13 research outputs found
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
The classical umbral calculus: Sheffer sequences
Following the approach of Rota and Taylor \cite{SIAM}, we present an
innovative theory of Sheffer sequences in which the main properties are encoded
by using umbrae. This syntax allows us noteworthy computational simplifications
and conceptual clarifications in many results involving Sheffer sequences. To
give an indication of the effectiveness of the theory, we describe applications
to the well-known connection constants problem, to Lagrange inversion formula
and to solving some recurrence relations
Symbolic Calculus in Mathematical Statistics: A Review
In the last ten years, the employment of symbolic methods has substantially
extended both the theory and the applications of statistics and probability.
This survey reviews the development of a symbolic technique arising from
classical umbral calculus, as introduced by Rota and Taylor in The
usefulness of this symbolic technique is twofold. The first is to show how new
algebraic identities drive in discovering insights among topics apparently very
far from each other and related to probability and statistics. One of the main
tools is a formal generalization of the convolution of identical probability
distributions, which allows us to employ compound Poisson random variables in
various topics that are only somewhat interrelated. Having got a different and
deeper viewpoint, the second goal is to show how to set up algorithmic
processes performing efficiently algebraic calculations. In particular, the
challenge of finding these symbolic procedures should lead to a new method, and
it poses new problems involving both computational and conceptual issues.
Evidence of efficiency in applying this symbolic method will be shown within
statistical inference, parameter estimation, L\'evy processes, and, more
generally, problems involving multivariate functions. The symbolic
representation of Sheffer polynomial sequences allows us to carry out a
unifying theory of classical, Boolean and free cumulants. Recent connections
within random matrices have extended the applications of the symbolic method.Comment: 72 page
Polinómios de Appell multidimensionais e sua representação matricial
Doutoramento em MatemáticaNesta dissertação é apresentada uma abordagem a polinómios de Appell
multidimensionais dando-se especial relevância à estrutura da sua função
geradora. Esta estrutura, conjugada com uma escolha adequada de ordenação
dos monómios que figuram nos polinómios, confere um carácter unificador Ã
abordagem e possibilita uma representação matricial de polinómios de Appell
por meio de matrizes particionadas em blocos. Tais matrizes são construÃdas a
partir de uma matriz de estrutura simples, designada matriz de criação,
subdiagonal e cujas entradas não nulas são os sucessivos números naturais.
A exponencial desta matriz é a conhecida matriz de Pascal, triangular inferior,
onde figuram os números binomiais que fazem parte integrante dos
coeficientes dos polinómios de Appell.
Finalmente, aplica-se a abordagem apresentada a polinómios de Appell
definidos no contexto da Análise de Clifford.In this thesis an approach to multidimensional Appell polynomials is presented
with special relevance for the structure of their generating function. This
structure, together with an adequate choice of an ordering for the monomials
that are present in the polynomials, gives a unifying nature to our approach and
allows the representation of Appell polynomials by means of block matrices.
Such matrices are constructed from another matrix with simple structure, called
creation matrix, which is a sub-diagonal matrices whose nonzero entries are
the successive natural numbers. The exponential of this matrix is the well
known lower triangular Pascal matrix, lower triangular, where the binomial
numbers appear as part of the coefficients of Appell polynomials.
Finally, the presented approach is applied to Appell polynomials defined in the
context of Clifford Analysis