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

Abstract

In 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 t n 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.