Extended Bernoulli and Stirling matrices and related combinatorial identities

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.Comment: Accepted for publication in Linear Algebra and its Application

Wick's theorem for q-deformed boson operators

In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated ``Feynman diagrams'' the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick's theorem in the present context.Comment: 9 page

Recursively minimally-deformed oscillators

A recursive deformation of the boson commutation relation is introduced. Each step consists of a minimal deformation of a commutator [a,\ad]=f_k(\cdots;\no) into [a,\ad]_{q_{k+1}}=f_k(\cdots;\no), where $\cdots$ stands for the set of deformation parameters that $f_k$ depends on, followed by a transformation into the commutator [a,\ad]=f_{k+1}(\cdots,\, q_{k+1};\no) to which the deformed commutator is equivalent within the Fock space. Starting from the harmonic oscillator commutation relation [a,\ad]=1 we obtain the Arik-Coon and the Macfarlane-Biedenharn oscillators at the first and second steps, respectively, followed by a sequence of multiparameter generalizations. Several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained. The ``generic'' form consists of a linear combination of exponentials of the number operator, and the various recursive families can be classified according to the number of free linear parameters involved, that depends on the form of the initial commutator.Comment: 19 pages, LateX, no figur