Knuth wrote in Concrete Math that hypergeometric functions are useful because they allow the construction of a "database" of identities, since any sum with the property that the ratio between successive terms is a rational function of $z$ can be put in the canonical form $F(a_0 \ldots a_n; b_0 \ldots b_n; z)$. Yet Wikipedia says "There is no known system for organizing all of the identities", and looking online I couldn't find any actual database of hypergeometric identities. Why is this the case?
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1what is the question ? – KonKan Mar 31 '16 at 02:13
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1Are you just looking for special cases of the hypergeometric function? For example, is this on the right track?: http://dlmf.nist.gov/15.4 – Xoque55 Mar 31 '16 at 02:33
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@Xoque55 I'm looking for a website where I can put in the a's and b's and get identities – Elliot Gorokhovsky Mar 31 '16 at 02:41
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1A good question. I suggest this database. It's not as functional as you want, but it contains a lot of information – Yuriy S Apr 18 '16 at 09:55
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@YuriyS Both are nice; however, since the hypergeometric function is defined by two sequences of integers, why not make a database where you can just input the integers and get a closed form, if it's known? – Elliot Gorokhovsky Apr 18 '16 at 15:59
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1@RenéG, Mathematica can do that. It acts exactly like any such database would, only better, since it allows computations – Yuriy S Apr 18 '16 at 16:12
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I refer you to section 3.7 of the book A=B by Petkovšek, Wilf and Zeilberger (available for free at Wilf's website). It is very well explained there.
Anyway, it is not possible to elaborate such a database. Suppose you think you did it. Paraphrasing the mentioned book:
"Your database should first look to see if your sum lives in the data, and if not it should next try to transform your sum into another one that does live in the data. If that succeeds, great. If it fails, well maybe there’s a sequence of two transformations that will do it. Or maybe three — you see the problem. Besides sequences of transformations, one can also use various substitutions for the parameters, and it may be hard to recognize that a certain identity is a specialization of a database entry.
There is no algorithm that will discover whether your sum is or is not transformable into an identity that lives in the database."
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