Preliminary results

We tested our method using machine arc051 in Busch computer lab on the following classical hypergeometric summation formulas, all of which can be found in [2, Appendix III, pp. 243-245]:

Theorem 1 (Gauss)

$\displaystyle {}_{2}\mathrm{F}_{1} \left[ \genfrac{}{}{0pt}{}{a,b}{c} ; {1} \right] = \frac{(c-1)!(c-a-b-1)!}{(c-a-1)!(c-b-1)!}$

(4.1)

Theorem 2 (Gauss)

$\displaystyle {}_{2}\mathrm{F}_{1} \left[ \genfrac{}{}{0pt}{}{a,b}{(1+a+b)/2} ; {1/2} \right] = \frac{(-1/2)!((a-b-1)/2)!}{((a-1)/2)!((b-1)/2)!}$

(4.2)

Theorem 3 (Chu-Vandermonde)

$\displaystyle {}_{2}\mathrm{F}_{1} \left[ \genfrac{}{}{0pt}{}{a,-n}{c} ; {1} \right] = \frac{(c-a)_n}{(c)_n}$

(4.3)

Theorem 4 (Pfaff-Saalschuetz)

$\displaystyle {}_{3}\mathrm{F}_{2} \left[ \genfrac{}{}{0pt}{}{a,b,-n}{c,a+b-c-n+1} ; {1} \right] = \frac{(c-a)_n(c-b)_n}{(c)_n(c-a-b)_n}$

(4.4)

Theorem 5 (Kummer)

$\displaystyle {}_{2}\mathrm{F}_{1} \left[ \genfrac{}{}{0pt}{}{a,b}{1+a-b} ; {-1} \right] = \frac{(a-b)!(a/2)!}{a!(a/2-b)!}$

(4.5)

Theorem 6

$\displaystyle {}_{4}\mathrm{F}_{3} \left[ \genfrac{}{}{0pt}{}{a,1+a/2,b,c}{a/2,1+a-b,1+a-c} ; {-1} \right] = \frac{(a-b)!(a-c)!} {a!(a-b-c)!}$

(4.6)

Theorem 7 (Bailey)

$\displaystyle {}_{2}\mathrm{F}_{1} \left[ \genfrac{}{}{0pt}{}{a,1-a}{c} ; {«} \right] = \frac{(c/2-1)!(c/2-1/2)!}{(c/2+a/2-1)!}{(c/2-a/2-1/2)!}$

(4.7)

Theorem 8 (Dixon)

$\displaystyle {}_{3}\mathrm{F}_{2} \left[ \genfrac{}{}{0pt}{}{a,b,-n}{1+a-b,1+a+n} ; {1} \right] = \frac{(1+a)_n(1+a/2-b)_n}{(1+a/2)_n(1+a-b)_n}$

(4.8)

Theorem 9

$\displaystyle {}_{4}\mathrm{F}_{3} \left[ \genfrac{}{}{0pt}{}{a,1+a/2,b,-n}{a/2,1+a-b,1+a+n} ; {-1} \right] = \frac{(1+a)_n}{(1+a-b)_n}$

(4.9)

Theorem 10

$\displaystyle {}_{3}\mathrm{F}_{2} \left[ \genfrac{}{}{0pt}{}{1+a/2,a,-n}{a/2,b} ; {1} \right] = (b-a-n-1)\frac{(b-a)_{n-1}}{(b)_n}$

(4.10)

Identity	Our time (sec)	EKHAD's time (sec)	ratio
(4.1)	0.0156	0.1898	12.167
(4.2)	0.0068	0.2367	34.81
(4.3)	0.0184	0.2358	12.82
(4.4)	0.5814	0.5546	0.9539
(4.5)	0.0164	0.2681	16.348
(4.6)	0.2717	0.5342	1.966
(4.7)	0.1035	0.2559	2.472
(4.8)	0.0877	0.3921	4.471
(4.9)	0.1759	0.5613	2.850
(4.10)	0.905	0.220	0.243