.pdf
.ps
.tex
Written: Feb. 17, 2006.
There are lots of ways to evaluate (symbolic) determinants,
and there are quite a few humans (Tewodros Amdeberhan, George Andrews,
Mihai Ciucu, Christian Krattenthaler and Herb Wilf, for example),
who are very good at it.
In my
Lieber Opa
article, I taught Shalosh how to evaluate determinants
using Charles Dodgson's Condensation method, but its scope was
rather limited, since it was restricted to the case
of what I called the hyperhypergeometric framework.
Moving up to the holnomic ansatz can do many
more determinant evaluations. Alas, not all of them!
Of course, the main point of this article is not
the results themselves but as a case study
and metaphor for the shape of mathematics to come.
Important: This article is accompanied by Maple
package
DET
that automatically guesses and proves
(rigorously or semirigorously) holonomic
determinant evaluations.
Sample Input and Output for DET
 Input and Output for Rproof

To (symbolically!) evaluate and (prove!)
the determinant of the famous Hilbert matrix,
the
input
yields the output.

To (symbolically!) evaluate and (prove!)
a special case [the general case is below] of
the determinant in Theorem 33 of Christian Krattenthaler's
"Advanced Determinant Claculus: a Complement"
the
input
yields the output.

To (symbolically!) evaluate and (prove!)
the determinant (1/(i+j+1)!),
the
input
yields the output.
 Input and Output for RproofP

To (symbolically!) evaluate and (prove!)
the determinant of an arbitrary minor of the Hilbert matrix,
the
input
yields the output.

To (symbolically!) evaluate and (prove!)
(essentially) the determinant (1.2) in
Christian Krattenthaler's "Advanced Determinant Calculus"
the
input
yields the output.

To (symbolically!) evaluate and (prove!)
a determinant conjectured by Greg Kuperberg and
Jim Propp and first proved, in a
cute paper
by Shalosh B. Ekhad and Tewodros Amdberhan,
the
input
yields the output.

To (symbolically!) evaluate and (prove!)
the determinant in Theorem 33 of Christian Krattenthaler's
"Advanced Determinant Claculus: a Complement"
the
input
yields the output.

To (symbolically!) evaluate and (prove!), as an
explicit expression in n and p,
the determinant of the (n+1) by (n+1) matrix whose
(i,j) entry is binomial(2p+2i+2j,p+i+j)
the
input
yields the output.

To (symbolically!) evaluate and (prove!), as an
explicit expression in n and p,
the determinant of the (n+1) by (n+1) matrix whose
(i,j) entry is (i+j+p)!,
the
input
yields the output.

And, finally, TaTa!,
to (symbolically!) evaluate and (prove!), as an
explicit expression in n and p,
the determinant of the (n+1) by (n+1) matrix whose
(i,j) entry is binomial(p+i+j,2*ij+1),
which is essentially the MillsRobbinsRumsey
determinant, (MRR) mentioned in the body of the article,
the
input
yields the output.
 Input and Output for SRproof

To (symbolically!) evaluate and provide a semirigorous
proof of seven determinant evaluations of
Ira Gessel and Guoce Xin, reproduced in
Theorem 31 of Christian Krattenthaler's
"Advanced Determinant Claculus: a Complement"
the
input
yields the output.
 Input and Output for SRproofI

To (symbolically!) evaluate and provide a semirigorous
proof of the determinant of the (n+1) by (n+1) matrix whose
(i,j) entry is delta(i,j)+binomial(i+j,j),
where delta(i,j) is 1 if i=j and 0 otherwise
(the celeberated determinant, first evaluated in
George Andrews's article "Plane Partitions (III): The
Weak Macdoland conjecture", published in the
"prestigious" journal Invent. Math. v. 53 (1979), 193225,
and used by him to enumerate cyclically symmetric
plane partitions [note that the qanalog of this,
that would be doable with the forthcoming
qanalog, of DET, qDET, is the determinant that made
William Mills, Dave Robbins and Howard Rumsey famous (proving the
"strong" form of Ian Macdonald's conjecture about
the generating function of Cyclically Symmetric Plane
Partitions, that Richard Stanely, at the time, called
"the most interesting open problem in enumeration")
the
input
yields the output.
Doron Zeilberger's List of Papers
Doron Zeilberger's Home Page