Polynomial Triangles Revisited

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to fill this gap. We describe some aspects of these arrays, which generalize similar properties of the binomial coefficients. Some combinatorial models enumerated by polynomial coefficients, including lattice paths model, spin chain model and scores in a drawing game, are introduced. Several known binomial identities are then extended. In addition, we calculate recursively generating functions of column sequences. Interesting corollaries follow from these recurrence relations such as new formulae for the Fibonacci numbers and Hermite polynomials in terms of trinomial coefficients. Finally, properties of the entropy density function that characterizes polynomial coefficients in the thermodynamical limit are studied in details.Comment: 24 pages with 1 figure eps include

An Exclusion zone for Massive MIMO With Underlay D2D Communication

Fifth generation networks will incorporate a variety of new features in wireless networks such as data offloading, D2D communication, and Massive MIMO. Massive MIMO is specially appealing since it achieves huge gains while enabling simple processing like MRC receivers. It suffers, though, from a major shortcoming refereed to as pilot contamination. In this paper we propose a frame-work in which, a D2D underlaid Massive MIMO system is implemented and we will prove that this scheme can reduce the pilot contamination problem while enabling an optimization of the system spectral efficiency. The D2D communication will help maintain the network coverage while allowing a better channel estimation to be performed