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By Wilf, Zeilberger.

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2n − 1)(k − n − 1) = 1, 3. For each of the four parts of Problem 2 above, write out the complete proof of the identity, using the full text of the standardized WZ proof together with the appropriate rational function certificate. 4. For each of the parts of Problem 2 above, say exactly what the standardized summand F (n, k) is, and in each case evaluate lim F (n, k) k→∞ and lim F (n, k). n→∞ 5. Write a procedure, in your favorite programming language, whose input will be the summand t(n, k), and the right hand side rhs(n), of a claimed identity k t(n, k) = rhs(n), as well as a claimed WZ proof certificate R(n, k).

The rest of the proof is standardized. The rational function R(n, k) certifies the proof. Here is the standardized WZ proof algorithm: 1. Suppose that you wish to prove an identity of the form k t(n, k) = rhs(n), and let’s assume, for now, that for each n it is true that the summand t(n, k) vanishes for all k outside of some finite interval. 2. Divide through by the right hand side, so the identity that you wish to prove now reads as k F (n, k) = 1, where F (n, k) = t(n, k)/rhs(n). 3. Let R(n, k) be the rational function that the WZ method provides as the proof of your identity (we’ll discuss how to find this function in Chapter 7).

This book contains both mathematics and software, the former being the theoretical underpinnings of the latter. 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.

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

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