| The Holonomic Ansatz II: Automatic DISCOVERY(!) and PROOF(!!) of Holonomic Determinant Evaluations, in preparation (2009) | |||||||||||||||||
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| In a wonderful essay[W] on Experimental Mathematics, Herb Wilf outlines the four steps of doing Experimental Mathematics, in the way it is usually practiced today. 1. Wondering, by a human, what a “particular situation looks like in detail”. 2. Some computer experimentation to show the structure of that situation for a selection of small values of the parameters. 3. The [human] mathematician gazes at the computer output, attempting to see or to codify some pattern, that hopefully leads him or her to formulate a conjecture. 4. Human-made proof of the human-made conjecture (that was computer-inspired). Under this scheme, only step 2 employs the computer. In the present series of articles, I illustrate, by example, how computers can be used, without any human intervention, to also do steps 3 and 4. As for step 1, the wondering, this can also be done by machinekind-it is not too hard to teach the computer how to wonder. All that we, humans, ultimately would have to do is metawonder. In other words, make up new ansatzes and write once and for all computer programs teaching the computer how to wonder in these ansatzes, then gaze at the pattern, then formulate a conjecture (within the given ansatz) and then, finally, prove the conjecture, all by itself, without any human intervention! No longer just computer-assisted but fully computer-generated. The Art of Determinant Evaluations To find out about the state of the art in contemporary explicit determinant evaluations, by homo sapiens, the reader should consult Christian Krattenthaler’s beautiful surveys [K1] and [K2]. | |||||||||||||||||
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