# Mathematica implementation of Zeilberger's algorithm (previously done in Maple)

I have this Mathematica code:

G = (4^(-3 - 2 α) Gamma[5/2 + 3 α] Gamma[2 + 5
α] ((-54 + α (39 + 5 α (628 + 25 α (161 +
2 α (-581 + 740 α)))))
HypergeometricPFQ[{
1, 2/5 + α, 3/5 + α, 4/
5 + α, 5/6 + α, 7/6 + α,
6/5 + α}, {13/10 + α,
3/2 + α, 17/10 + α, 19/10 + α, 2 + α, \
21/10 + α}, 27/64] + (347274 + 5 α (-312019 + 25 α (22255 + 8
α (-2431 + 925 α)))) HypergeometricPFQ[{2, 2/5 + α, 3/
5 + α, 4/5 + α, 5/6 +
α, 7/6 + α, 6/5 + α}, {13/10 + α, 3/2 + α,
17/10 + α, 19/10 + α, 2 + α, 21/10 + α}, 27/
64] + 10 ((-769797 + 25 α (66227 + 4 α (-12843 + 3700
α))) HypergeometricPFQ[{3, 2/5 + α, 3/5 + α, 4/5 +
α, 5/6 + α, 7/6 + α, 6/5 +
α}, {13/10 + α, 3/2 + α, 17/
10 + α, 19/10 + α, 2 + α, 21/10 + α}, \
27/64] + 75 ((44133 + 8 α (-6131 + 1850 α)) HypergeometricPFQ[{4, 2/5 + α, 3/
5 + α, 4/5 + α, 5/6 + α, 7/6 + α, 6/5 + α}, {13/10 + α, 3/
2 + α, 17/10 + α, 19/10 + α, 2 + α, 21/10 + α}, 27/64] +
8 ((-7981 + 3700 α) HypergeometricPFQ[{5, 2/
5 + α, 3/5 + α, 4/5 + α, 5/6 + α, 7/6 + α,
6/5 + α}, {13/10 + α, 3/2 +
α, 17/10 + α, 19/10 + α, 2 +
α, 21/10 + α}, 27/64] +
3700 HypergeometricPFQ[{6, 2/5 + α, 3/5 \
+ α, 4/5 + α, 5/6 + α, 7/6 + α, 6/5 + α}, {13/10 + α, 3/2 + α, 17/
10 + α, 19/10 + α, 2 + α, 21/10 +
α}, 27/64])))))/(3
Gamma[1 + α] Gamma[3 + 2 α]
Gamma[13/2 + 5 α]);


In my J. Phys. A paper, in Figs. 5 and 6 I include two Maple worksheets of Qing-Hu Hou applying the famous Zeilberger ("creative telescoping") algorithm to this expression, with the resultant very interesting set of relations shown in eqs. (1)-(3) of the paper.

Now, I'd like to perform the same analysis and related ones in Mathematica ("my native language"). There is an available Mathematica package to do this, it would seem. However, I haven't gotten it to succeed. Don't know if the output is too large or not, or if I'm doing things exactly right. So, what I want is an implementation in Mathematica of what Qing-Hu Hou accomplished with Maple. (I do have Maple 14--not very recent I think--and did try implementing his package on my machine, but have not really succeeded.)

• What exactly goes wrong with the package code? Perhaps there's no need to re-implement everything if an implementation if already available. – MarcoB Jun 24 '16 at 18:21
• Thanks MarcoB! When I issue the command, in various forms, I simply get \$Failed as a response, so it's hard to know exactly what is not working. It's certainly a possibility that I'm not proceeding entirely correctly, since I have no previous experience with the RISC package (and I've sent this posting to the authors). If there is a genuine problem, I suspect it may be that the input is too large. – Paul B. Slater Jun 24 '16 at 18:46
• Paul, could you include the exact expression that failed? – MarcoB Jun 24 '16 at 19:14
• Well, here's what I think should serve as input to the Zb command in the RISC package. – Paul B. Slater Jun 24 '16 at 20:45

There is also a newer package, HolonomicFunctions, that has an implementation of Chyzak's generalization of Zeilberger's algorithm. To perform the desired task, use the following commands:

smnd = Simplify[
G /. HoldPattern[HypergeometricPFQ[pl_List, ql_List, x_]] :>
(Times @@ (Pochhammer[#, k] & /@ pl)) /
(Times @@ (Pochhammer[#, k] & /@ ql)) * x^k / k!];

CreativeTelescoping[smnd, S[k] - 1, S[α]]


As answer, you get exactly the same output as Hou's Maple package.

• Since you work at RISC, I just wanted to point out packagedata.net, a relatively new Mathematica package aggregator. RISCErgoSum is already added there. You can edit it if appropriate and feel free to post more packages as they released (as each package separately). The package list is moderated but anyone may edit it. – Szabolcs Jun 24 '16 at 20:34
• Christoph, Thanks very much for making that package available. – Daniel Lichtblau Jun 24 '16 at 20:35
• Thanks, Szabolcs, for the link. I have now submitted my package info there. – Christoph Koutschan Jun 24 '16 at 21:12