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In Preparation
Meanwhile you can read the following
excerpt from my ISSAC 96 paper. There will
be a quiz afterwards...
Results
and
Conclusion
The main accomplishment of our algorithm
is the essential reproduction of 1504 formulas in 9 tables of representations
of 
listed
in Integrals and Series, Volume 3: More Special Functions.
The total number of formulas in each of these tables is neatly summarized by
the following table:
The 
,
, and 
entries are covered by general formulas. The remaining 9 tables occupy most
of the 186 pages of Chapter 7 material on hypergeometric functions. Our algorithm
can be used to extend these tables to values of parameters very far out from
those given by Integrals and Series, Volume 3: More Special
Functions. The only limits on distance are the computer
resources of time and memory.
The next table indicates the proportion of pFq formulas with parameters in 
that can be reduced by our algorithm.
This table means, for example,
that our algorithm was able to compute 51.186% of the 
's.
(Our algorithm does reduce other instances of 
and 
, but
none with the parameters mentioned here.)
In more recent work, our algorithm has been extended to compute representations
for 
,
therefore making our algorithm encompass even more elementary and special functions.
Gallery
We now present a gallery of formulas produced by our
algorithm. While our algorithm has been used to compute thousands
of representations for F, we must limit ourselves here to putting
on display just a small number of these representations. To make
a point of the strength of our algorithm, we've selected examples
which are not listed in Integrals and
Series, Volume 3: More Special Functions,
cannot be computed by Mathematica 2.2's HypergeometricPFQ
function, and cannot be computed by Maple 5.3's hypergeom
function. Macsyma 419.0's hgfred function is able to make
progress on the first, third, and fourth examples (for the latter
two choosing representations in terms of whittaker_m and
alegendre_p) but is unable to eliminate hyper_f from the
remaining examples. These examples are all quite typical of the
formulas that can be produced by our algorithm.
References
A. P. Prudnikov,
Yu. A. Brychkov, O. I. Marichev, Integrals
and Series, Volume 3: More Special Functions,
Gordon and Breach Science Publishers, 1990.