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\begin{document} 
\vspace*{-60pt} 
\centerline{\smalltt INTEGERS: 
 \smallrm ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 5 
(2005), \#A03} 
\vskip 50pt

\begin{center}
{\bf TWO $\boldsymbol{q}$--IDENTITIES FROM THE THEORY OF FOUNTAINS AND
HISTOGRAMS PROVED WITH A TRI--DIAGONAL DETERMINANT}
\vskip 20pt
{\bf Helmut Prodinger\footnote{Dedicated to Kathy Driver for six
years of
 loyal support.\\ \indent
Thanks are due to a friendly referee who pointed out some trouble with signs.\\
}}\\
{\smallit 
The John Knopfmacher
Centre for Applicable Analysis and Number Theory,
 School of Mathematics,
University of the Witwatersrand, P.~O. Wits, 
2050 Johannesburg, South Africa
{\tt helmut@maths.wits.ac.za}}
\end{center}
\vskip 30pt
\centerline{\smallit Received: 8/27/04, 
 Accepted: 2/3/05, Published: 2/8/05}
\vskip 30pt 

\centerline{\bf Abstract}


\noindent
Two identities required in the theory of fountains and histograms are easily proved by
expanding a tri--diagonal determinant (reminiscent of Schur's) in two different ways.

\pagestyle{myheadings}
\markright{\smalltt INTEGERS: \smallrm ELECTRONIC
 JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 5 (2005), \#A03\hfill}

\thispagestyle{empty} 
\baselineskip=15pt 
\vskip 30pt 
















We consider the following infinite tri-diagonal determinant
(elements not displayed are zero)
\begin{equation*}   
\schur(x):=\left|
\begin{matrix}   
1&&\overset{p-2}{\overbrace{0\dots0}}&&xq^{1}&&&&\dots\\
-1&1&&0\dots0&&xq^{2}&&&\dots\\
&-1&1&&0\dots0&&xq^{3}&&\dots\\
&&-1&1&&0\dots0&&xq^{4}&\dots\\
&&&\ddots&\ddots&\qquad \ddots
\end{matrix}
\right|.
\end{equation*}

Schur, when providing his proof of the Rogers--Ramanujan identities
in 1917 \cite{Schur17} used a similar determinant; since I am advocating 
that Schur's work   deserves to be better known, I use the name
$\schur(x)$. This short note shows that
two identities that were
required in the study of fountains and histograms \cite{MeSp02}
are most easily proved by expanding the determinant in two different ways.



Expanding the determinant with respect to the first column
(``top--recursion'') we get
\begin{equation*}   
\schur(x)=\schur(xq)+(-1)^pxq\,\schur(xq^{p}).
\end{equation*}
Setting
\begin{equation*}   
\schur(x)=\sum_{n\ge0}a_nx^n, 
\end{equation*}
we get, upon comparing coefficients, 
\begin{equation*}   
a_n=q^na_n+(-1)^pq^{1+p(n-1)}a_{n-1}=\frac{(-1)^pq^{1+p(n-1)}}{1-q^n}.
\end{equation*}
Since $a_0=1$, iteration leads to
\begin{equation*}   
a_n=\frac{q^{n+p\binom{n}{2}}(-1)^{pn}}{(1-q)(1-q^2)\dots(1-q^n)}.
\end{equation*} 
Therefore
\begin{align*}
\schur((-q)^{p-1})&=\sum_{n\ge0}\frac{(-1)^nq^{n+p\binom{n}{2}}}{(1-q)(1-q^2)\dots(1-q^n)}
q^{(p-1)n}\\&=\sum_{n\ge0}\frac{(-1)^nq^{p\binom{n+1}{2}}}{(1-q)(1-q^2)\dots(1-q^n)}.
\end{align*}

Now consider the {\it finite\/}
determinants $\schur_n(x)$, obtained from $\schur(x)$ by taking the first $n$ 
rows and columns.
Expanding this determinant with respect to the last
row 
(``bottom--recursion'') we get
\begin{equation*}   
\schur_n(x)=\schur_{n-1}(x)+(-1)^pxq^{n-p+1}\schur_{n-p}(x).
\end{equation*}
In particular,
\begin{equation*}   
\schur_n((-q)^{p-1})=\schur_{n-1}((-q)^{p-1})-q^{n}\schur_{n-p}((-q)^{p-1}),
\end{equation*}
and $\schur_j((-q)^{p-1})=1$ for $j=0,\dots,p-1$. The quantities
$\schur_n((-q)^{p-1})$ were called $E_n$ in \cite{MeSp02}
(with matching initial conditions $E_j=1$ for $j=0,\dots,p-1$).
 Whence we proved
\begin{equation*}
\lim_{m\to\infty}{E_m}=\sum_{n\ge0}\frac{(-1)^nq^{p\binom{n+1}{2}}}{(1-q)(1-q^2)\dots(1-q^n)}.
\end{equation*}
Merlini and Sprugnoli had asked for a direct proof, which was given
in \cite{PaPr03}, by showing an explicit form for $E_m$. The present proof
avoids this and is thus simpler.

A second (similar) formula was also requested, namely
\begin{equation*}
\lim_{m\to\infty}{D_m}=\sum_{n\ge0}\frac{(-1)^nq^{n+p\binom{n}{2}}}{(1-q)(1-q^2)\dots(1-q^n)}.
\end{equation*}
for $D_n=D_{n-1}-q^nD_{n-p}$ and (different) initial values
$D_j=1-\sum_{i=1}^jq^i$ for $j=0,\dots,p-1$. This follows immediately by
setting $D_n=\schur_{n+p-1}((-1)^{p-1})$.





\bibliographystyle{plain}
\begin{thebibliography}{1} \footnotesize


\bibitem{MeSp02}
D.~Merlini and R.~Sprugnoli, \emph{Fountains and histograms}, J. Algorithms
  \textbf{44} (2002), no.~1, 159--176.


\bibitem{PaPr03}
P.~Paule and H.~Prodinger, \emph{Fountains, histograms, and $q$--identities}, 
Discrete Mathematics and Theoretical Computer Science
  \textbf{6} (2003), 101--106.


\bibitem{Schur17}
I.~Schur.
\newblock 
Ein Beitrag zur additiven Zahlentheorie und zur
Theorie der Kettenbr\"uche.
\newblock {\em S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl.},  1917,
302--321, reprinted in I. Schur, Gesammelte Abhandlungen, vol. 2,
pp. 117--136, Springer, 1973.

\end{thebibliography}





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