By Wilf, Zeilberger.

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7 Is there really a hypergeometric database? would be honest, but not very useful. In fact, the act of checking whether the given sum lives among the data is only the first step that any competent human analyst would take. If the sum could not be located, the next step that the analyst would probably take would be to try out some hypergeometric transformation rules. A transformation rule is a relation between two hypergeometric functions whose parameter sets are different, which shows, nonetheless, that if you know one of them then you know both of them.

Chapter 5 is about the fundamental algorithm of Gosper, which is to summation as finding antiderivatives is to integration. This algorithm allows us to do indefinite hypergeometric sums in simple closed form, or it furnishes a proof of impossibility if, in a given case, that cannot be done. Beyond its obvious use in doing indefinite sums, it has several nonobvious uses in executing the WZ method, in finding recurrences for definite sums, and even for finding the right hand side of a definite sum whose evaluation we are seeking.

Then we need to know how to recognize when a higher order recurrence has simple solutions of a certain form, and when it does not. The fundamental algorithm of this subject, due to Petkovˇsek, is in Chapter 8 (see page 152). 7 Exercises 1. Let f (n) = (3n + 1)! 2 . Use a computer algebra program to exhibit 3 f (n − k) f (n) k=0 explicitly as a quotient of two polynomials in n. 2. Use a computer algebra program to check the following pairs. Each pair consists of an identity and its WZ proof certificate R(n, k): (−1)k k k k x+1 2k + 1 n k n x = k k+x 1 x+n n , x n+x = , k+r n+r x − 2k 2k+1 2x + 2 2 = , n−k 2n + 1 k(k + x) (n + 1)(k − n − 1) k(k + r) (n + x + 1)(k − n − 1) k(2k + 1)(x − 2n − 1) (k − n − 1)(2n − 2x − 1)(n − x) 32 Tightening the Target (1 − 2n) n k 4 k k (−1) 2k k k k(2k − 1) .

### A=B (symbolic summation algorithms) by Wilf, Zeilberger.

by James

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