The Markov-WZ Method
By Mohamud Mohammed and Doron Zeilberger
Written: May 28, 2004.
Margo Kondratieva and Sergey Sadov's
amazing historical discovery
that Andrei Markov, of Markov Chains fame,
"scooped" WZ (well, not quite!, read the article),
turned out to be more than of historical value, since, while
Markov lacked some of the central ideas of WZ theory, he also
had an idea that WZ didn't have! If you add this on, you get
a new-improved concept of WZ-pair, that we call MWZ pairs.
(Plain ) .tex version (12
pages)
.dvi version (for previewing)
.ps version
.pdf version
IMPORTANT: This article is accompanied by the Maple packages
-
MarkovWZ,
To use it, first download it, saving it as MarkovWZ, then
stay in the same directory, go into Maple, and type:
read MarkovWZ:
and follow the on-line instructions.
Added July 19, 2004: try out the enhanced package
MarkovWZdiag, that also does diagonal contours. To find out
specifics about the new features, type ezraD(); in that package.
-
MarkovAZ,
To use it,
stay in the same directory, go into Maple, and type:
read MarkovAZ:
and follow the on-line instructions.
-
ContMarkovWZ,
To use it, first download it, saving it as ContMarkovWZ, then
stay in the same directory, go into Maple, and type:
read ContMarkovWZ:
and follow the on-line instructions.
Sample Input and Output files for MarkovWZ
-
To see some `good' kernels for Zeta(2) through Zeta(8),
complete with the MWZ pairs and the Acceleration schemes,
use
input,
to get
output.
-
To see the best delta (empirically) with n+k=40, with
kernel ((z+x)!/(2*x+z+1)!)^r for appx. Zeta(r) (r between 2 and 6)
use
input,
to get
output.
-
To see the various delta's one gets with various kernles for
Zeta(5) use the
input,
to get
output.
-
To see the best delta (empirically) with n+k=40, with
kernel ((z+x)!/(3*x+z+1)!)^r for appx. Zeta(r) (r between 2 and 5),
use
input,
to get
output.
-
To get some best delta's (for cnk's, not the weighted average),
for Apery's kernels for log(2), Zeta(2) and Zeta(3), use
input,
to get
output.
-
To get the MWZ pairs whose kernels are the powers of the
binomial coefficients, from the first to the eighth, use
input,
to get
output.
-
To get some empirical delta's for Apery's appx. seq for
Zeta(2) and Zeta(3) (with n+k=20), use
input,
to get
output.
-
To get some empirical delta's for Apery's appx. seq for
Zeta(2) and Zeta(3) (with n+k=40), use
input,
to get
output.
-
To see the appx. for Zeta(r) (r=2..9) for using
((z+x)!/(2*x+z+1)!)^r, use
input,
to get
output.
-
To see the appx. for Zeta(r) (r=2..6) for using
((z+x)!/(3*x+z+1)!)^r, use
input,
to get
output.
Doron Zeilberger's List of Papers
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