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\vskip 1cm{\LARGE\bf 
Linear Recurrences for $r$-Bell Polynomials
}
\vskip 1cm
\large
Miloud Mihoubi\footnote{This research is supported by the PNR project
8/u160/3172.} and Hac\`{e}ne Belbachir\footnote[2]{The research is partially
supported by the LITIS Laboratory of Rouen University and the PNR project
8/u160/722.}\\
USTHB, Faculty of Mathematics \\
RECITS Laboratory,  DG-RSDT \\
P. O. Box 32 \\ 
16111 El-Alia \\
Bab-Ezzouar \\
Algiers \\
Algeria\\
\href{mailto:mmihoubi@usthb.dz}{\tt mmihoubi@usthb.dz} \\
\href{mailto:miloudmihoubi@gmail.com}{\tt miloudmihoubi@gmail.com} \\
\href{mailto:hbelbachir@usthb.dz}{\tt hbelbachir@usthb.dz} \\
\href{mailto:hacenebelbachir@gmail.com}{\tt hacenebelbachir@gmail.com} \\
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\begin{abstract}
Letting\textbf{\ }$B_{n,r}$ be the $n$-$th$ $r$-Bell polynomial, it is well known
that $B_{n}(x)$ admits\ specific integer coordinates in the 
two bases $ \{x^{i}\}  _{i}$\ and $ \{  xB_{i}(
x )    \}  _{i}$ according to, respectively, the Stirling numbers and the
binomial coefficients. Our aim is to prove that the sequences $B_{n+m,r}
 (  x )  $ and $B_{n,r+s} (  x )  $ admit a binomial recurrence coefficient
 in different bases of the
$\mathbb{Q}$-vector space formed by polynomials of $\mathbb{Q} [X].$
\end{abstract}

\section{Introduction}

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