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# The 5F4 Summation (Part II): The Wilf-Petkovsek Summation.

Several months after the end of the ``contest'' described in Section 5, Wilf and Petkovsek [in a paper published in this volume] using the WZ methodology proved in a direct and efficient way the most important special case of (5.1) (namely when z is restricted to a certain class of integers). While this is not the full (5.1), it is nonetheless quite adequate to do the entire evaluation of the Mills-Robbins-Rumsey determinant as presented in Section 7 of [1].

Given the parallelism that we have thus far described between Pfaff proofs and WZ-proofs, it is natural to ask if, inspired by the success of Wilf and Petkovsek, we can find a correspondingly easier proof of the following special case of (5.1). Let n and v be non-negative integers with ,

Then for v and n nonnegative integers,

The proof of (6.2) is now quite straightforward. The bottom line follows easily from classical summations, and the top line follows directly from an application of Pfaff's method. To see the bottom line we note

(by [6, p. 30, eq.(1),] (Click here to see this equation) with , , c=1, m=n-1, )

(by [6, p. 16, Section 3.3, eq.(1),] (Click here to see this equation) with a = 1, , )

Hence the bottom line of (6.2) is proved.

For the top line of (6.2), we proceed using Pfaff's method:

Once (6.4) is established we may set v = n and use the bottom line of (6.2) to conclude

Then by induction on i using (6.4) recursively, we conclude that for :

i.e. the top line of (6.2) is valid.

This, then, is the result obtained by Wilf and Petkovsek and is sufficient to effect the evaluation of the Mills-Robbins-Rumsey determinant in [1].

Contents Next: A Paean to Up: Pfaff's Method (III): Comparison Previous: The 5F4 Summation.