On a sequence of sparse binomial-type polynomials

Abstract

a b s t r a c tThe sequence of sparse polynomials  n k=0  n k  z k(k−1)/2 arises naturally in a graph theoretic setting. Here we study these polynomials in their own right; we derive some identities and study both algebraic and analytic properties, among them the distribution of real and complex zeros.

On a conjecture on sparse binomial-type polynomials by Brown, Dilcher and Manna

“…Moreover, those polynomials may also be considered to be interesting because of their close connection with the Jacobi Theta functions investigated in [4]. In the paper [2] the authors prove algebraic and analyic properties, and in addition to those, in [1] they obtain upper and lower bounds for the values of these polynomials for 0 < z < 1. Furthermore, in [1] the asymptotic relation log f n 1 y ∼ 1 2 log y log 2 n, n → ∞, (1.2) has been established.…”

Divisibility and Arithmetic Properties of a Class of Sparse Polynomials

“…In [3] it was shown that for any integer k ≥ 1, the polynomial f 2,2k+1 (z) is divisible by z k + 1. This gives rise to the question whether there are similar divisibility results for polynomials f m,n (z) with other parameters m. Computations indicate that this is indeed the case when m is a power of 2, with certain additional restrictions.…”

“…Proof of Proposition 3.1. We use the basic idea of the proof of Proposition 2.1 in [3], which is actually our case µ = 1. Using the definition (1.2), we have…”

“…arises naturally from a graph theoretic question related to the expected number of independent sets of a graph [2]. Various properties, including asymptotics, zero distribution, and arithmetic properties, can be found in [1], [2], [3], and [9]. More recently, in [4], we extended the polynomials in (1.1) by introducing the class of polynomials where we typically fix the integer parameter m ≥ 1 and consider the sequence (f m,n (z)) n ; obviously f 2,n (z) = f n (z).…”

Some Properties of a Class of Sparse Polynomials

“…In [3] it was shown that the roots of each polynomial f n (z), as defined by (1.1), all lie in the annulus…”