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By Petkovsek M., Wilf H.S., Zeilberger D.

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4 A Mathematica session For our next example of the use of the WZ proof algorithm we’ll take some of the pain out by using Mathematica to do the routine algebra. To begin, let’s try to simplify some expressions that contain factorials. /n! then what we get back is (1 + n)! , n! which doesn’t help too much. ] then we also get back (1 + n)! , n! so we must be doing something wrong. Well, it turns out that if you would really like to simplify ratios of factorials then the thing to do is to read in the package RSolve, because in that package there lives a command FactorialSimplify, which does the simplification that you would like to see.

2n (−2n)! n! (−n − 12 )! (− 12 )! This is a rather distressing development. We were expecting an answer in simple form, but the answer that we are looking at contains some factorials of negative numbers, 46 The Hypergeometric Database and some of these negative numbers are negative integers, which are precisely the places where the factorial function is undefined. Fortunately, what we have is a ratio of two factorials at negative integers; if we take an appropriate limit, the singularities will cancel, and a pleasant limiting ratio will ensue.

From among the letters n + 1, . . , 2n. n , n. There are n−k Hence there are nk ways of of the 2n n n k ways to choose k ways to choose n−k letters n n−k = n 2 k ways to make choosing n letters from the such a pair of choices. But every one 2n letters 1, 2, . . , 2n corresponds uniquely to such a pair of choices, for some k. ✷ We must pause to remark that that one is a really nice proof. So as we go through this book whose main theme is that computers can prove all of these identities, please note that we will never7 claim that computerized proofs are better than human ones, in any sense.

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A=B by Petkovsek M., Wilf H.S., Zeilberger D.

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