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Daniel E. Loeb

Publications and source records attributed to Daniel E. Loeb.

16 recordsLinked to original sources
arXiv1995

Recent contributions to the calculus of finite differences: a survey

We retrace the recent history of the Umbral Calculus. After studying the classic results concerning polynomial sequences of binomial type, we generalize to a certain type of logarithmic series. Finally, we demonstrate numerous typical examples of our theory. Nous passons en revue ici les resultats recents du calcul ombral. Nous nous interessons tout d'abord aux resultats classique appliqués aux suites de polynômes de type binomial, pius elargions le champ d'étude aux series logarithmiques. Enfin nous donnons de nombreaux exemples types d'application de cette théorie.

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arXiv1995

DX-operator expansion

We characterize those linear operators that can be expressed as a sum over k of terms of the form f_k(D) x^k and give several examples.

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arXiv1995

Maple umbral calculus package

We are developing a Maple package of functions related to Rota's Umbral Calculus. A Mathematica version of this package is being developed in parallel.

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arXiv1995

Getting results with negative thinking

Given a universe of discourse $U$, a {\em multiset} can be thought of as a function $M$ from $U$ to the natural numbers ${\bf N}$. In this paper, we define a {\em hybrid set} to be any function from the universe $U$ to the integers ${\bf Z}$. These sets are called hybrid since they contain elements with either a positive or negative multiplicity. Our goal is to use these hybrid sets {\em as if} they were multisets in order to adequately generalize certain combinatorial facts which are true classically only for nonnegative integers.

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arXiv1995

A simpler characterization of Sheffer polynomial

We characterize the Sheffer sequences by a single convolution identity $$ F^{(y)} p_{n}(x) = \sum _{k=0}^{n}\ p_{k}(x)\ p_{n-k}(y)$$ where $F^{(y)}$ is a shift-invariant operator. We then study a generalization of the notion of Sheffer sequences by removing the requirement that $F^{(y)}$ be shift-invariant. All these solutions can then be interpreted as cocommutative coalgebras. We also show the connection with generalized translation operators as introduced by Delsarte. Finally, we apply the same convolution to symmetric functions where we find that the ``Sheffer'' sequences differ from ordinary full divided power sequences by only a constant factor.

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arXiv1995

Series with general exponents

We define the Artinian and Noetherian algebra which consist of formal series involving exponents which are not necessarily integers. All of the usual operations are defined here and characterized. As an application, we compute the algebra of symmetric functions with nonnegative real exponents. The applications to logarithmic series and the Umbral calculus are deferred to another paper. On définit ici les algèbres Artinienne et Noetherienne comme étant des algèbres constituées des séries formelles à exposants pas nécessairement entiers. On definit sur ces algèbres toutes les opérations classiques et on les caracterise. Comme exemple d'exploitation de cette théorie, on s'interesse à algèbre de fonctions symétriques &agrave exponsants rèels en nonnégatifs. Une autre publication est consacrée aux applications aux series logarithmiques et au calcul ombral.

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arXiv1995

A generalization of Stirling numbers

We generalize the Stirling numbers of the first kind $s(a,k)$ to the case where $a$ may be an arbitrary real number. In particular, we study the case in which $a$ is an integer. There, we discover new combinatorial properties held by the classical Stirling numbers, and analogous properties held by the Stirling numbers $s(n,k)$ with $n$ a negative integer. On généralise ici les nombres de Stirling du premier ordre $s(a,k)$ au cas où $a$ est un réel quelconque. On s'interesse en particulier au cas où $a$ est entier. Ceci permet de mettre en evidence de nouvelles propriétés combinatoires aux quelles obeissent les nombres de Stirling usuels et des propriétés analougues auquelles obeissent les nombres de Stirling $s(n,k)$ où $n$ est un entier nègatif.

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arXiv1995

A generalization of the binomial coefficients

We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici à déterminer est la meilleure généralisation possible des factorielles et des coefficients du binôome. On s'interesse à plusieurs exemples, à leurs propriétés combinatoires, et aux differentes relations qu'ils mettent en jeu.

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arXiv1995

The iterated logarithmic algebra

We generalize the Umbral Calculus of G-C. Rota by studying not only sequences of polynomials and inverse power series, or even the logarithms studied in, but instead we study sequences of formal expressions involving the iterated logarithms and x to an arbitrary real power. Using a theory of formal power series with real exponents, and a more general definition of factorial, binomial coefficient, and Stirling numbers to all the real numbers, we define the Iterated Logarithmic Algebra I. Its elements are the formal representations of the asymptotic expansions of a large class of real functions, and we define the harmonic logarithm basis of I which will be interpreted as a generalization of the powers x^n since it behaves nicely with respect to the derivative We classify all operators over I which commute with the derivative (classically these are known as shift-invariant operators), and formulate several equivalent definitions of a sequence of binomial type. We then derive many formulas useful towards the calculation of these sequences including the Recurrence Formula, the Transfer Formula, and the Lagrange Inversion Formula. Finally, we study Sheffer sequences, and give many examples.

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arXiv1995

The iterated logarithmic algebra II: Sheffer sequences

An extension of the theory of the Iterated Logarithmic Algebra gives the logarithmic analog of a Sheffer or Appell sequence of polynomials. This leads to several examples including Stirling's formula and a logarithmic version of the Euler-MacLaurin summation formula. Grâce à une généralisation de la théorie de l'algèbre des logarithmes itérés, on definit un analogue logarithmique des suites de polynômes de Sheffer et d'Appell. Quelques exemples d'applications permettent de déduire la formule de Stirling ainsi qu'un version logarithmique de la formule de sommation de Euler--MacLaurin.

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arXiv1995

Sequences of symmetric functions of binomial type

We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the results of Umbral Calculus in this context including a Taylor's expansion and a binomial identity for symmetric functions. Surprisingly, the delta operators for all the sequences of binomial type correspond to the same operator on symmetric functions. On s'appuie ici sur les interprétations combinatoires de nombreuses suites de polynômes de type binomial pour définir une suite de fonctions symétriques associée à chque suite de polynômes de type binomial. On retrouve dans ce cadre, de nombreaux résultats du calcul ombral, en particulier une version de la formule de Taylor et la formule d'identité du binôme pour les fonctions symétriques. On s'aper\oit que les opérateurs differentiels de degré un pour toutes les suite de polynômes de type a binomial correspondent à un opérateur unique sur les fonction symétriques.

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arXiv1995

Richman games

A Richman game is a combinatorial game in which, rather than alternating moves, the two players bid for the privilege of making the next move. We consider both the case where the players pay each other and the case where the players pay a neutral third party. We find optimal strategies considering both the case where the players know how much money their opponent has and the case where they do not.

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arXiv1995

A new proof of Monjardet's median theorem

New proofs are given for Monjardet's theorem that all strong simple games (i.e., ipsodual elements of the free distributive lattice) can be generated by the median operation. Tighter limits are placed on the number of iterations necessary. Comparison is drawn with the $χ$ function which also generates all strong simple games.

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arXiv1995

Symmetric chain decompositions of B_n and Pi_n

We review the Green/Kleitman/Leeb interpretation of de Bruijn's symmetric chain decomposition of ${\cal B}_{n}$, and explain how it can be used to find a maximal collection of disjoint symmetric chains in the nonsymmetric lattice of partitions of a set.

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arXiv1995

The combinatorics of Mancala-type games: Ayo, Tchoukaitlon, and 1/pi

Certain endgame considerations in the two-player Nigerian Mancala-type game Ayo can be identified with the problem of finding winning positions in the solitaire game Tchoukaitlon. The periodicity of the pit occupancies in $s$ stone winning positions is determined. Given $n$ pits, the number of stones in a winning position is found to be asymptotically bounded by $n^{2}/π$.

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