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The WZ Method

The WZ method is explained in [1, Chapter 7]. Consider

$\displaystyle \sum_k \tilde{F}(n,k) = \mathrm{rhs}(n). $

Let $ F(n,k) = \tilde{F}(n,k)/\mathrm{rhs}(n).$ Then

$\displaystyle \sum_k F(n,k) = 1$ (3.1)

(3.1) can be proven by finding a rational function $ R(n,k)$ such that

$\displaystyle F(n+1,k) - F(n,k) = R(n,k+1)F(n,k+1) - R(n,k)F(n,k)$ (3.2)

Using (3.2) when $ k = -1$, we can find that $ R(n,0) = 0$. Solving (3.2) for $ R(n,k+1)$, we obtain:

$\displaystyle R(n,k+1) = \frac{F(n,k)}{F(n,k+1)} \left[\frac{F(n+1,k)}{F(n,k)} + R(n,k) - 1 \right].$ (3.3)

From (3.3), we can find $ R(n,1)$, $ R(n,2)$ and so on.

Then we get a system of equations:

0 $\displaystyle =$ $\displaystyle a_0$  
$\displaystyle (b_0 + b_1 + b_2 + b_3) R(n,1)$ $\displaystyle =$ $\displaystyle a_0 + a_1 + a_2 + a_3$  
$\displaystyle (b_0 + 2b_1 + 4b_2 + 8b_3) R(n,2)$ $\displaystyle =$ $\displaystyle a_0 + 2a_1 + 4a_2 + 8a_3$  
$\displaystyle (b_0 + 3b_1 + 9b_2 + 27b_3) R(n,3)$ $\displaystyle =$ $\displaystyle a_0 + 3a_1 + 9a_2 + 27a_3$  
  $\displaystyle \vdots$    

next up previous
Next: Preliminary results Up: Automated Proofs of Combinatorial Previous: Examples
Math Guest 1 2004-08-10