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 appliqu\'es aux suites de polyn\^omes de type binomial, pius elargions le champ d'\'etude aux series logarithmiques. Enfin nous donnons de nombreaux exemples types d'application de cette th\'eorie.

We apply the invariant umbral calculus to obtain a coordinate-free approach to multivariate exponential families.

There are several applications of maximal intersecting families (MIFs) and different notions of fairness. We survey known results regarding the enumeration of MIFs, and we conclude the enumeration of the 207,650,662,008 maximal families of intersecting subsets of X whose group of symmetries is transitive for |X|<13.

ARichman gameis a combinatorial game in which, rather than alternating moves, the two players bid for the privilege of making the next move. The theory of such games is a hybrid between the classical theory of games (von Neumann, Morgenstern, …) and the combinatorial theory of games (Berlekamp, Conway, Guy, …). We expand upon our previous work by considering games with infinitely many positions as well as several variants including thePoorman variantin which the high bidder pays a third party (rather than the other player).Journal of Economic LiteratureClassification Number: C7.

Umbral calculus in its modern form [28, 46] is a powerful tool for calculations with polynomials. Applications of the umbral calculus include combinatorics (e.g. [14, 16, 22, 29, 31, 36, 39, 40, 42, 45, 50, 55]), special function theory [10, 21], approximation theory [17, 19, 26, 47], statistics (e.g. [7, 12, 30, 32]), probability theory (e.g. [6, 49, 51]), topology (e.g. [35, 37, 38]), and physics (e.g. [3, 4, 57]).

We use the Umbral Calculus to investigate the relation between natural exponential families and Sheffer polynomials. As a corollary, we obtain a new transparent proof of Feinsilver's theorem which says that natural exponential families have a quadratic variance function if and only if their associated Sheffer polynomials are orthogonal.

We prove the following conjecture of Narayana: there are no nontrivial dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime. We also count the number of natural significance levels of the Smirnov two-sample test in terms of the sample sizes and relate this to the Narayana conjecture. In particular, Smirnov tests with relatively prime sample sizes turn out to have many more natural significance levels than do Smirnov tests whose sample sizes are not relatively prime (for example, equal sample sizes).

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.

We introduce the notion of a stable winning coalition in a multiplayer game as a new system of classification of games. An axiomatic refinement of this classification for three-player games is also presented. These classifications are compared in light of a probabilistic model and the existing literature.

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.

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.

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.

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 $n^{2}/\pi$.

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.

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.

We prove the following conjecture of Narayana: there are no dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime.

We generalize the Umbral Calculus of G.-C. Rota (Adv. in Math.27, 1978, 95–188) by studying not only sequences of polynomials and inverse power series, or even the logarithms studied by D. Loeb and G.-C. Rota (Adv. in Math.75, 1989, 1–118), 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 J. 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 J which will be interpreted as a generalization of the powers xn since it behaves nicely with respect to the derivative. We classify all operators over J which commute with the derivative (classically there 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. In the sequel [22], we study Sheffer sequences and give many examples.

An extension of the theory of the Iterated Logarithmic Algebra [1] 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.