
\documentclass{article} \def\emptyline{\vspace{12pt}} \begin{document}
\noindent {\bf {\Large A note on the paper : Closed Form (pun intended),1993 by
Doron Zeilberger}}\\\\
\noindent Mohamud Mohammed \footnote{Department of Mathematics, Temple University, Philadelphia, PA}
 \\\\
\noindent 1.{\bf Definition}: A WZ-form of degree k , k an
integer, is a closed difference form that can be written as
 $$ w = f(m_1,...,m_n)(\sum_{I}P_I(m_1,..,m_n)\delta_I)$$ where
$P_I(m_1,..m_n)$ is a polynomial in the variables $m_1,....m_n$, and
$f(m_1,...,m_n)$ is a closed form.(i.e. $\delta f(m_1,..,m_n) = 0$).\\\\
\noindent In this paper , Closed Form(pun intended, 1993) several examples of
WZ forms of degree one are given. Eq.(7.6) in the paper as stated is not closed
just for what is assumed in eq. (7.5). But it might be possible under special
cases it turns out to be closed. What is more general under the assumption in
equation (7.5) is that one can get ${r \choose 2} $ possible WZ-forms of degree
one from the r parameters : 
 $$ w_i = F(n_1,n_2,...,n_r,k)\delta k +G_i(n_1,n_2,...,n_r,k)\delta n_i, i=1..r$$
To find the corresponding $G_i$ we apply Zeilberger's algorithm to  
$F(n_1,...,m_r,k)$ with respect to each variable $ m_i$ and find the WZ-mate
$G_i$.

\noindent Using the above procedure for Dixon's identity,

$$ \sum_{k}(-1)^k
\frac{(a+b)!(a+c)!(b+c)!a!b!c!}{(2(a+k)!(a-k)!(b+k)!(b-k)!(c+k)!(c-k)!(a+b+c)!} = 1 $$

\noindent  we get the following three of the possible six WZ-forms of degree
one.   $$W_{Dixon} = (-1)^k
\frac{(a+b)!(a+c)!(b+c)!a!b!c!}{(2(a+k)!(a-k+1)!(b+k)!(b-k+1)!(c+k)!(c-k+1)!(a+b+c+1)!}$$ 
$(2(a-k+1)(b-k+1)(c-k+1)(a+b+c+1)\delta k + (b-k+1)(b+k)(c-k+1)(c+k)\delta
a).$
 $$ W_{Dixon} = (-1)^k
\frac{(a+b)!(a+c)!(b+c)!a!b!c!}{(2(a+k)!(a-k+1)!(b+k)!(b-k+1)!(c+k)!(c-k+1)!(a+b+c+1)!}$$
$(2(a-k+1)(b-k+1)(c-k+1)(a+b+c+1)\delta k + (a-k+1)(a+k)(c-k+1)(c+k)\delta
b).$

$$W_{Dixon} = (-1)^k
\frac{(a+b)!(a+c)!(b+c)!a!b!c!}{(2(a+k)!(a-k+1)!(b+k)!(b-k+1)!(c+k)!(c-k+1)!(a+b+c+1)!}$$
$(2(a-k+1)(b-k+1)(c-k+1)(a+b+c+1)\delta k+(a-k+1)(a+k)(b-k+1)(b+k)\delta
c).$ \pagebreak

\noindent on the other hand there are no $G_a, G_b$ and $G_c $ such that
$$({\bf {A}-{1}}) F(k,a,b,c) = ({\bf {B}-{1}})G_a(k,a,b,c) $$
$$({\bf {A}-{1}}) F(k,a,b,c) = ({\bf {C}-{1}})G_c(k,a,b,c) $$
$$({\bf {B}-{1}}) F(k,a,b,c) = ({\bf {C}-{1}})G_c(k,a,b,c) $$
\noindent Where {\bf A}, {\bf B}, {\bf C} are forward shift operator in a,b
and c respectively by 1 and {\bf{1}} is the identity operator. Even in his
case the discussion in eq. 7.3 shows an operator,say
$R(A,a)$, can exist such that(similarly for the remaining cases) 
$$(A-{\bf{1}})(R(A,a)F(k,a,b,c))=(B-{\bf{1}})G_b(k,a,b,c)$$
\noindent Which then gives a wz form of degree 1 
$$w = R(A,a)F(k,a,b,c)\delta a + G_b (k,a,b,c)\delta b $$  
\noindent In the same way,from Saalschut's identity
$$\sum_{k}\frac{(m-r+s)!(n+r-s)!(r+k)!m!n!(r-m)!(s-n)!}{k!(m-r+s-k)!(n-k)!(r-s+k)!(m+n)!(r+k-m-n)!r!s!} = 1$$  \noindent we get four wz-forms of degree one.

$$W_{Saalscutz's} =
\frac{(m-r+s)!(n+r-s)!(r+k)!(m!n!(r-m)!(s-n)!}{k!(m-r+s-k)!(n-k)!(r-s+k)!(m+n)!(r+k-m-n)!r!s!{(n-k+1)}}.$$
$\frac{1}{(m+n+1)(n-s)}({(n-k+1)(m+n+1)(n-s)}\delta
k+{-k(r-s+k)(-r-k+m+n)}\delta n)$

$$W_{Saalscutz's} =
\frac{(m-r+s)!(n+r-s)!(r+k)!(m!n!(r-m)!(s-n)!}{k!(m-r+s-k)!(n-k)!(r-s+k)!(m+n)!(r+k-m-n)!r!s!{(r+1)(-m+r-s)}}.$$
$({(r+1)(-m+r-s)}\delta k + {(-2r-m+1-s-k)k}\delta r)$

$$W_{Saalscutz's} =\frac{(m-r+s)!(n+r-s)!(r+k)!(m!n!(r-m)!(s-n)!}{k!(m-r+s-k)!(n-k)!(r-s+k)!(m+n)!(r+k-m-n)!r!s!{(m-r+s-k+1)}}.$$
$\frac{1}{(s+1)(-n-r+s)}({(m-r+s-k+1)(s+1)(-n-r+s)}\delta
k{-k(-r+s-k)(r+k-m-n)}\delta s)$

$$W_{Saalscutz's}
=\frac{(m-r+s)!(n+r-s)!(r+k)!(m!n!(r-m)!(s-n)!}{k!(m-r+s-k)!(n-k)!(r-s+k)!(m+n)!(r+k-m-n)!r!s!(m-r)}$$
 $\frac{1}{(m-r+s-k+1)(m+n+1)}({(m-r+s-k+1)(m+n+1)(m-r)}\delta
k-k(r-s+k)(-r-k+m+n)\delta m)$\\


\noindent 2. The generalization given after the proof of Theorem 9 is also 
seen in the same line as above and the discussion following reframed.
i.e. one can start with a known multi-variables  $$\sum C(n,k_1,...,k_r) =
cont.$$ and get ${r \choose 2}$ additional  potential function for a WZ-1
forms.\\ 

$$ w = F(n,k_1,...,k_r) \delta n + \sum_{i=1}^{r}
G_i(n,k_1,...,k_r)\delta k_i$$ to be closed we need stronger general
conditions than equation (13.21).\\

\noindent I would like to thank Prof. Boris Datskovsky who pointed out the
error first in a related discussion I had with him.

\end{document}


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