
Contents
Next: A Paean to
Up: Pfaff's Method (III): Comparison
Previous: The 5F4 Summation.
![[Annotate]](/organics/icons/sannotate.gif)
![[Shownotes]](../gif/annotate/sshow-61.gif)
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.