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Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$). ...

Let $B(n,x) = \sum_{k=0}^n {n\brace k}x^k$ be the Bell polynomials and $B_n = B(n,1)$ be the Bell numbers. I recently proved a nice relation between the two: $$ B(n,x)^{1/n}/x \ge B_{n/x}^{x/n}, $$ ...

Recall $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ and the Gaussian polynomial $\binom{a}{b}_q=\frac{[a]!_q}{[b]!_q[a-b]!_q}$ with $[0]!_q:=1$. Given two polynomials $U(q)=\sum_k\alpha_kq^...

On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$. One may ...