Factorials and Stirling numbers in the algebra of formal Laurent series II: za−zb=t

AbstractIn Part I, Stirling numbers of both kinds were used to define a binomial (Laurent) series of integer degree in the formal variable x. The binomial series in turn served as coefficient of tn in a formal series that reasonably well reflects the properties of (1+t)x. Analogously, generalized Stirling numbers (like central factorial numbers) are now used to define a kind of generalized Catalan series. By a different method, the Catalan series can be shown to generate a formal series that has the properties of z(t)x, where z(t)a−z(t)b=t. As in the case of ordinary Stirling numbers, not all the necessary coefficients can be described by generalized Stirling numbers alone. But they can all be explicitly expressed as an ordinary double sum of powers and factorials

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