Central and local limit theorems for the coefficients of polynomials of binomial type

AbstractWe introduce the problem of establishing a central limit theorem for the coefficients of a sequence of polynomials Pn(x) of binomial type; that is, a sequence Pn(x) satisfying exp(xg(u)) = ∑n=0∞ Pn(x)(unn!) for some (formal) power series g(u) lacking constant term. We give a complete answer in the case when g(u) is a polynomial, and point out the widest known class of nonpolynomial power series g(u) for which the corresponding central limit theorem is known true. We also give the least restrictive conditions known for the coefficients of Pn(x) which permit passage from a central to a local limit theorem, as well as a simple criterion for the generating function g(u) which assures these conditions on the coefficients of Pn(x). The latter criterion is a new and general result concerning log concavity of doubly indexed sequences of numbers with combinatorial significance. Asymptotic formulas for the coefficients of Pn(x) are developed

Twisted modules for vertex operator algebras

This contribution is mainly based on joint papers with Lepowsky and Milas, and some parts of these papers are reproduced here. These papers further extended works by Lepowsky and by Milas. Following our joint papers, I explain the general principles of twisted modules for vertex operator algebras in their powerful formulation using formal series, and derive general relations satisfied by twisted and untwisted vertex operators. Using these, I prove new "equivalence" and "construction" theorems, identifying a set of sufficient conditions in order to have a twisted module for a vertex operator algebra, and a simple way of constructing the twisted vertex operator map. This essentially combines our general relations for twisted modules with ideas of Li (1996), who had obtained similar construction theorems using different relations. Then, I show how to apply these theorems in order to construct twisted modules for the Heisenberg vertex operator algebra. I obtain in a new way the explicit twisted vertex operator map, and in particular give a new derivation and expression for the formal operator $\Delta_x$ constructed some time ago by Frenkel, Lepowsky and Meurman. Finally, I reproduce parts of our joint papers. The untwisted relations in the Heisenberg vertex operator algebra are employed to explain properties of a certain central extension of a Lie algebra of differential operators on the circle, in relation to the Riemann Zeta function at negative integers. A family of representations for this algebra are constructed from twisted modules for the vertex operator algebra, and are related to the Bernoulli polynomials at rational values.Comment: 41 pages, contribution to proceedings of the workshop "Moonshine - the First Quarter Century and Beyond, a Workshop on the Moonshine Conjectures and Vertex Algebras" (Edinburgh, 2004) v2: 43 pages, presentation, discussion and proofs improve