Laguerre-type derivatives: Dobiński relations and combinatorial identities

Penson, K. A.; Blasiak, P.; Horzela, A.; Duchamp, G. H. E. and Solomon, A. I. (2009). Laguerre-type derivatives: Dobiński relations and combinatorial identities. Journal of Mathematical Physics, 50(8) 083512.

DOI: https://doi.org/10.1063/1.3155380

Abstract

We consider properties of the operators D(r,M)=ar(a†a)M (which we call generalized Laguerre-type derivatives), with r=1,2,…, M=0,1,…, where a and a† are boson annihilation and creation operators, respectively, satisfying [a,a†]=1. We obtain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation that generalizes the Dobiński formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified

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