polynomials in terms of their associated Riordan arrays. We use these means to calculate the Hankel transforms of the associated polynomial sequences.

math.CO↗
arXiv2011

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.

math.CO↗