We solve the boson normal ordering problem for (q(a)a + v(a)) with arbitrary functions q and v and integer n, where a and a are boson annihilation and creation operators, satisfying [a, a] = 1. This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state ...

In this paper, by using the p-adic q-integral on Zp which was defined by Kim, we define the weighted Carlitz q-Bernoulli polynomials and investigate some identities of these polynomials. In particular, we define the weighted degenerate Carlitz’s q-Bernoulli polynomials and numbers and give some interesting properties that are associated with these numbers and polynomials. AMS subject classifica...

By using the fermionic p-adic q-Volkenborn integral, we construct generating functions of higher-order (h, q)-extension of Euler polynomials and numbers. By using these numbers and polynomials, we give new approach to the complete sums of products of (h, q)-extension of Euler polynomials and numbers one of which is given by the following form: are the multinomial coefficients and E (h) m,q (y) ...

School of Mathematical and Statistical Sciences, Arizona State University Abstract. Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings on [2n] := {1, 2, . . . , 2n} using the cyclic group C2n generated by a cyclic shift of order 2n and the q-Cata...

The paper applies the pseudo-linear algebra to unify the results on reducibility, reduction and transfer equivalence for continuousand discrete-time nonlinear control systems. The necessary and sufficient condition for reducibility of nonlinear input-output equation is presented in terms of the greatest common left factor of two polynomials describing the behaviour of the ‘tangent linearized sy...

We solve the boson normal ordering problem for ( q(a†)a+ v(a†) )n with arbitrary functions q(x) and v(x) and integer n, where a and a† are boson annihilation and creation operators, satisfying [a, a†] = 1. This consequently provides the solution for the exponential e †)a+v(a†)) generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle...

Abstract In this paper we observe the behavior of complex roots of q-Euler numbers and polynomials Ẽ n,q (x) with weak weight α, using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of q-Euler numbers and polynomials Ẽ n,q (x) with weak weight α. Finally, we give a table for the solutions of q-Euler numbers and poly...

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges. Kergin interpolation is a generalization of both Lagrange interpolation in the one dimensional case, and the Taylor polynomial in the case where all interpolation points coincide. In several vari...