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In our ongoing work to speed up symbolic summation and other similar algorithms in Sagemath, we notice that naive implementations of Gosper and Wilf-Zeilberger (a.k.a. WZ) algorithms are usually quite slow.

What is the state of the art here? E.g. is there software available that can derive Clausen $_4 F_3$ identity (see (V) on p.43 of the A=B book) quickly (few seconds on a laptop, say?) using the the WZ algorithm?

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The group at RISC has been working aggressively towards developing improved algorithms for many computer algebra problems, including the Zeilberger's. So, it is a good place to ask.

Meantime, I just wanted to say this regarding Clausen $_4 F_3$ identity that the OP mentioned. If you read on in the book A=B, on page 127 there is a rewrite of the identity and a procedure outlined as well. I just checked it myself using a much earlier version from Zeilberger, called EKHAD, and executed in Maple. The verification of Clausen is instant!

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  • $\begingroup$ p.127 gives $R(n,k)$. Do you mean to say that actually computing $R(n,k)$ is instant, or that carrying on if $R(n,k)$ is already known is instant? $\endgroup$
    – Dima Pasechnik
    Jan 27, 2017 at 21:50
  • $\begingroup$ I mean computing $R(n,k)$ is instant and hence proving the identity is instant, if you know how the WZ machinary works. $\endgroup$
    – T. Amdeberhan
    Jan 27, 2017 at 22:44

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