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\topmatter
\title
``Trivializing'' Generalizations
of Some Izergin-Korepin-type determinants 
\endtitle
\author 
Tewodros Amdeberhan, Doron Zeilberger
\endauthor
\affil
Tulane Univ., Dept of Math., New Orleans, LA 70118, USA\\
Rutgers Univ., Dept of Math., Piscataway, NJ 08854, USA
\endaffil
\abstract
We generalize (and hence trivialize and routinize)
numerous explicit evaluations of determinants and pfaffians due to
Kuperberg, as well as a determinant of Tsuchiya.
The level of generality of our statements render their proofs
easy and routine, by using Dodgson Condensation and/or
Krattenthaler's factor exhaustion method.
\endabstract
\endtopmatter
\def\({\left(}
\def\){\right)}
\def\R{\bold R}
\document
%\bf Key words: \rm 
%\bf MSC codes: \rm 
\smallskip
\noindent
All our matrices will be assumed to be
embedded inside an infinite matrix. 
\smallskip
\noindent
The first theorem adds \it parameters \rm to the 
determinant formulas found in Kuperberg [Ku] (Theorem 15), 
as well as older determinants, mentioned there, due to
Cauchy, Stembridge, Laksov-Lascoux-Thorup, and Tsuchiya [T]. This way, the formulation is suited to the method of [AZ]. Our proofs are much more {\bf succinct} and {\bf automatable}, since their generality enables an easy induction using Dodgson's rule [D, AZ],
or by employing Krattenthaler's elegant factor exhaustion method [Kr1]. Relevant background for this paper can found in [Ku], and references thereof.


%\midspace{.1in}
\bigskip
\noindent
\bf Theorem 1: \it
$$\det\left(\frac{1}{x_{i}+y_{j}+Ax_{i}y_{j}}\right)_{i,j}^{1,n}=
\frac{\prod_{i<j}
(x_{j} -x_{i})(y_{j} -y_{i})}{\prod_{i,j}^n(x_{i}+y_{j}+Ax_{i}y_{j})}.$$

$$\align
&\det\left(\frac{1}{x_{i}+y_{j}}-\frac{1}{1+x_{i}y_{j}} \right)_{i,j}^{1,n}\\
&=\frac{\prod_{i<j}(1-x_{i}x_{j})(1-y_{i}y_{j})
(x_{j}-x_{i})(y_{j}-y_{i}) }
{\prod_{i,j}^n(x_{i}+y_{j})(1+x_{i}y_{j})}
\cdot \prod_{i=1}^n(1-x_{i})(1-y_{i}).\endalign$$

$$\align
&\det\left(\frac{Ay_{j}+Bx_{i}}{y_{j}+x_{i}}\right)_{i,j}^{1,n}\\
&=(A-B)^{n-1}\frac{\left(A\prod_j^ny_{j}+(-1)^{n-1}B\prod_i^nx_{i}\right)
\prod_{i<j}(x_{i}-x_{j})(y_{i}-y_{j}) }
{\prod_{i,j}^n(x_{i}+y_{j})}.\endalign$$

$$\align
&\det\left(\frac{1-x_{i}y_{j}}{y_{j}-x_{i}}\right)_{i,j}^{1,n}\\
&=(-1)^{\binom{n}2}\frac{\left(\prod_j^n(1+x_{j})(1-y_{j})+\prod_i^n(1-x_{i})(1+y_{i})\right)
\prod_{i<j}(x_{i}-x_{j})(y_{i}-y_{j})}
{2\prod_{i,j}^n(y_{j}-x_{i})}.\endalign$$ \rm
\bf Sketch of Proof: 
\rm An automatic application of Dodgson Condensation [D, AZ]. $\square$
\bigskip
\noindent
\bf Corollary (Cauchy, Stembridge, Laksov-Lascoux-Thorup) \it
$$\det\left(\frac{1}{x_{i}+y_{j}}\right)_{i,j}^{1,n}=
\frac{\prod_{i<j}
(x_{j}-x_{i})(y_{j}-y_{i})}{\prod_{i,j}^n(x_{i}+y_{j})}.$$

%$$\det\left( \frac{1}{x_{i}+y_{j}}-\frac{1}{1+x_{i}y_{j}} \right)_{i,j}^{1,n}
%=\frac{\prod_{i<j}(1-x_{i}x_{j})(1-y_{i}y_{j})
%(x_{j}-x_{i})(y_{j}-y_{i}) }
%{\prod_{i,j}^n(x_{i}+y_{j})(1+x_{i}y_{j})}
%\cdot \prod_{i=1}^n(1-x_{i})(1-y_{i}).$$

$$Pf^2=\det\left(\frac{x_{j}-x_{i}}{x_{j}+x_{i}}\right)_{i,j}^{1,2n}
=\prod_{i<j\leq 2n}\frac{(x_{i}-x_{j})^2 }{(x_{i}+x_{j})^2}.$$

$$Pf^2=\det\left(\frac{x_{j}-x_{i}}{1-x_{i}x_{j}}\right)_{i,j}^{1,2n}
=\prod_{i<j\leq 2n}\frac{(x_{i}-x_{j})^2}{(1-x_{i}x_{j})^2}.$$ \rm

\bigskip
\noindent
\bf Remark: \rm Notice that the latter two statements apply only to even-dimensional matrices. An error from [Ku] in the second formula has been corrected here. $Pf$ stands for Pfaffian of a matrix.

\medskip
\noindent
The next theorem generalizes, and presents variations of, 
several of the determinants that appear in Theorems 16 and  17 [Ku] (typos corrected in [Kr2, Theorems 13 and 14]).
 Below, $Z_1,Z_2,Z_3,Z_4,Z_5$ are defined by
(here $\gamma(a,b)=a-b, \tau(a,b)=a+b$)
$$\align
&Z_1(p,q;x,y)_{i,j}=\frac{\gamma(q^{j-i},x^{j-i})}{\tau(p^{j-i},y^{j-i})}, \qquad{}\qquad {  }
Z_2(p,q;x,y)_{i,j}=\frac{\gamma(q^{1+j-i},x^{1+j-i})}{\tau(p^{1+j-i},y^{1+j-i})}, \\
&Z_3(p,q;x,y)_{i,j}=\frac{\gamma(q^{-1+j-i},x^{-1+j-i})}{\tau(p^{-1+j-i},y^{-1+j-i})}, \qquad
Z_4(q,q;x,x)_{i,j}=\frac{\gamma(q^{a+j-i},x^{a+j-i})}{\tau(q^{a+j-i},x^{a+j-i})},\\
&Z_5(q,q;x,x)_{i,j}=\frac{\tau(q^{b+j-i},x^{b+j-i})}{\gamma(q^{b+j-i},x^{b+j-i})},\endalign$$
for $a\in\Bbb{Z}, b=\pm n,\pm (n+1),\dots$.
\smallskip
\noindent
Let $\delta_{e,n}=\frac{1+(-1)^n}2$ denote \it Kronecker`s delta \rm function centered at the even integers, and let $\lambda_{i,j}=1$, if $i\neq j$ and $\lambda_{i,i}=0$.
\smallskip

\noindent
\bf Theorem 2: \it Write $\gamma(a,b)=a-b, \tau(a,b)=a+b$, we have the matrix determinants
$$\det\left(\frac{\gamma(q^{n+j-i},x^{n+j-i})}{\gamma(p^{n+j-i},y^{n+j-i})}\right)_{i,j}^{1,n}=
(py)^{\binom{n}3}\gamma(q,x)^n\frac{\prod_{j>i}
\gamma(p^{j-i},y^{j-i})^2\gamma(qp^{j-i},xy^{j-i})\gamma(xp^{j-i},qy^{j-i})}
{\prod_{i,j}\gamma(p^{n+j-i},y^{n+j-i})}.$$

$$\det\left(\frac{\tau(q^{j-i},x^{j-i})}
{\tau(p^{j-i},y^{j-i})}\right)_{i,j}^{1,n}=
\frac{\prod_{2|j-i>0}
\gamma(p^{j-i},y^{j-i})^2\prod_{2\nmid j-i>0}
\gamma(qp^{j-i},xy^{j-i})\gamma(xp^{j-i},qy^{j-i})}
{(qx)^{\lfloor{n^2/4}\rfloor}\prod_{j>i}\tau(p^{j-i},y^{j-i})^2}.$$

$$\det(Z_1)_{i,j}^{1,n}=
\delta_{e,n}
\frac{\gamma(q,x)^n(py)^{n/2}}{(qx)^{\lfloor{n^2/4}\rfloor}}
\frac{\prod_{2|j-i>0}
\gamma(p^{j-i},y^{j-i})^2
\gamma(qp^{j-i},xy^{j-i})\gamma(xp^{j-i},qy^{j-i})}
{\prod_{j>i}
\tau(p^{j-i},y^{j-i})^2}.$$

$$\det(Z_2)_{i,j}^{1,n}=
\frac{\gamma(q,x)^n}{(qx)^{\lfloor{(n-1)^2/4}\rfloor}}
\frac{\prod_{2|j-i>0}
\gamma(p^{j-i},y^{j-i})^2
\gamma(qp^{j-i},xy^{j-i})\gamma(xp^{j-i},qy^{j-i})}
{\tau(p^n,y^n)^{1-\delta_{e,n}}\prod_{j>i}
\tau(p^{j-i},y^{j-i})^2}.$$

$$\det(Z_3)_{i,j}^{1,n}=
\frac{(-py)^n\gamma(q,x)^n}{(qx)^{\lfloor{(n+1)^2/4}\rfloor}}
\frac{\prod_{2|j-i>0}
\gamma(p^{j-i},y^{j-i})^2
\gamma(qp^{j-i},xy^{j-i})\gamma(xp^{j-i},qy^{j-i})}
{\tau(p^n,y^n)^{1-\delta_{e,n}}\prod_{j>i}
\tau(p^{j-i},y^{j-i})^2}.$$


$$\det(Z_4)_{i,j}^{1,n}=2^{n-1}
\frac{q^{na}+(-1)^nx^{na}}
{(qx)^{n(n-1)(n+1-3a)/6}} 
\frac{\prod_{j\neq i}
\gamma(q^{|j-i|},x^{|j-i|})}
{\prod_{i,j}\tau(q^{a+j-i},x^{a+j-i})}.$$

$$\det(Z_5)_{i,j}^{1,n}=(-1)^{\binom{n}2}2^{n-1}
\frac{q^{nb}+x^{nb}}
{(qx)^{n(n-1)(n+1-3b)/6}}
\frac{\prod_{j\neq i}
\gamma(q^{|j-i|},x^{|j-i|})}
{\prod_{i,j}\gamma(q^{b+j-i},x^{b+j-i})}.$$
	

$$\align 
Pf^2=&\det\left(\lambda_{i,j}\frac{\gamma(q^{j-i},x^{j-i})\gamma(r^{j-i},z^{j-i})}
{\gamma(p^{j-i},y^{j-i})}\right)_{i,j}^{1,2n}=
\frac{(yp)^n}{(qxrz)^{n^2}}
\frac{\prod_{j\neq i}^{1,n}
\gamma(p^{|j-i|},y^{|j-i|})^2}
{\prod_{i,j}^{1,n}\gamma(p^{n+j-i},y^{n+j-i})^2}\times \\
&\prod_{i,j}^{1,n}
\gamma(qp^{|j-i|},xy^{|j-i|})\gamma(xp^{|j-i|},qy^{|j-i|})
\gamma(rp^{|j-i|},zy^{|j-i|})\gamma(zp^{|j-i|},ry^{|j-i|}).
\endalign$$ \rm



\noindent
\bf Sketch of Proof: 
\rm Identities $Z_4$ and $Z_5$ are directly amenable to Dodgson`s Condensation technique [AZ]. For the remaining assertions, 
use the factor exhaustion method [Kr1] (see also [Ku]): 
the essential idea is to 
compare zeros and poles on both sides of the equation at hand. 
We leave the straightforward details to the reader.
$\square$

\bigskip
\noindent
\bf Acknowledgment: \rm The authors wish to thank Christian Krattenthaler for helpful suggestions.


%\pagebreak
\midspace{.1in}


\Refs 
\widestnumber\key{MRR}

\ref\key AZ \by T. Amdeberhan, D. Zeilberger 
\paper \it  Determinants Through the Looking Glass
\jour Adv. Appl. Math. (Foata special issue) 2/3
\vol 27\yr2001 \pages 225-230
\endref

\ref\key D \by  C.L. Dodgson
\paper \it Condensation of Determinants
\jour Proceedings of the Royal Society of London 
\vol 15\yr1866 \pages 150-155
\endref

\ref\key Kr1 \by C. Krattenthaler\paper \it Advanced determinant calculus \jour S\`eminaire Lotharingien Combin. (B42q)\vol 42\yr1999 
\endref 

\ref\key Kr2 \by C. Krattenthaler\paper \it Advanced determinant calculus: A Complement \jour Linear Alg. Appl. \vol 411\yr2005\pages68-166 
\endref 

\ref\key Ku \by G. Kuperberg\paper \it Symmetry classes of alternating-sign matrices under one roof \jour Ann. of Math. (2)\vol 156\yr2002 \pages835-866
\endref

\ref\key T \by O. Tsuchiya\paper \it Determinant formula for the six-vertex model with reflecting end \jour J. Math. Phys. \vol 39\yr1998 \pages5946-5951
\endref


\endRefs

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