Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Formal calculus and umbral calculus. | Formal calculus and umbral calculus Thomas J. Robinson Department of Mathematics Rutgers University New Brunswick Piscataway USA thomasro@ Submitted Mar 12 2010 Accepted Jun 28 2010 Published Jul 10 2010 Mathematics Subject Classification 05A40 17B69 Abstract 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 Faa 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. 1 Introduction We present from first principles certain aspects of the classical umbral calculus concluding with a connection to the Virasoro algebra. One of our main purposes is to show connections between the classical umbral calculus and certain central considerations in vertex operator algebra theory. The first major connection is an analogue noted in FLM of those authors original argument showing that lattice vertex operators satisfy a certain fundamental .
