Extended Bell and Stirling numbers from hypergeometric exponentiation

Abstract

Exponentiating the hypergeometric series gives a recursion relation for integer sequences which are generalizations of conventional Bell numbers. The corresponding associated Stirling numbers of the second kind are also generated and investigated. For the lowest order generalisation, one can give a combinatorial interpretation of these 'Bell' numbers, and of some Stirling numbers associated with them. We also consider these analogues of Bell numbers in the case of restricted partitions.