1202.0228 (N-E. Fahssi)

Polynomial Triangles Revisited    [PDF]

N-E. Fahssi

The theme of our study is extremely simple. It consists in investigating
several aspects of coefficients in integral powers of polynomials. These
coefficients generate an array, called polynomial triangle, whose $k$-th row
consists of the coefficients of the power of $t$ in $p(t)^k$, for a given
polynomial $p(t)$. The table naturally resembles Pascal's triangle and reduces
to it when $p(t)=1+t$. 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 a restricted occupancy
model (Gentile-type statistics) and a lattice paths model, are introduced.
Several known binomial identities are then extended. For instance, the symmetry
property of the binomial coefficients is generalized to an interesting identity
reflecting Particle-Hole duality. In addition, asymptotic of polynomial
coefficients in the thermodynamical limit is characterized by a function called
entropy density function. Properties of this function are studied in details.

View original: http://arxiv.org/abs/1202.0228