Тне asymptotic of the Bell‘s numbers sequence

V. E. Firstov

Saratov State University named after N. G. Chernyshevsky

Abstract: Bell`s numbers $B(s)$ defines the amount partitions of $s$-element set and with growth $s$ they have an exponentiale growth. That`s why the asymptotic`s investigation $s >>1$ of sequence $\{B(s)\}$ of Bell`s numbers $B(s)$ becomes actual, for example, if do the following combinatorial sum. Let`s take a discrete space of elementary event containing $s$ points with given law of probability distribution $p_1;…;p_s$, $p_1+\ldots+p_s=1$. On configurations of partitions one should define such a partition at which minimum of informational Shanon`s entropy is gained. One can face with this problem when the optimization of block-control of difficult cybernetic systems is present.
In this work some asymptotic properties of sequence of Bell`s numbers are considered. The main result of work represents the correlation:
$\lim\limits_{s\to\infty}\dfrac{B(s)B(s+2)}{B^2(s+1)}=1$, where $B(s)$; $B(s+1)$; $B(s+2)$ — Bell`s numbers with numerals $s$; $s+1$; $s+2$. This result shows that asymptotical sequence of Bell`s numbers behaved themselves geometrical progression with denominator $x*= B(s+1) / B(s)$. In the frames of additive presentation of Bell`s numbers with the help of Stirling`s numders the asymptotics is set up
$B(s) ~ St(s; n*) ~(n^*)^s/(n^*)! $, where $n*= [x*]$. Thus, a new class of sequences is up, the topology of which is characterized by the asymptotics in the form of the geometrical progression. Thus, a new class of sequences is established, the topology of wich is characterized by asymptotics in the form of geometrical progression.

Keywords: Bell's numbers, course of value function, saddle-point method, Stirlig`s numbers, asymptotic sequence.

UDC: 519.6

Received: 09.02.2014