34 research outputs found
Limit theorems for numbers satisfying a class of triangular arrays
The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays, defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain the partial differential equation and special analytical expressions for the numbers using a semi-exponential generating function. We apply the results to prove the asymptotic normality of special classes of the numbers and specify the convergence rate to the limiting distribution. We demonstrate that the limiting distribution is not always Gaussian
Why Delannoy numbers?
This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete
Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of
Statistical Planning and Inference
Identities involving Narayana polynomials and Catalan numbers
We first establish the result that the Narayana polynomials can be
represented as the integrals of the Legendre polynomials. Then we represent the
Catalan numbers in terms of the Narayana polynomials by three different
identities. We give three different proofs for these identities, namely, two
algebraic proofs and one combinatorial proof. Some applications are also given
which lead to many known and new identities.Comment: 13 pages,6 figure
Appell and Sheffer sequences: on their characterizations through functionals and examples
The aim of this paper is to present a new simple recurrence for Appell and
Sheffer sequences in terms of the linear functional that defines them, and to
explain how this is equivalent to several well-known characterizations
appearing in the literature. We also give several examples, including integral
representations of the inverse operators associated to Bernoulli and Euler
polynomials, and a new integral representation of the re-scaled Hermite
-orthogonal polynomials generalizing the Weierstrass operator related to the
Hermite polynomials
A selected survey of umbral calculus
We survey the mathematical literature on umbral calculus (otherwise known as the calculus of finite differences) from its roots in the 19th century (and earlier) as a set of "magic rules" for lowering and raising indices, through its rebirth in the 1970’s as Rota’s school set it on a firm logical foundation using operator methods, to the current state of the art with numerous generalizations and applications. The survey itself is complemented by a fairly complete bibliography (over 500 references) which we expect to update regularly