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It is known that the Stirling numbers of the second kind are related to normal ordering in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. Recently, Kim-Kim introduced a λ\lambda -analogue of the unsigned Stirling numbers of the first kind and that of the rr-Stirling numbers of the first kind. In this article, we introduce a λ\lambda -analogue of the shift algebra (called λ\lambda -shift algebra) and investigate normal ordering in the λ\lambda -shift algebra. From the normal ordering in the λ\lambda -shift algebra, we derive some identities about the λ\lambda -analogue of the unsigned Stirling numbers of the first kind.

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a ‘degenerate version’ of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials.

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a 'degenerate version' of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials. [ABSTRACT FROM AUTHOR]

Abstract The observation of neutrinoless double beta decay would allow to establish lepton number violation and the Majorana nature of neutrinos. The rate of this process in the case of 3-neutrino mixing is controlled by the neutrinoless double beta decay effective Majorana mass | 〈 m 〉 |. For a neutrino mass spectrum with normal ordering, which is favoured over the spectrum with inverted ordering by recent global fits, | 〈 m 〉 | can be significantly suppressed. Taking into account updated data on the neutrino oscillation parameters, we investigate the conditions under which | 〈 m 〉 | in the case of spectrum with normal ordering exceeds 10 − 3 (5 × 10 − 3) eV : | 〈 m 〉 | NO > 10 − 3 (5 × 10 − 3) eV. We analyse first the generic case with unconstrained leptonic CP violation Majorana phases. We show, in particular, that if the sum of neutrino masses is found to satisfy Σ > 0.10 eV , then | 〈 m 〉 | NO > 5 × 10 − 3 eV for any values of the Majorana phases. We consider also cases where the values for these phases are either CP conserving or are in line with predictive schemes combining flavour and generalised CP symmetries. [ABSTRACT FROM AUTHOR]

We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for q -commuting variables x and y satisfying y x = q x y . In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we derive results for an elliptic derivative of elliptic commuting variables, and finally study weight-dependent extensions of the Weyl algebra which we connect to rook theory. [ABSTRACT FROM AUTHOR]

Dowling showed that the Whitney numbers of the first kind and of the second kind satisfy Stirling number-like relations. Recently, Kim-Kim introduced the degenerate r -Whitney numbers of the first kind and of the second kind, as degenerate versions and further generalizations of the Whitney numbers of both kinds. The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. In this paper, it is noted that the normal ordering of a certain quantity involving the number operator is expressed in terms of the degenerate r -Whitney numbers of the second kind. We derive some properties, recurrence relations, orthogonality relations and several identities on those numbers from such normal ordering. In addition, we consider the degenerate r -Dowling polynomials as a natural extension of the degenerate r -Whitney numbers of the second kind and investigate their properties. • Introduction of the degenerate r -Whitney numbers of the second kind and of the first kind. • The normal ordering of the quantity (m a + a + r) k , λ in terms of the degenerate r -Whitney numbers of the second kind. • The generating function of r -Dowling polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give a modified approach to the generalized Stirling numbers of the second kind and . These numbers were firstly defined by Carlitz and recently studied extensively by Blasiak, Penson and Solomon. This approach depends on the previous results obtained by Carlitz, Toscano and Cakić. We show that Blasiak’s results can be investigated from Carlitz and Cakić’s results. Some interesting combinatorial identities are obtained. [Copyright &y& Elsevier]

We investigate the combinatorics of the general formulas for the powers of the operator h ∂ d , where ∂ is a differential operator on an arbitrary noncommutative ring in which h is central. New formulas for the generalized Stirling numbers are obtained, as well as results on the divisibility by primes of the coefficients which occur in the normally ordered form of h ∂ d. All of the above applies to the theory of formal differential operator rings. [ABSTRACT FROM AUTHOR]