Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

Eulerian polynomials and Quasi-Birth-Death processes with time-varying-periodic rates

A Quasi-Birth-Death (QBD) process is a stochastic process with a two dimensional state space, a level and a phase. An ergodic QBD with time-varying periodic transition rates will tend to an asymptotic periodic solution as time tends to infinity . Such QBDs are also asymptotically geometric. That is, as the level tends to infinity, the probability of the system being in state ( k , j ) (k,j) at time t t within the period tends to an expression of the form f j ( t ) α − k Π j ( k ) f_j(t)\alpha ^{-k}\Pi _j(k) where α \alpha is the smallest root of the determinant of a generating function related to the generating function for the unbounded (in the level) process, Π j ( k ) \Pi _j(k) is a polynomial in k k , the level, that may depend on j j , the phase of the process, and f j ( t ) f_j(t) is a periodic function of time within the period which may also depend on the phase. These solutions are analogous to steady state solutions for QBDs with constant transition rates. If the time within the period is considered to be part of the state of the process, then they are steady-state solutions. In this paper, we consider the example of a two-priority queueing process with finite buffer for class-2 customers. For this example, we provide explicit results up to an integral in terms of the idle probability of the queue. We also use this asymptotic approach to provide exact solutions (up to an integral equation involving the probability the system is in level zero) for some of the level probabilities.

Eulerian polynomials via the Weyl algebra action

Through the action of the Weyl algebra on the geometric series, we establish a generalization of the Worpitzky identity and new recursive formulae for a family of polynomials including the classical Eulerian polynomials. We obtain an extension of the Dobiński formula for the sum of rook numbers of a Young diagram by replacing the geometric series with the exponential series. Also, by replacing the derivative operator with the q-derivative operator, we extend these results to the q-analogue setting including the q-hit numbers. Finally, a combinatorial description and a proof of the symmetry of a family of polynomials introduced by one of the authors are provided.

Convolutions on the complex torus

“Quasi-elliptic” functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials closely tied to the Eulerian polynomials.