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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.