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\begin{center}
\vskip 1cm{\LARGE\bf
A $(p, q)$-Deformed Recurrence for \\
\vskip .1in
the Bell Numbers
}
\vskip 1cm
\large
Lahcen Oussi \\
Institute of Mathematics\\
University of Wroc{\l}aw\\
Pl.\ Grunwaldzki 2/4\\
50-384 Wroc{\l}aw\\
Poland \\
\href{mailto:lahcen.oussi@math.uni.wroc.pl}{\tt lahcen.oussi@math.uni.wroc.pl} \\
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\begin{abstract}
We obtain a $(p, q)$-deformation of the recurrence formula for the
Bell numbers, using algebraic techniques. Specializing to the case
$p=1$ and the case $p=q=1$, respectively, we recover the generalized
recurrence formula for Bell numbers as obtained by Katriel and Spivey,
in related papers.
\end{abstract}

\section{Introduction}

The Bell numbers, denoted by $B_{n}$, are given by the following formula:
\begin{equation}\label{cbn}
    B_{n}=\sum_{k=0}^{n}S(n, k),
\end{equation}
where $S(n, k)$ are the Stirling numbers of the second kind,
which appear as  coefficients in the expansion of
\begin{equation*}
x^n=\sum_{k=0}^{n}S(n, k)\prod_{i=0}^{k-1}(x-i).
\end{equation*}
The Stirling numbers of the second kind $S(n, k)$ count the number of ways to partition a set of size $n$ into $k$ nonempty subsets.

The Bell numbers \eqref{cbn} satisfy the following recursive formula:
\begin{equation}\label{crbn}
    B_{n+1}=\sum_{k=0}^{n}\binom{n}{k}B_{k}.
\end{equation}
In 2008, Spivey \cite{Sp2008} obtained the following generalization of the recurrence formula \eqref{crbn}:
\begin{equation}\label{Spformula}
B_{n+m}=\sum_{j=0}^{n}\sum_{k=0}^{m}k^{n-j}S(m, k)\binom{n}{j}B_{j},
\end{equation}
using techniques from combinatorics. This formula is known in the literature as ``Spivey's Bell number formula''. Subsequently, Katriel \cite{JK2008} proved a $q$-deformed version of Spivey's Bell number formula using algebraic techniques. The resulting formula can be written as follows:
\begin{equation}\label{JKformula}
B_{n+m}(q)=\sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}S_{q}(m, k)[k]_{q}^{n-j}q^{jk}B_{j}(q),
\end{equation}
where
\begin{equation}\label{qstirling}
  S_{q}(m, k)=q^{k-1}S_{q}(m-1, k-1)+[k]_{q}S_{q}(m-1, k),
\end{equation}
are the $q$-Stirling numbers of the second kind with the initial value $S_{q}(0, 0)=1$, and $[k]_{q}=\frac{1-q^k}{1-q}$. Here,
\begin{equation}\label{qBN}
B_{n}(q)=\sum_{k=0}^{n}S_{q}(n, k)
\end{equation}
are the $q$-Bell numbers.

An alternative derivation of \eqref{JKformula} was obtained by Mangontarum \cite{MMM2018}, using techniques based on the analysis of the creation, annihilation and number operators in the $q$-Boson-Fock space \cite{MADC}. It is worthwhile to mention that, Eq.~(\ref{cbn}) has also been extended in several ways by several authors, including various generalizations of Stirling and Bell numbers. We refer the reader to \cite{HBMM, HWGJQ, IM, AX} for the details.

In the present paper, we follow the techniques by Katriel \cite{JK2008}, and obtain a $(p,q)$-deformed version of Spivey's Bell number formula \eqref{Spformula}, using algebraic methods.

\section{A $(p, q)$-deformation of Spivey's Bell number formula}

For $0<q<p \leq 1$, define the following operators:
\begin{enumerate}
\item The operator $X$ of multiplication by the variable $x$:
\begin{equation*}
Xf(x)=xf(x).
\end{equation*}
\item The $(p, q)$-derivative operator:
\begin{equation*}
D_{p, q}f(x)=\frac{f(px)-f(qx)}{x(p-q)}.
\end{equation*}
\item The Fibonacci operator \cite{MH}:
\begin{equation*}
N_{p}f(x)=f(px).
\end{equation*}
\end{enumerate}
These operators satisfy the $(p, q)$-commutation relation:
\begin{equation}\label{eq1}
D_{p, q}X-qXD_{p, q}=N_{p}.
\end{equation}
We can check easily that
\begin{equation}\label{eq2}
N_{p}X=pXN_{p},
\end{equation}
and
\begin{equation}\label{eq2'}
D_{p, q}N_{p}=pN_{p}D_{p, q}.
\end{equation}
\begin{proposition}
For any positive integer $n$, we have
\begin{equation}\label{eq3}
D_{p,q}X^n=q^nX^nD_{p, q}+[n]_{p, q}X^{n-1}N_{p},
\end{equation}
where $[n]_{p, q}=\frac{p^n-q^n}{p-q}$.
\end{proposition}
\begin{proof}
The proof follows from an easy computation involving an induction on $n$, and using Eqs.~(\ref{eq1}) and \eqref{eq2}.
\end{proof}
\begin{corollary}\label{cor1}
One can rewrite \eqref{eq3} as follows:
\begin{equation}\label{eq4}
(XD_{p, q})X^n=X^n\bigl([n]_{p, q}N_{p}+q^n(XD_{p, q})\bigr).
\end{equation}
\end{corollary}
\begin{proposition}
Let $n$ be a nonnegative integer, then the following relation holds:
\begin{equation}\label{meq}
(XD_{p, q})^n=\sum_{k=0}^{n}S_{p, q}(n, k)X^kN_{p}^{n-k}D_{p, q}^k,
\end{equation}
where
\begin{equation}
\label{pqstr}
S_{p, q}(n, k)=p^{n-k}q^{k-1}S_{p, q}(n-1, k-1)+[k]_{p, q}S_{p, q}(n-1, k),
\end{equation}
are the $(p, q)$-Stirling numbers of the second  kind with the initial value $S_{p, q}(0, 0)=1$, and consequently the numbers
\begin{equation}
\label{pqbell}
B_{n}(p, q)=\sum_{k=0}^{n}S_{p, q}(n, k),
\end{equation}
can be considered as the $(p, q)$-Bell numbers.
\end{proposition}
\begin{proof}
 The proof follows by induction with respect to $n$, using Eqs.~(\ref{eq1}), \eqref{eq2'}, and \eqref{eq4}.
\end{proof}
\begin{remark}
For $p=1$, Eqs.~(\ref{pqstr}) and \eqref{pqbell} reduce to the $q$-Stirling numbers of the second kind  \eqref{qstirling} and $q$-Bell numbers \eqref{qBN}, respectively.
\end{remark}
\begin{definition}
The $(p, q)$-exponential function is defined by the formula:
\begin{equation*}
e_{p, q}(x):=\sum_{n=0}^{\infty}p^{\binom{n}{2}}\frac{x^n}{[n]_{p, q}\,!},
\end{equation*}
where $[n]_{p, q}\,!=\displaystyle{\prod_{i=1}^{n}[i]_{p, q}}$, and $[0]_{p, q}\,!=1$.
\end{definition}
Sadjang \cite{PNS} showed that the $(p, q)$-exponential function $e_{p, q}(x)$ satisfies the following differential equation:
\begin{equation}
\label{pqexpn}
 D_{p, q}^n e_{p, q}(x)=p^{\binom{n}{2}}e_{p, q}(p^nx),
\end{equation}
where $D_{p, q}^n$ is the $n$th $(p, q)$-derivative.

Applying the $(p, q)$-exponential function $e_{p, q}(x)$ to both sides of Eq.~(\ref{meq}), and dividing by $e_{p, q}(p^nx)$, we arrive at the identity:
\begin{equation}\label{maineq}
  \frac{1}{e_{p, q}(p^nx)}(XD_{p, q})^ne_{p, q}(x)=\sum_{k=0}^{n}\tilde{S}_{p, q}(n, k)x^k =\tilde{B}_{n}(p, q;x),
\end{equation}
where $\tilde{S}_{p, q}(n, k)=p^{\binom{k}{2}}S_{p, q}(n, k)$ can be considered as new versions of the $(p, q)$-Stirling numbers of the second kind, with $\tilde{B}_{n}(p, q; x)$ being the associated $(p,q)$-Bell polynomials, which give for $x=1$ the new version of the $(p, q)$-Bell numbers $\tilde{B}_{n}(p, q; 1)=\sum_{k=0}^{n}\tilde{S}_{p, q}(n, k)$. Moreover, putting $x=-1$ in the Equation \eqref{maineq}, we obtain $(p, q)$-R\'enyi numbers $R_{n}(p, q)=\sum_{k=0}^{n}(-1)^{k}\tilde{S}_{p, q}(n, k)$ which reduce to the usual $q$-R\'enyi number \cite{AEF} for $p=1$.

\section{Main result}
We are now in a position to prove the main result of this paper.
\begin{theorem}
Let $m$ and $n$ be nonnegative integers. Then we have the following $(p, q)$-deformation of Spivey's Bell number formula:
\begin{equation}
\label{lasteq}
\tilde{B}_{n+m}(p, q; 1)=\sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}\tilde{S}_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk}\tilde{B}_{j}(p, q; p^{n+m-j}).
\end{equation}
\end{theorem}
\begin{proof}
Using Eqs.~(\ref{meq}) and \eqref{eq4}, we obtain
\begin{align*}
(XD_{p, q})^{n+m}&=\sum_{k=0}^{m}S_{p, q}(m, k)(XD_{p, q})^n X^{k}N_{p}^{m-k}D_{p, q}^{k}\\
&=\sum_{k=0}^{m}S_{p, q}(m, k)X^{k}\left([k]_{p, q}N_{p}+q^{k}(XD_{p, q})\right)^{n}N_{p}^{m-k}D_{p, q}^{k}\\
&=\sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}S_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk}X^{k}N_{p}^{n-j}(XD_{p, q})^{j}N_{p}^{m-k}D_{p, q}^k,
\end{align*}
where in the last equality, we have used the binomial expansion.
Applying the above identity to the $(p, q)$-exponential function $e_{p, q}(x)$, and using Eqs.~(\ref{pqexpn}) and \eqref{maineq}, we get the following:
\begin{align*}
(XD_{p, q})^{n+m}e_{p, q}(x) &= \sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}S_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk} X^{k} N_{p}^{n-j} (XD_{p, q})^{j} N_{p}^{m-k}D_{p, q}^ke_{p, q}(x)\\
&= \sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}\tilde{S}_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk}X^{k}N_{p}^{n-j}(XD_{p, q})^{j}N_{p}^{m-k}e_{p, q}(p^kx)
\\
&= \sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}\tilde{S}_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk}X^{k}N_{p}^{n-j}(XD_{p, q})^{j}e_{p, q}(p^mx)\\
&= \sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}\tilde{S}_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk}X^{k}N_{p}^{n-j}\tilde{B}_{j}(p, q, p^mx)e_{p, q}(p^{m+j}x)\\
&= \sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}\tilde{S}_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk}x^{k}\tilde{B}_{j}(p, q; p^{n+m-j}x)e_{p, q}(p^{n+m}x).
\end{align*}
Dividing both sides of the above identity  by $e_{p, q}(p^{n+m}x)$ and setting $x=1$ gives
\begin{equation*}
\tilde{B}_{n+m}(p, q; 1)=\sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}\tilde{S}_{p, q}(m, k)[k]_{p, q}^{n-j}q^{jk}\tilde{B}_{j}(p, q; p^{n+m-j}).
\end{equation*}
\end{proof}
\begin{remark}
For $p=1$, Eq.~(\ref{lasteq}) reduces to the $q$-deformation of Spivey's Bell number formula \eqref{JKformula}, and for $p=q=1$, it reduces to Spivey's Bell number formula \eqref{Spformula}.
\end{remark}
\begin{proposition}[Dobinski formula]
The $(p, q)$-Bell polynomials $\tilde{B}_{n}(p, q; x)$, satisfy the following identity:
\begin{equation}\label{DF}
 \tilde{B}_{n}(p, q; x)=\frac{1}{e_{p, q}(p^nx)}\sum_{k=0}^{\infty}p^{\binom{k}{2}}\frac{[k]_{p, q}^{n}}{[k]_{p, q}\,!}x^k.
\end{equation}
Consequently, the $(p, q)$-Bell numbers $\tilde{B}_{n}(p, q; 1)$, are given by:
\begin{equation*}
 \tilde{B}_{n}(p, q; 1)=\frac{1}{e_{p, q}(p^n)}\sum_{k=0}^{\infty}p^{\binom{k}{2}}\frac{[k]_{p, q}^{n}}{[k]_{p, q}\,!}.
\end{equation*}
\end{proposition}
\begin{proof}
Since $(XD_{p, q})x^m=[m]_{p, q}x^m$, it follows that
\begin{equation*}
(XD_{p, q})^nx^m=[m]_{p, q}^{n}x^m,
\end{equation*}
and
\begin{equation}\label{XDn}
 (XD_{p, q})^{n}e_{p, q}(x)=\sum_{k=0}^{\infty}p^{\binom{k}{2}}\frac{[k]_{p, q}^{n}}{[k]_{p, q}\,!}x^k.
\end{equation}
Multiplying both sides of Eq.~(\ref{XDn}) with $\frac{1}{e_{p, q}(p^nx)}$ and taking into account Eq.~(\ref{maineq}) proves the desired result.
\end{proof}
\begin{remark}
When $p=1$, Eq.~(\ref{DF}) reduces to the Dobinski formula for $q$-Bell polynomials given by Katriel \cite{JK2008}. Furthermore, when $p=q=1$, we obtain the Dobinski formula for the ordinary Bell polynomials:
\begin{equation*}
B_{n}(x)=\frac{1}{e}\sum_{k\geq 0}\frac{k}{k\,!}x^k.
\end{equation*}
Therefore, we can think of Eq.~(\ref{DF}) as the Dobinski formula for $(p, q)$-Bell polynomials $\tilde{B}_{n}(p, q; x)$.
\end{remark}

\section{Final remarks}
It seems likely that following the techniques due to Mangontarum \cite{MMM2018}, one can obtain an alternative proof of \eqref{lasteq}, using the creation, annihilation and number operators in the $(p, q)$-Fock space \cite{NB}. Our speculation stems from the following observation:

Let $l^{2}(\mathbb{N})$ be a Hilbert space with the standard orthonormal basis $(\delta_{n})_{n=0}^{\infty}$. We define the creation, annihilation and number operators as follows:
\begin{enumerate}
\item The creation operator $a^+$ defined by
\begin{equation*}
a^+\delta_{n}=\sqrt{[n+1]_{p, q}}\cdot \delta_{n+1}, \quad n\geq 0.
\end{equation*}
\item The annihilation operator (adjoint of $a^+$) $a^-$ defined by
\begin{equation*}
a^-\delta_{0}=0, \quad a^-\delta_{n}=\sqrt{[n]_{p, q}} \cdot\delta_{n-1}, \quad n\geq 1.
\end{equation*}
\item The number operator $a^{\circ}$ defined by
\begin{equation*}
a^{\circ}\delta_{n}=p^n\delta_{n}, \quad n\geq 0.
\end{equation*}
\end{enumerate}
It follows by an easy computation that these operators satisfy the $(p, q)$-commutation relation
\begin{equation*}
a^-a^+-qa^+a^-=a^{\circ}.
\end{equation*}

\section{Acknowledgments}
Research partially supported by the National Agency for Academic Exchange (NAWA) POLONIUM project PPN/BIL/2018/1/00197/U/00021 and by the Polish National Science Center (NCN) grant 2016/21/B/ST1/00628.

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\end{thebibliography}


\bigskip
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\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B73;  Secondary 11B83, 05A19, 05A30.

\noindent \emph{Keywords: }
$(p, q)$-Bell number, $(p, q)$-Stirling number, Spivey's Bell number formula, $(p, q)$-Fock space, $(p, q)$-calculus.

\bigskip
\hrule
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\noindent (Concerned with sequences
\seqnum{A000110} and \seqnum{A008277}.)

\bigskip
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\vspace*{+.1in}
\noindent
Received December 4 2019;
revised versions received  December 13 2019; February 7 2020; 
March 25 2020; May 4 2020; May 6 2020.
Published in {\it Journal of Integer Sequences}, May 7 2020.

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