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%\section{preamble}
\title[On the moments of a distribution defined by the 
Gaussian polynomials]
{On the moments of a distribution defined by the 
Gaussian polynomials}

\author{Helmut Prodinger}
%\thanks{}

\address{ Helmut Prodinger,
The John Knopfmacher
Centre for Applicable Analysis and Number Theory,
 Department of Mathematics,
University of the Witwatersrand, P.~O. Wits, 
2050 Johannesburg, South Africa, email:
{\tt helmut@gauss.cam.wits.ac.za},\newline
homepage: {\tt http://www.wits.ac.za/helmut/index.htm}
}


\date{June 12, 2000}









\begin{document}


\maketitle

Di Bucchianico \cite{Bucchianico99} has considered the
probability distribution defined by the (probability)
generating function
\begin{equation*}   
F_{m,n}(q)=\frac{\gauss {m+n}m}{\binom{m+n}{m}},
\end{equation*}
with the Gaussian polynomials
\begin{equation*}   
\gauss {m+n}m=\frac{(1-q)(1-q^2)\dots(1-q^{m+n})}
{(1-q)(1-q^2)\dots(1-q^{m})\cdot
(1-q)(1-q^2)\dots(1-q^{n})}.
\end{equation*}
He discussed several methods to compute the moments of
this distribution. 

In this short note, I want to draw the attention of the reader
to another (potentially superior) method that is
due to Panny \cite{Panny86}. This method first computes the 
{\it cumulants\/} and translates them into the 
{\it moments\/} by
a standard formula. This yields  {\it explicit\/} formul\ae{}
for  all the moments.

Note that 
\begin{equation*}   
F_{m,n}(q)=\frac{g_1(q)\dots g_{m+n}(q)}
{g_1(q)\dots g_{m}(q)\cdot g_1(q)\dots g_{n}(q)}
\end{equation*}
with
\begin{equation*}   
g_k(q)=\frac{1-q^k}{k(1-q)},
\end{equation*}
where $g_k(q)$ is a probability generating function of
a random variable $X_k$.
The function $\log g_k(e^{it})$ is the generating function of
the cumulants\footnote{We write $\log x$ for the
(natural) logarithm with base $e$.}. Panny has computed that as
\begin{equation*}   
\log g_k(e^{it})=\frac{(k-1)it}{2}+
\sum_{j\ge1}\frac{B_{2j}}{2j}(k^{2j}-1)\frac{(it)^{2j}}
{(2j)!},\qquad |t|<\frac{2\pi }{k},
\end{equation*}
where $B_i$ denotes the $i$th Bernoulli number.  

Reading off coefficients we find the cumulants of $X_k$ as
\begin{equation*}   
\kappa_1=\frac{k-1}{2},\qquad
\kappa_{2r}=\frac{(k^{2r}-1)B_{2r}}{2r},
\qquad \kappa_{2r+1}=0,\qquad r=1,2,3,\ldots.
\end{equation*}

Since 
\begin{equation*}   
\log F_{m,n}(e^{it})=\sum_{k=1}^{m+n}\log g_k(e^{it})
-\sum_{k=1}^{m}\log g_k(e^{it})-
\sum_{k=1}^{n}\log g_k(e^{it}),
\end{equation*}
the cumulants $\kappa_r^{m,n}$ are obtained by summing the
$\kappa_r$'s.

So we get
\begin{align*}   
\kappa_1^{m,n}&=\frac{\binom{m+n}{2}}{2}
-\frac{\binom{m}{2}}{2}-\frac{\binom{n}{2}}{2}=
\frac{mn}{2},\\
\kappa_{2r}^{m,n}&=\frac{B_{2r}}{2r(2r+1)}\big(
B_{2r+1}(m+n+1)-B_{2r+1}(m+1)-B_{2r+1}(n+1)\big),\\
\kappa_{2r+1}^{m,n}&=0.
\end{align*}
 
Here is a little list:
\begin{align*}
\kappa_1^{m,n}&=\frac{mn}{2},\\
\kappa_2^{m,n}&=\frac{mn(m+n+1)}{12},\\
\kappa_4^{m,n}&=-\frac{mn(m+n+1)\big(m(m+1)+mn+n(n+1)\big)}{120}.
\end{align*}

The formula to compute the moments $\mu_r$ from this is
\begin{equation*}   
\mu_r=\sum_{\pi_1+2\pi_2+\dots+r\pi_r=r}
\Big(\frac{\kappa_1}{1!}\Big)^{\pi_1}
\Big(\frac{\kappa_2}{2!}\Big)^{\pi_2}\dots
\Big(\frac{\kappa_r}{r!}\Big)^{\pi_r}
\frac{r!}{\pi_1!\pi_2!\dots\pi_r!},
\end{equation*}
which is a finite sum since $\pi_k\in\{0,1,\dots,r\}$.
Hence
\begin{align*}
\mu_1&=\frac{mn}{2},\\
\mu_2&=\kappa_1^2+\kappa_2=\frac{mn(m+n+3mn+1)}{12},\\
\mu_3&=\kappa_1^3+3\kappa_1\kappa_2+\kappa_3=
\frac{m^2 (m+1)n^2(n+1)}{8},\qquad \&c.
\end{align*}
We finish by mentioning that, as in \cite{Panny86}, 
one could also get asymptotic formul\ae.       

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

\bibitem{Bucchianico99}
A.~Di~Bucchianico.
\newblock 
Combinatorics, computer algebra, and the Wilcoxon--Mann--Whitney test.
\newblock {\em Journal of Statistical Planning
and Inference}, 79:349--364, 1999.

\bibitem{Panny86}
W.~Panny.
\newblock 
A note on the higher moments of the expected behavior of
straight insertion sort.
\newblock {\em Information Processing Letters},
 22:175--177, 1986.


\end{thebibliography}






\end{document}
