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

Abstract

We prove a conjecture by Brown, Dilcher and Manna on the asymptotic behavior of sparse binomial-type polynomials arising naturally in a graph theoretical context in connection with the expected number of independent sets of a graph.

On the asymptotics of a sequence of lacunary binomial-type polynomials

“…it is seen that the leading terms of (3.7) agree with the result (1.3) (with y = x 2 ) obtained by Gawronski and Neuschel [3].…”

“…into (1.1), followed by an interchange in the order of summation and integration and use of the binomial theorem, we obtain the integral representation derived in [3] ℘ n (…”

“…Recently, Gawronski and Neuschel [3] considered an integral representation for ℘ n (z) and employed a path displacement argument in the complex plane combined with a non-standard version of the saddle-point method to obtain the asymptotic formula when y > 1 It was established in [3] that the expression in front of the curly braces in (1.3) is asymptotically equivalent to the conjectured approximation in (1.2) in the limit n → ∞.…”

“…In this paper, we consider the same integral representation for ℘ n (z) as derived in [3] and employ the method of steepest descents to determine a more accurate expansion for moderately large values of n. We also discuss the case of complex z satisfying |z| < 1. We present numerical computations to demonstrate the accuracy of our expansion and the approximation in (1.3).…”

Divisibility and Arithmetic Properties of a Class of Sparse Polynomials

“…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).…”