Catalan Numbers, Their Generalization, and Their Uses

Hilton, Peter; Pedersen, Jean

doi:10.1007/bf03024089

Cited by 210 publications

(189 citation statements)

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“…On the other hand, when θ is an integer, the even moments C and (θ + 2)(θ + 3) · · · (2θ + 1)C θ k is an integer multiple of θ!. We refer to Gessel [16] or Hilton and Pedersen [20] for the study of this kind of generalized Catalan numbers.…”

Section: C) If θ Is An Integermentioning

confidence: 99%

On the non-classical infinite divisibility of power semicircle distributions

Arizmendi¹,

Pérez‐Abreu²

2010

COSA

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Abstract. The family of power semicircle distributions defined as normalized real powers of the semicircle density is considered. The marginals of uniform distributions on spheres in high-dimensional Euclidean spaces are included in this family and a boundary case is the classical Gaussian distribution. A review of some results including a genesis and the so-called Poincaré's theorem is presented. The moments of these distributions are related to the super Catalan numbers and their Cauchy transforms in terms of hypergeometric functions are derived. Some members of this class of distributions play the role of the Gaussian distribution with respect to additive convolutions in non-commutative probability, such as the free, the monotone, the anti-monotone and the Boolean convolutions. The infinite divisibility of other power semicircle distributions with respect to these convolutions is studied using simple kurtosis arguments. A connection between kurtosis and the free divisibility indicator is found. It is shown that for the classical Gaussian distribution the free divisibility indicator is strictly less than 2.

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Section: C) If θ Is An Integermentioning

confidence: 99%

On the non-classical infinite divisibility of power semicircle distributions

Arizmendi¹,

Pérez‐Abreu²

2010

COSA

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“…It is known that I (n, k), defined by (3.5), is the generalized Catalan number, and that I (n, k) = (nk)!/(n(k − 1) + 1)!n! (see [9]). Therefore, we conclude that…”

Section: Abstract Resultsmentioning

confidence: 96%

“…There is a large amount of literature on the Cauchy problem and the scattering theory for (1.1) (see, for instance, [2][3][4][5][6][7][8][9][10][11][12][13][15][16][17], and references therein). The usual scattering theory for (1.1) compares the full dynamics given by the solutions to (1.1) and the free dynamics described by the free propagator U(t) = exp it .…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Expansion of Solutions to the Nonlinear Schrödinger Equation With Power Nonlinearity

Masaki

2009

Kyushu J. Math.

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Abstract. We study an asymptotic expansion near t = ∞ of the solution to the Cauchy problem for the nonlinear Schrödinger equation with repulsive short-range nonlinearity of power type. We construct two kinds of approximate solution with asymptotic expansions. The first is an accurate approximate solution and of abstract form. The second is the approximation of the first and of explicit form. The sharpness of these approximations strongly depend on the fractional part of the power of the nonlinearity. In particular, if the power is an integer, we obtain a complete expansion of the solution.

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“…is a generalised Catalan number [7]. The Catalan number is the number of different trees with n internal nodes of arity a (and, of course, (a − 1)n + 1 leaves).…”

Section: Branching Processes and Lagrange Distributionmentioning

confidence: 99%