User:Mmhassany/sandbox

Mehdi Hassani (born on July 23, 1979) in Zanjan) is an Iranian mathematician, specializing in analytic number theory. He is an associate professor at the University of Zanjan.

Education
Hassani received his PhD in 2010 at the Institute for Advanced Studies in Basic Sciences and University of Bordeaux.

Contributions
In 2003 he showed that the number of derangements of an n-Element Set is given by


 * $$ d_n = \begin{cases}

\lfloor \frac{n!}{e} + r_1 \rfloor, & n \text{ is odd} , \quad \  r_1 \in [0, \frac{1}{2}], \\ \lfloor \frac{n!}{e} + r_2 \rfloor, & n \text{ is even} , \quad r_2 \in [\frac{1}{3}, 1], \end{cases}$$

where $$\left\lfloor x \right\rfloor$$ is the floor function. So, for any integer $n &ge; 1$, and for any real number $r ∈ [1⁄3, 1⁄2]$,
 * $$ d_n = \left\lfloor \frac{n!}{e} + r \right\rfloor, \quad \ n \ge 1 , \quad  r \in \left[\frac{1}{3}, \frac{1}{2}\right]. $$

Therefore, as $e = 2.71828…$, $1⁄e ∈ [1⁄3, 1⁄2]$, then
 * $$ d_n = \left\lfloor \frac{n!+1}{e} \right\rfloor , \quad \ n \ge 1. $$

Also, he proved that


 * $$ d_n = \left\lfloor(e+e^{-1})n!\right\rfloor-\lfloor en!\rfloor, \quad n\geq 2.$$

In 2020 he found an asymptotic expansion for the number of derangements in terms of Bell numbers. Given any positive integer $$m$$ he proved that


 * $$ d_n = \frac{n!}{e}+\sum_{k=1}^m \left(-1\right)^{n+k-1}\frac{B_k}{n^k}+O\left(\frac{1}{n^{m+1}}\right),$$

where $$B_k$$ denotes the $$k$$-th Bell number and the constant of $$O$$-term does not exceed $$B_{m+1}$$.

Category:20th-century Iranian mathematicians Category:21st-century Iranian mathematicians Category:Living people Category:1979 births