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

Zk(r)\mathbb{Z}_k^{(r)}-Algebras, FQH Ground States, and Invariants of Binary Forms

A prominent class of model FQH ground states is those realized as correlation function of $\mathbb{Z}_k^{(r)}$-algebras. In this paper, we study the interplay between these algebras and their corresponding wavefunctions. In the hopes of realizing these wavefunctions as a unique densest zero energy state, we propose a generalization for the projection Hamiltonians $\mathscr{H}_{k+1}^{2r-1}$. Finally, using techniques from invariants of binary forms, an ansatz for computation of correlations $\langle \psi(z_1)\cdots\psi(z_{2k})\rangle \prod_{i<j}(z_i-z_j)^{2r/k}$ is devised. We provide some evidence that, at least when $r=2$, our proposed Hamiltonian realizes $\mathbb{Z}_k^{(2)}$-wavefunctions as a \emph{unique} ground state