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Spivey's Bell number identity involving Stirling numbers of the second kind was derived by Spivey considering set partitions. Since its original derivation it has been derived (and generalized) by many authors using different techniques. In particular, Katriel gave a proof using normal ordering in the Weyl algebra. Mező derived the dual of Spivey's identitiy for factorials involving (unsigned) Stirling num- bers of the first kind considering permutations and cycles. The latter identity has also been been derived (and generalized) in different fashions. What is lacking in literature is a proof using normal ordering in the spirit of Katriel's proof of Spivey's identity. In the present work, this gap is filled by providing a new proof of the dual of Spivey's identity using normal ordering in the shift algebra. [ABSTRACT FROM AUTHOR]

In this paper, the bijection between k-rook placements on a Ferrers board and k-matchings in the associated bipartite graph is extended to a bijection between k-file placements and certain fractional matchings. Using the latter bijection, a new interpretation is given for the normal ordering coefficients in the shift algebra. Several further results concerning normal ordering in the shift algebra are derived. [ABSTRACT FROM AUTHOR]

The technique of the normal ordering of non-commuting operators is an important tool in the solution of problems involving creation and annihilation operators in quantum physics, such as in many-body theory or quantum optics. We point out the inconsistencies in previous definitions of the two standard normal ordering procedures for such operators, and show how consistent definitions may be made. [ABSTRACT FROM AUTHOR]