A New Approach to the -Whitney Numbers by Using Combinatorial Differential Calculus
arXiv:1702.06519
Abstract
In the present article we introduce two new combinatorial interpretations of the -Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar . By specializing we obtain also a new combinatorial interpretation of the -Stirling numbers of the second kind. Again, by specializing to the case we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard's. Moreover, we show several well-known identities involving the -Dowling polynomials and the -Whitney numbers using the combinatorial differential calculus. Finally we prove that the -Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce -Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities involving the -Whitney number of both kinds, Bernoulli and Euler polynomials.