The Hankel transform of the sum of consecutive generalized Catalan numbers

Rajković, Predrag M.; Petković, Marko D.; Barry, Paul

doi:10.1080/10652460601092303

Cited by 29 publications

(27 citation statements)

References 4 publications

(4 reference statements)

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“…As follows from e.g., [23,52] the stated moment relations for the Catalan numbers may be derived by induction from the following recursive relations…”

Section: Rank-one Perturbationsmentioning

confidence: 99%

Spectral Theory for Discrete Laplacians

Dutkay

Jørgensen

2009

Complex Anal. Oper. Theory

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We give the spectral representation for a class of selfadjoint discrete graph Laplacians Δ, with Δ depending on a chosen graph G and a conductance function c defined on the edges of G. We show that the spectral representations for Δ fall in two model classes, (1) tree-graphs with N -adic branching laws, and (2) lattice graphs. We show that the spectral theory of the first class may be computed with the use of rank-one perturbations of the real part of the unilateral shift, while the second is analogously built up with the use of the bilateral shift. We further analyze the effect on spectra of the conductance function c: How the spectral representation of Δ depends on c.Using ΔG, we introduce a resistance metric, and we show that it embeds isometrically into an energy Hilbert space. We introduce an associated random walk and we calculate return probabilities, and a path counting number.

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“…As follows from e.g., [23,52] the stated moment relations for the Catalan numbers may be derived by induction from the following recursive relations…”

Section: Rank-one Perturbationsmentioning

confidence: 99%

Spectral Theory for Discrete Laplacians

Dutkay

Jørgensen

2009

Complex Anal. Oper. Theory

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“…An important transform on integer sequences that has been intensively studied recently is the Hankel transform .…”

Section: Introductionmentioning

confidence: 99%

Hankel transforms of generalized Motzkin numbers

Bojičić¹,

Petković

Rajković

2017

Math Methods in App Sciences

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We study Hankel transform of the sequences (u,l,d),t, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed‐form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (u,l,d)‐Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd.

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“…For example, the papers [7], [22] deal with the Hankel transform of the sequences of the Catalan and adjusted Catalan numbers. Both the evaluations are done by using the method based on the orthogonal polynomials.…”

Section: Hankel Transform and Series Reversionmentioning

confidence: 99%

“…Many other Hankel transform evaluations are given in the papers [8], [9], [14], [26]. In this paper we also use a method based on orthogonal polynomials, as used in [7], [22].…”

Section: Hankel Transform and Series Reversionmentioning

confidence: 99%