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By Marko Petkovsek, Herbert S. Wilf

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For those who have not previously used them, the programs will likely be a revelation. Imagine the convenience of being able to input a sum that one is interested in and having the program print out a simple formula that evaluates it! Think also of inputting a complicated sum and getting a recurrence formula that it satisfies, automatically. We hope you’ll enjoy both the mathematics and the software. Taken together, they are the story of a sequence of very recent developments that have changed the field on which the game of discrete mathematics is played.

Our conclusion is that the Bessel function is indeed hypergeometric, and it is in fact Jp (x) = ( x2 )p x2 ··· F ; − . 0 1 p! p+1 4 ✷ 2 We hope to convince you that a better first step is to reach for your computer! 4 Software that identifies hypergeometric series 39 We will use the notation  a(a + 1)(a + 2) · · · (a + n − 1), def (a)n = 1, if n ≥ 1; if n = 0. for the rising factorial function. In terms of the rising factorial function, here is what the general hypergeometric series looks like: pFq (a1)k (a2 )k · · · (ap )k z k a1 a2 .

Again, this is an algorithm that finds recurrence relations that are satisfied by sums. It is in most cases much faster than the method of Sister Celine, and it has made possible a whole generation of computerized proofs of identities that were formerly inaccessible to these ideas. It is the cornerstone of the methods that we present for finding out if a given combinatorial sum can be simplified, and it is guaranteed to work every time. ” This method, which we have already previewed here, just to get you interested, provides by far the most compact certifications of combinatorial identities, though it does not exist in the full generality of the methods of Chapters 4 and 6.

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A=B by Marko Petkovsek, Herbert S. Wilf


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