Abstract
We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.
Highlights
A sparse polynomial is usually defined to be a polynomial in which the number of nonzero coefficients is small compared with its degree, where it often depends on the context what “small” means
As far as notation is concerned, sparse polynomials are normally given through its nonzero coefficients: n f (z) = cj zαj, j=0
Sparse polynomials are applied in computer algebra, cryptography, approximation and interpolation, and various areas of applied mathematics
Summary
A sparse polynomial (in one variable) is usually defined to be a polynomial in which the number of nonzero coefficients is small compared with its degree, where it often depends on the context what “small” means. The starting point of this paper is a specific sequence of sparse polynomials which arises naturally from a graph theoretic question related to the expected number of independent sets of a graph [2]. Where we will usually consider the integer m ≥ 1 as a fixed parameter, and study the sequence (fm,n(z))n≥0. It is clear from (1.2) that f1,n(z) = (1 + z)n and f2,n(z) = fn(z). We conclude this paper with some remarks and conjectures on the real roots of fm,n(z)
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