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\bf
\centerline
{
The Jiang Zeng-M.Ishikawa-H.Tagawa q-Catalan Hankel
Determinant Identity is Purely Routine
}
\rm
\bigskip
\centerline{ {\it
Doron 
ZEILBERGER}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/} .
 Dec. 22, 2011.
Exclusively published in the Personal Journal of Shalosh
B. Ekhad and Doron Zeilberger
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/pj.html} .
Supported in part by the NSF.
}
}

Would you imagine a mathematical article spending 17 pages on 
several proofs of the identity $23 \cdot 21=483$?
A first proof could be by explicitly drawing a rectangle of
23 by 21 dots, and asking the reader to count the 
number of dots.
A more advanced proof could be 
$$
23 \cdot 21=(20+3)\dot(20+1)= 20 \cdot 20+ 20 \cdot 1 + 3 \cdot 20+ 3\cdot 1=
400+20+60+3=400+80+3=483 \quad,
$$
and a really clever and elegant proof, using the advanced algebraic
identity $(a-b)(a+b)=a^2-b^2$ is as follows:
$$
23 \cdot 21=(22+1)\dot(22-1)=22^2-1^2=(2 \cdot 11)^2-1=
4 \cdot 11^2 -1=4  \cdot 121 -1=484-1 =483  \quad .
$$
Of course not! {\it numerical} identities, and even {\it algebraic} identities
(e.g. $(a+b)^2=a^2+2ab+b^2$) and
even {\it trig} identities (e.g. $\sin^2 x+ \cos^2 x=1$) are
{\it nowadays} considered {\bf routine}, since there
exist {\bf algorithms} for proving them
(learned in third grade in the US and first grade in China).

Yet something analogous appeared in the recent article
{\it ``A q-analogue of Catalan Hankel determinants''}
by M. Ishikawa, H. Tagawa, and Jiang Zeng,
that was published in: {\it  RIMS Kôkyûroku Bessatsu, B11 (2009), 19--42}
and also posted in
{\tt http://arxiv.org/abs/1009.2004} .
The main ``theorem'' (1.1) (or rather the more general
corollary 1.2) follows {\it immediately} and {\bf routinely}
from Dodgson's condensation identity. Indeed calling
the left side and right side of Eq. (1.14) of that paper
$L(n,t)$ and $R(n,t)$ respectively, 
it follows, thanks to Rev. Charles, that
$L(n,t)=(L(n-1,t)L(n-1,t+2)-L(n-1,t+1)^2)/L(n-2,t+2)$,
and it is purely routine to check that the same identity holds
with $L(n,t)$ replaced by $R(n,t)$,
since this boils
down to a certain routine polynomial identity in the variables $a,b,q^n$.
Once this is done the ``theorem'' follows by induction since
$L(0,t)=R(0,t)$ and $L(1,t)=R(1,t)$ (check!).

I recommend that the authors of this paper, and other people too,
who wax insightful combinatorics on such routinely provable identities,
read my article:

{\tt http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/opa.html} ,

as well as the excellent paper by Tewodros Amdeberhan and myself:

{\tt http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/greg.html} .
\halmos 

{\bf Added Dec. 23, 2011}: Jiang Zeng just drew my attention to the fact that the above
comment is actually mentioned in their paper! So they give several proofs to an identity that
they {\it actually} knew was utterly trivial. They should have mentioned it in the abstract,
and not bury it in a comment on p.12 .
\end


