We use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a very short proof which naturally arose as a motivating computation related to a certain crucial "associativity" property of an important class of vertex operator algebras. Very similar (somewhat forgotten) proofs had appeared by the 19-th century, of course without any motivation related to vertex operator algebras. Using this formula, we derive certain results, including especially the calculation of certain adjoint operators, of the classical umbral calculus. This is, roughly speaking, a reversal of the logical development of some standard treatments, which have obtained formulas for the higher derivatives of a composite function, most notably Faà di Bruno's formula, as a consequence of umbral calculus. We also show a connection between the Virasoro algebra and the classical umbral shifts. This leads naturally to a more general class of operators, which we introduce, and which include the classical umbral shifts as a special case. We prove a few basic facts about these operators.
Abstract. We discuss certain aspects of the formal calculus used to describe vertex algebras. In the standard literature on formal calculus, the expression (x + y) n , where n is not necessarily a nonnegative integer, is defined as the formal Taylor series given by the binomial series in nonnegative powers of the second-listed variable (namely, y). We present a viewpoint that for some purposes of generalization of the formal calculus including and beyond "logarithmic formal calculus," it seems useful, using the formal Taylor theorem as a guide, to instead take as the definition of (x + y) n the formal series which is the result of acting on x n by a formal translation operator, a certain exponentiated derivation. These differing approaches are equivalent, and in the standard generality of formal calculus or logarithmic formal calculus there is no reason to prefer one approach over the other. However, using this second point of view, we may more easily, and in fact do, consider extensions in two directions, sometimes in conjunction. The first extension is to replace x n by more general objects such as the formal variable log x, which appears in the logarithmic formal calculus, and also, more interestingly, by iterated-logarithm expressions. The second extension is to replace the formal translation operator by a more general formal change of variable operator. In addition, we note some of the combinatorics underlying the formal calculus which we treat, and we end by briefly mentioning a connection to Faà di Bruno's classical formula for the higher derivatives of a composite function and the classical umbral calculus. Many of these results are extracted from more extensive papers [R1] and [R2], to appear. IntroductionOur subject is certain aspects of the formal calculus used, as presented in [FLM], to describe vertex algebras, although we do not treat any issues concerning "expansions of zero," which is at the heart of the subject. An important basic result which we describe in detail is the formal Taylor theorem and this along with some variations is the topic we mostly consider. It is well known, and we recall the simple argument below, that if we let x and y be independent formal variables, then the formal exponentiated derivation e
We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of logarithmic intertwining operators and logarithmic tensor category theory for modules for a vertex operator algebra. This extension has a variety of interesting arithmetic properties. We develop one such result here, the aforementioned recursive identity. We have applied this identity elsewhere to certain formal series expansions related to a general formal Taylor theorem and these series expansions in turn yield a sequence of combinatorial identities which have as special cases certain classical combinatorial identities involving (separately) the Stirling numbers of the first and second kinds.
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