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

Question

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.)

Answer 1

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.