The \(\alpha\)-analogues of r-Whitney Numbers via Normal Ordering

The normal ordering of an integral power of the number operator a\(^\dagger\)a in terms of boson annihilation a and creation a\(^\dagger\) operators is expressed with the help of the Stirling numbers of the second kind. The normal ordering problems directly links the problems to combinatorics. Whit this in mind, in this paper, we define the \(\alpha\)-analogues of r-Whitney numbers of the first kind and those of second kind, which are different from degenerate r-Whitney numbers. We show that \(\alpha\)-analogues falling factorial of the number operator is expressed in terms of the \(\alpha\)-analogues of r-Whitney numbers of the first kind and its inverse formula is expressed as those of the second kind. We also derived some properties, recurrence relations and several identities on those numbers arising from Boson annihilation a and creation operators a\(^\dagger\), number operators \(\hat{h}\) and coherent states.