On a Conjecture on Unimodal Sequences Gholamhassan Shirdel, Maryam Sarabadan

# On a Conjecture on Unimodal SequencesGholamhassan Shirdel, Maryam Sarabadan

Abstract:  Wang and Yeh proved that if $P(x)$ is a polynomial with non-negative and nondecreasing coefficients, then $P(x + d)$ is unimodal for any $d > 0$. A mode of a unimodal polynomial $f(x) = a_0 + a_1x +\cdots+a_mx^m$ is an index $k$ such that $a_k$ is the maximum coefficient. Suppose that $M_*(P, d)$ is the smallest mode of $P(x + d)$, and $M^*(P, d)$ the greatest mode. Wang and Yeh conjectured that if $d_2 > d_1 > 0$, then $M_*(P, d_1)\geq M_*(P, d_2)$ and $M^*(P, d_1) \geq M^*(P, d_2)$. This conjecture has already been proved in [7] but we give a different proof of this conjecture. We also show that if $\{d_j:\,\,0\leq j\leq m\}$ is a unimodal sequence, then there is a polynomial $p(x)=\sum_{i=0}^m a_ix^i$ with nonnegative and non-decreasing coefficients such that $p(x+n)=\sum_{j=0}^m d_jx^j$, where $n$ is a positive integer number. Furthermore, we define the almost unimodal sequences and prove that under some conditions the polynomial $p(x^k+d)$ for any positive real number $d$ and integer number $k\geq2$, is almost unimodal.

Keywords: Almost unimodality; unimodality; unimodal polynomial; mode.