Proof of a conjecture on unimodality

Details Proof of a conjecture on unimodality Proof of a conjecture on unimodality

2008-09-09

arxiv.org/abs/0809.1586v1

European J. Combin. 26 (2005) 617--627

Mathematics

Combinatorics

11 pages

Scientific paper

Let $P(x)$ be a polynomial of degree $m$, with nonnegative and non-decreasing coefficients. We settle the conjecture that for any positive real number $d$, the coefficients of $P(x+d)$ form a unimodal sequence, of which the special case $d$ being a positive integer has already been asserted in a previous work. Further, we explore the location of modes of $P(x+d)$ and present some sufficient conditions on $m$ and $d$ for which $P(x+d)$ has the unique mode $\lceil{m-d\over d+1}\rceil$.

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