
  
%strange.tex:  FORTY  ``STRANGE'' Computer-Discovered HYPERGEOMETRIC SERIES EVALUATIONS...
%%a Plain TeX file by Shalosh B. Ekhad (13 pages)
 
%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\Tilde{\char126\relax}
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
 0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
 height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
%\headline={\rm  \ifodd\pageno  \RightHead  \else 
% \LeftHead  \fi}
%\def\RightHead{\centerline{
%Strange Hypergeometric Evaluations
%}}
%\def\LeftHead{ \centerline{Shalosh B. Ekhad}}
%end macros
 
\bf
\centerline
{
FORTY  ``STRANGE'' 
}
\centerline
{
COMPUTER-DISCOVERED
[and COMPUTER-PROVED (of course!)]
}
\centerline
{
HYPERGEOMETRIC SERIES EVALUATIONS
}
\rm
\bigskip
\centerline{ {\it 
Shalosh B. EKHAD}\footnote{$^1$}
{\eightrm  \raggedright
c/o D. Zeilberger,
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
\hfill \break
E-mail: {\eighttt 
zeilberg at math dot rutgers dot edu} (Subject: For Shalosh), 
\hfill \break
url: {\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/ekhad/ekhad.html .}
First version: Oct. 12, 2004 .
Supported in part by the NSF.
} 
}

{\bf Abstract:} We state and prove forty hypergeometric
(${}_2 F_1$) ``strange'' series evaluations that were found
by a systematic search for `nice' identities inspired
by the simplified and annotated Zeilberger algorithm,
implemented in the Maple package {\tt twoFone}.

\bigskip

{\bf Foreword} (by Human Doron Zeilberger)

There are infinitely many mathematical facts, some shallow but interesting (like $2+2=4$) and some
deep (like the fact that the product of two random million-digit integers is
some specific two-million-digit integer) but utterly boring (to humans, at least).
A (human) {\it theorem} is an {\it interesting} (i.e. {\it simple}) fact
that is not too trivial. Granted, a computer can discover, and even prove,
lots of facts, but can it discover {\bf theorems}?


Until now, WZ theory was able to {\it prove} `nice' identities,
but these had to be first {\it conjectured} (i.e. discovered) by humans.
WZ theory could also {\it discover} lots of {\it new} `theorems',
usually much deeper than the human-discovered ones, 
by systematically running the Zeilberger algorithm on many different
summands, finding, and simultaneously proving, recurrences satisfied by the
definite sums.
But the recurrences outputted
are (usually) horrendous to human eyes. Humans like their facts to be {\it simple}.
In [Z] I showed how to use the simplified Zeilberger algorithm [MZ] in order
to look out for hypergeometric terms that yield recurrences of
lower-than-expected order, in particular, of order one, which imply
{\it closed-form evaluations}.

The novelty of the present compilation is not so much in its {\it substance}.
It is very possible that all (or at least most) of these forty identities
are either already known, or 
are derivable, via classical ${}_2F_1$ transformations, and more modern
quadratic ones, from known evaluations. In particular Theorems $1$ and $2$ have recently
been discovered and proved, by a completely different (and very interesting, analytical) approach
by Robert Maier[M]. Others were conjectured by Bill Gosper and first proved by Ira Gessel
and Dennis Stanton[GS].

Indeed the great  novelty of this article  is in its {\bf form}.
A computer discovering {\it ab initio}, humanly-nice results, worthy of
the name {\it theorems}, {\bf all by itself}, without any human intervention
(except the initial one of programming the
{\bf ansatz} and the meta-algorithm). Let it serve as a {\bf paradigm}
and {\bf iconic example} for future {\bf computer-generated} {\it interesting}
mathematics, as opposed to mere computer-assisted one.

{\bf Introduction}

The classical {\it hypergeometric} series
$$
F(a\, , \, b \, , \, c \, , \,x) \, = \,
\sum_{k=0}^{\infty}
{ {(a)_k (b)_k}  \over {k! (c)_k } } x^k \quad,
$$
(where $(z)_k:=z(z+1)(z+2) \cdots (z+k-1)$),
that nowadays is more commonly denoted by
$$
{}_2 F_1 \left ( {{a,b} \atop {c}} \, ; \, x \right ) \quad,
$$
enjoys several {\it closed-form} evaluations.
The most celebrated one is the {\it three-parameter}
Gauss evaluation at $x=1$ (see, e.g. [AAR])
$$
F(a\, ,\, b \, , \, c \, , \, 1) \, =  \,
{ {\Gamma(c) \Gamma(c-a-b)} \over {\Gamma(c-a) \Gamma(c-b)}} \quad .
\eqno(Gauss)
$$
Next comes Kummer's {\it two-parameter } exact evaluation at
$x=-1$
$$
F(a\, ,\, b \, , \, 1+a-b \, , \, -1)= 
{{\Gamma(1+a-b) \Gamma(1+a/2)} \over {\Gamma(1+a) \Gamma(1+a/2-b)}} \quad .
\eqno(Kummer)
$$
and two other ones at $x=1/2$, also due to Gauss. 
In addition, Gosper conjectured,
and Gessel and Stanton[GS] proved, several `strange'
{\it one-parameter} evaluations at other values of $x$.
Here I will state and prove {\it all} 
such {\bf inequivalent} strange evaluations
that are of the form
$$
F(-an \, , \, bn+b_1 \, , \, cn+c_1,x) \quad 
$$
in the {\bf range}
$$
1 \leq a \leq 2 \quad , \quad
-4 \leq b \leq 4 \quad , \quad
-4 \leq c \leq 4 \quad , \quad
$$
with $a,b,c$ all {\bf integers}. None of them is a specialization
of the classical identities, and none of them can be derived from
other ones in the list via the Euler and Pfaff transformations 
([AAR] Theorem 2.2.5) and their iterations (up to depth $3$).

They were found by running the Maple package
{\tt twoFone} written by my beloved mentor, Doron Zeilberger,
that accompanies his article [Z]. 
I ran out of time and memory for such identities
when $b$ and/or $c$ were allowed to be $5$.
It would be interesting to know whether there exist
infinitely many such strange evaluations. Another
curious surprise was that the numbers $b_1,c_1$, that are,
in general, along with $x$, solutions of a system of {\it 
polynomial}  equations, and hence are only guaranteed to be {\it algebraic},
turn out, at least in all the examples discovered so far, to be
{\it rational numbers}. On the other hand, $x$ is sometimes
irrational, but so far, it was, at worst, a {\it quadratic}
irrationality.

\bigskip
{\bf The Forty Identities}


The proofs below are given in the famous, succinct, WZ (Wilf-Zeilberger)
format([WZ], see also [PWZ]), that only states a rational function,
the so-called {\it certificate}, of $(n,k)$.
Recall that in order to translate it into a traditional proof, 
one has to divide
the summand by the right side, and call it $F(n,k)$.
Calling the supplied certificate $R(n,k)$, one defines
$G(n,k):=R(n,k)F(n,k)$ and verifies the purely-routine high-school
algebra identity
$$
F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k) \quad.
\eqno(WZ)
$$
$(WZ)$ is indeed routine, in every given case, since dividing
throughout by $F(n,k)$ and simplifying, yields an identity
amongst {\it rational functions}.
Finally summing $(WZ)$ over $k$ proves that
$\sum_k F(n,k)$ is identically constant, as a function of $n$,
and the proof is completed by checking that it equals $1$ when
$n=0$.

\bf
Theorem 1 : 
$$
F(-n \, , \, -4n-1/2 \, , \, -3n \, , \, -1 )\,=\,
\left ( {{2^6} \over {3^3}} \right )^n
\, \cdot \,
{{(3/8)_n(5/8)_n} \over {(1/3)_n(2/3)_n}} \quad .
$$

Proof : 
$$
{
{-2(296n^3-216n^2k+771n^2+651n-372nk+52nk^2-155k+177-4k^3+45k^2)(3n-k+1)k}
\over
{(8n+9-2k)(8n+7-2k)(8n+5-2k)(8n+3-2k)(n-k+1)}} \quad 
\halmos
$$

Theorem 2 : 
$$
F(-n \, , \, -4n-5/2 \, , \, -3n-1 \, , \, -1 )\,=\,
\left ( {{2^6} \over {3^3}} \right )^n
\, \cdot \,
{{(7/8)_n(9/8)_n} \over {(2/3)_n(4/3)_n}} \quad.
$$
Proof : 
$$
(296n^3+1215n^2-216n^2k+1660n+52nk^2-
586nk+756-394k-4k^3+71k^2)
\cdot
$$
$$
{{
-2
(3n-k+2)k} \over
{(8n+13-2k)(8n+11-2k)(8n+9-2k)(8n+7-2k)(n-k+1)}}  \quad \halmos
$$

Theorem 3 : 
$$
F(-n \, , \, -3n-1/2 \, , \, -4n \, , \, 4 )\,=\,
\left ( {{-3^3} \over {2^4}} \right )^n
\, \cdot \,
{{(1/3)_n(5/6)_n} \over {(1/4)_n(3/4)_n} } \quad .
$$

Proof : 
$$
{{(344n^3-216n^2k+768n^2+522n-282nk+34nk^2-75k+104+13k^2)
(4n-k+1)k} \over
{4 (6n+7-2k)(6n+5-2k)(6n+3-2k)(n-k+1)(3n+1)}}  \quad \halmos.
$$

Theorem 4 : 
$$
F(-n \, , \, -3n-3/4 \, , \, -4n-1 \, , \, -8 )\,=\,
\left ( {{3^3} \over {2^2}} \right )^n
\, \cdot \,
{{(2/3)_n(7/12)_n} \over {(1/2)_n(3/4)_n}} .
$$

Proof : 
$$
{
{-(2208n^3-1488n^2k+6272n^2+5762n-2620nk+252nk^2-1082k+1707+167k^2)
(4n-k+2)k}
\over
{
12(12n+15-4k)(12n+11-4k)(12n+7-4k)(n-k+1/(3n+2)}}  \quad \halmos
$$

Theorem 5 : 
$$
F(-n \, , \, -3n-3/2 \, , \, -4n-2 \, , \, -8 )\,=\,
\left ( {{3^3} \over {2^2}} \right )^n
\, \cdot \,
{
{(5/6)_n(7/6)_n}
\over
{(3/4)_n(5/4)_n}} \quad.
$$

Proof : 
$$
-(1/24)
{{(184n^2+492n-124nk+326+21k^2-167k)(4n-k+3)k}
\over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n-k+1)}}  \quad \halmos
$$


Theorem 6 : 
$$
F(-n \, , \, -3n-1/4 \, , \, -4n  \, , \, -8 )\,=\,
\left ( {{3^3} \over {2^2}} \right )^n
\, \cdot \,
{{(1/3)_n(5/12)_n} \over {(1/2)_n(1/4)_n}} .
$$

Proof : 
$$
{{(-1/4)(736n^3-496n^2k+1600n^2+1058n+84nk^2-624nk-149k+200+27k^2)(4n-k+1)k}
\over
{(12n+13-4k)(12n+9-4k)(12n+5-4k)(n-k+1)(3n+1)}}  \quad \halmos
$$

Theorem 7 : 
$$
F(-n \, , \, -3n-1/2 \, , \, -4n \, , \, -8 )\,=\,
\left ( {{3^3} \over {2^2}} \right )^n
\, \cdot \,
{{(1/3)_n(2/3)_n} \over
{ (1/4)_n(3/4)_n}} .
$$
Proof : 
$$
(-1/32)(2208n^4-1488n^3k+6640n^3+252n^2k^2-2988n^2k+7104n^2+248nk^2-1812nk+
3156n+53k^2-321k+478) \cdot
$$
$$
{{
(4n-k+1)k} \over
{(6n+7-2k)(6n+5-2k)(6n+3-2k)(n-k+1)(3n+2)(3n+1)}}  \quad \halmos
$$

Theorem 8 : 
$$
F(-n \, ,\,  -3n-5/4 \, , \, -4n-1 \, , \, -8 )\,=\,
\left ( {{3^3} \over {2^2}} \right )^n
\, \cdot \,
{{(2/3)_n(13/12)_n} \over {(1/2)_n(5/4)_n}} .
$$
Proof : 
$$
{
{(-1/4)(736n^3-496n^2k+2336n^2+2382n-956nk+84nk^2-414k+767+55k^2)(4n-k+2)k}
\over
{(12n+17-4k)(12n+13-4k)(12n+9-4k)(n-k+1)(3n+2)}}  \quad \halmos
$$

Theorem 9 : 
$$
F(-n \, , \, -3n-3/2 \, , \, -4n-2 \, , \, 4 )\,=\,
\left ( {{-3^3} \over {2^4}} \right )^n
\, \cdot \,
{{(4/3)_n(5/6)_n} \over {(3/4)_n(5/4)_n }}.
$$

Proof : 
$$
{{
(1/12)(1032n^3-648n^2k+4024n^2+5202n-1690nk+102nk^2-1103k+2228+135k^2)
(4n-k+3)k} \over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n-k+1)(3n+4)}}  \quad \halmos
$$

Theorem 10 : 
$$
F(-n \, , \, -3n-7/4 \, , \, -4n-2 \, , \, -8 ) \, = \,
\left ( \frac{3^3}{2^2} \right )^n
{{(4/3)_n(11/12)_n} \over {(3/2)_n(3/4)_n}} .
$$
Proof : 
$$
{{(-1/12)(2208n^3+9216n^2-1488n^2k+12806n+252nk^2-4112nk+5924-2839k+335k^2)
(4 n-k+3)k} \over {
(12n+19-4k)(12n+15-4k)(12n+11-4k)(n-k+1)(3n+4)}}  \quad \halmos
$$

Theorem 11 : Let $\omega$ be a primitive cubic root of unity
(i.e. $\omega\,=\,(1 \pm \sqrt{3} i)/2$).
$$
F(-n \, , \, -3n-1 \, , \, -2n \, , \, \omega )\,=\,
(-(3/2) \omega+3/4)^n \, \cdot \,
{{(2/3)_n} \over {(1/2)_n }} \quad .
$$
Proof : 
$$
(129n^2+289n-2\omega n+160-2\omega - 88kn+14k\omega n-
101k+19k \omega +13k^2-5 \omega k^2)
\cdot
$$
$$
{{(-1/129)
(-5+13\omega)
(2n-k+1)k} \over
{
(3n+4-k)(3n+3-k)(3n+2-k)(n-k+1) }}
\quad  \halmos
$$


Theorem 12 :
$$
F(-n \, , \, -3n-1/2 \, , \, -n+1/2 \, , \,4 )\, = \, (-3^3)^n .
$$

Proof : 
$$
{{
(1/9)k(2n+1-2k)(84n^2+172n-60nk+12k^2-64k+87)}
\over
{(6n+7-2k)(6n+5-2k)(6n+3-2k)(n+1-k)}} \quad  \halmos
$$


Theorem 13 : 
$$
F(-n \, , \, -3n-3/2 \, , \, -n+1/2 \, , \, 4 ) \, = \,
(-3^3)^n \, \cdot \,
{{(5/6)_n(7/6)_n}
\over
{(1/2)_n(3/2)_n}} \quad .
$$

Proof : 
$$
{{
(1/3)(28n^2+76n-20nk+51-28k+4k^2)(2n+1-2k)k}
\over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n+1-k)}}  \quad \halmos
$$

Theorem 14 :  Let $\alpha \, = \,9 \pm 4 \sqrt{5}$.
$$
F(-n \, , \, -3n-1/2  \, , \, 2n+3/2 \, ,  \,
\alpha  
)
\, = \,
\left ( {{2^7} \over {5^3}}(1+\alpha) \right )^n
\, \cdot \,
{{(3/4)_n(5/4)_n} \over
{(4/5)_n (6/5)_n}}\quad .
$$

Proof : 
$$
(79794+388827n+706989n^2+566136n^3+167796n^4-4518k-4875nk-51972k^2+11429
\alpha nk+
$$
$$
3754\alpha k+2132\alpha k^2+496k^4\alpha +
26472k^3-3696k^4-6816kn^3-6174kn^2-127434k^2n-72852k^2n^2+30768k^3n
$$
$$
-3336n^3\alpha -9466\alpha n^2-8839\alpha n-2718\alpha +10998k\alpha 
n^2+1060k^2\alpha n^2-2192k^3\alpha n+3336kn^3\alpha +3058k^2\alpha n-2584k^3
\alpha ) \cdot
$$
$$
{{
-(1/74576)(31 \alpha -327) k} \over
{
(6n+7-2k)(6n+5-2k)(6n+3-2k)(n-k+1)(4n+3+2k)
(\alpha -1) 
}} .
\quad \halmos
$$

Theorem 15 : Let $\alpha \, = \, 9 \pm 4 \sqrt{5}$.
$$
F(-n \, , \, -3n-3/2 \, , \, 2n+5/2 \, , \, \alpha  ) \, = \,
\left ( {{2^7} \over {5^3}}(1+\alpha) \right )^n
\, \cdot \,
{{(5/4)_n(7/4)_n}
\over
{(7/5)_n(8/5)_n}} \quad .
$$

Proof : 
$$
(384115+1282296n+1575413n^2+845796n^3+167796n^4-86193k-
153105nk-77610k^2+26963\alpha nk+
$$
$$
13383\alpha k+730\alpha k^2+
496k^4\alpha +34264k^3-3696k^4-6816kn^3-
$$
$$
74602kn^2-154458k^2n-
72852k^2n^2+30768k^3n-3336n^3\alpha -12126\alpha n^2-14517\alpha n-
$$
$$
5745\alpha +17114k\alpha n^2+1060k^2\alpha n^2-2192k^3\alpha n+
3336kn^3\alpha +1842k^2\alpha n-2984k^3\alpha ) \cdot
$$
$$
{{
k(-1/74576)
(31\alpha -327)
}
\over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n-k+1)(4n+5+2k)(\alpha -1)}}  \quad \halmos
$$

Theorem 16 : Let $\alpha \, =2 \, \pm \sqrt{3}$.
$$
F(-n \, , \, -2n-1/2 \, , \, -4n-1 \, , \,4\alpha  ) \, = \,
(3/4-(3/2)\alpha )^n 
%
\, \cdot \,
{{(2/3)_n} \over {(1/2)_n}} \quad .
$$

Proof : 
$$
(488n^2+978n-20\alpha n+472-23\alpha +94k^2-
25k^2\alpha -420kn+52kn\alpha -422k+72k\alpha )
\cdot
$$
$$
{{
(1/732)
(-68+19\alpha )(4n+2-k)k} \over {
(4n+5-2k)(4n+3-2k)(n-k+1)(3n+2)}}  \quad \halmos
$$

Theorem 17 : 
$$
F(-n \, , \, -2n-1/3 \, , \, -3n \, , \, 9 )\, = \,
(-2^2)^n \, \cdot \,
{{(1/2)_n} \over {(1/3)_n}} \quad .
$$

Proof : 
$$
{{
(1/9)k(3n-k+1)(45n^2-21nk+72n-11k+25)} \over
{(6n+7-3k)(6n+4-3k)(n-k+1)(2n+1)}}  \quad \halmos.
$$

Theorem 18 : 
$$
F(-n \, , \, -2n-2/3 \, ,\, -3n-1 \,, \,9 ) \, = \,
(-2^2)^n \, \cdot \,
{{(5/6)_n} \over {(2/3)_n}} \quad .
$$

Proof : 
$$
{{
(1/6)(15n+19-7k)(3n-k+2)k} \over
{
(6n+8-3k)(6n+5-3k)(n-k+1)}}  \quad \halmos
$$

Theorem 19 : 
$$
F(-n \, , \, -n-1/2 \, , \, 4n+9/2 \, , \, 1/5 )
\, = \,
\left ( {{2^{14}} \over {5^6}} \right )^n
\, \cdot \,
{{
(9/8)_n(11/8)_n(13/8)_n(15/8)_n } \over
{(6/5)_n(9/5)_n(13/10)_n(17/10)_n}} \quad.
$$

Proof : 
$$
(3036n^4+1256n^3k+16960n^3+35275n^2+4780n^2k-1184n^2k^2+6180nk-
$$
$$
3440nk^2-544nk^3+32445n+2720k+11154-2480k^2-64k^4-800k^3)
\cdot
$$
$$
{{
(-5/16)
k}
\over
{
(2n+3-2k)(n-k+1)(8n+13+2k)(8n+11+2k)(8n+9+2k)}}  \quad \halmos
$$

Theorem 20 : 
$$
F(-n \, , \, -n+1/2 \, , \, 4n+3/2 \, , \, 1/5 ) \, = \,
\left ( {{2^{14}} \over {5^6}} \right )^n
\, \cdot \,
{{(3/8)_n(7/8)_n(5/8)_n(9/8)_n}
\over
{(3/5)_n(2/5)_n(9/10)_n(11/10)_n}} \quad .
$$

Proof : 
$$
(3036n^4+1256n^3k+7852n^3+7409n^2+2844n^2k-1184n^2k^2+2252nk-1344nk^2+2993n-
$$
$$
544nk^3-336k^2+592k+430-352k^3-64k^4)
\cdot
$$
$$
{{
(-5/16)
k
} \over {
(2n+1-2k)(n-k+1)(8n+7+2k)(8n+5+2k)(8n+3+2k)}}  \quad \halmos
$$

Theorem 21 :  Let $\beta \, = \, (3 \pm \sqrt{5})/2$.
$$
F(-n \, , \, 2n+1 \, , \, -3n \, , \, \beta  )
\, = \,
\left ( {{5^2} \over {3^3}}(1+\beta) \right )^n
\, \cdot \,
{
{(2/5)_n(3/5)_n} \over
{(1/3)_n(2/3)_n}} \quad.
$$

Proof : 
$$
(1376n+308+2026n^2+958n^3-60k-23k^3\beta +38k^2\beta -
91k\beta +139\beta n+87\beta n^2
$$
$$
-201k\beta n-
87k\beta n^2+38k^2\beta n+48k^3+4k^2-122kn-110kn^2+4k^2n+52\beta )
\cdot
$$
$$
{{
(1/4790)
k(3n-k+1)(-78+17\beta )
} \over {
(n-k+1)(5n+3)(5n+2)(n+1)(2n+1)}}  \quad \halmos
$$

Theorem 22 : 
Let $\beta \, = \, (3 \pm \sqrt{5})/2$.
$$
F(-n \, , \, 2n+2 \, , \, -3n-1 \, , \, \beta  ) \, = \,
\left ( {{5^2} \over {3^3}}(1+\beta) \right )^n
\, \cdot \,
{{(4/5)_n(6/5)_n} \over
{(2/3)_n(4/3)_n}} \quad.
$$
Proof : 
$$
(4179n+3463n^2+958n^3+24k-317k\beta n-87k\beta n^2+48k^3+54k^2-
109kn-110kn^2+4k^2n+96\beta +
$$
$$
1674+183\beta n+
87\beta n^2+38k^2\beta n-23k^3\beta +34k^2\beta -251k\beta )
\cdot
$$
$$
{{
(1/4790)
k(3n-k+2)(-78+17\beta )
}
\over {
(n-k+1)(5n+6)(5n+4)(2n+3)(n+1)}}  \quad \halmos
$$

Theorem 23 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$.
$$
F(-n \, , \, 2n+1 \, , \, 4n+2 \, , \,\phi^{-1}  )
\, = \,
\left ( {{2^6} \over {5^3}}(2-\phi^{-1}) \right )^n
\, \cdot \,
{{
(3/4)_n(5/4)_n} \over
{(4/5)_n (6/5)_n}} \quad.
$$

Proof : 
$$
(2815n+3661n^2+1558n^3+14k^3\phi^{-1} -
148\phi^{-1} n-84\phi^{-1} n^2+579k+183k\phi^{-1} n+
$$
$$
1567kn+1013kn^2+88k\phi^{-1} +37k^2\phi^{-1} n+34k^2\phi^{-1} +
$$
$$
270k^2n+84k\phi^{-1} n^2+712+206k^2+39k^3-64\phi^{-1} )
\cdot
$$
$$
{{
(-1/779)k
(14\phi^{-1} -25)} \over
{(n-k+1)(4n+4+k)(4n+3+k)(4n+2+k)(-1+\phi^{-1} )}}  \quad \halmos
$$

Theorem 24 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$.
$$
F(-n \, , \,  2n+2 \, , \, 4n+4 \, ,  \, \phi^{-1}  ) \, = \,
\left ( {{2^6} \over {5^3}}(2-\phi^{-1}) \right )^n
\, \cdot \,
{{(5/4)_n(7/4)_n} \over
{(7/5)_n(8/5)_n}} \quad.
$$

Proof : 
$$
(7628n+5998n^2+1558n^3+14k^3\phi^{-1} -
178\phi^{-1} n
$$
$$
-84\phi^{-1} n^2+3188+1762k+ 309k\phi^{-1} n+2697kn+1013kn^2+253k\phi^{-1} +
$$
$$
37k^2\phi^{-1} n+79k^2\phi^{-1} +270k^2n+84k\phi^{-1} 
n^2+387k^2+39k^3-94\phi^{-1} )
\cdot
$$
$$
{{
(-1/779)
k
(14\phi^{-1} -25)} \over {
(n-k+1)(4n+6+k)(4n+5+k)(4n+4+k)(-1+\phi^{-1} )}}  \quad \halmos
$$

Theorem 25 : Let $\gamma \, = \, 3 \pm 2 \sqrt{2}$.
$$
F(-2n \, , \, -4n-1/2 \, , \, -2n+1/2 \, , \, \gamma  ) \, = \,
(-16+96\gamma )^n
{{
(3/8)_n(5/8)_n} \over
{(1/4)_n(3/4)_n}} \quad.
$$
Proof : 
$$
(25434+137162n+271600n^2+233440n^3+73472n^4+5216k^2\gamma n^2+
10144k^2\gamma n-21148k\gamma n-
$$
$$
6208k\gamma n^3-20304k\gamma n^2-1264k^3\gamma n-
142224kn^2-58944kn^3-111324kn+14304k^2n^2+
$$
$$
22536k^2n-1776k^3n-1372k^3+8594k^2+88k^4+4834k^2\gamma -
7067k\gamma -1204k^3\gamma -28145k+3837\gamma +
$$
$$
104k^4\gamma +13852\gamma n+6208\gamma n^3+16240\gamma n^2)
\cdot
$$
$$
{{
(-1/1148)(-89+13\gamma )
(4n+1-2k)k} \over {
(8n+9-2k)(8n+7-2k)(8n+5-2k)(8n+3-2k)(2n-k+1)(2n-k+2)(\gamma -1)}}  
\quad \halmos
$$


Theorem 26 : Let $\delta \, = \, 5 \pm 3 \sqrt{3}$.
$$
F(-2n \, , \, -4n-1/3 \, , \, 2/3 \, , \, 2\delta  ) \, = \,
(3^3(26 \delta +5))^n
\, \cdot \,
{{(7/12)_n} \over {(3/4)_n}}
 \quad .
$$
Proof : 
$$
(2059512+12099448n+766292\delta n+
25671232n^2+23410464n^3+7767936n^4+933000\delta n^2-
$$
$$
1239770k\delta n-404352kn^3\delta +506562k^2\delta 
n+268020k^2\delta n^2-62100k^3\delta n-1257180k\delta 
n^2
$$
$$
-2121648k-5589216kn^3-12542136kn^2-9090980kn+1684440k^2n^2+2508684k^2n
$$
$$
-240624k^3n-56934k^3\delta +232953k^2\delta 
-390660k\delta +370656n^3\delta 
$$
$$
+205020\delta +13446k^4-178812k^3+904542k^2+4941k^4\delta )
\cdot
$$
$$
{{
(1/364122)k(-1+3k)
(61\delta -776)} \over {
(12n+13-3k)(12n+10-3k)(12n+7-3k)(12n+4-3k)(2n-k+1)(2n-k+2)
2\delta -1)}}  \quad \halmos
$$

Theorem 27 : Let $\delta \, = \, 5 \pm 3 \sqrt{3}$.
$$
F(-2n \, , \, -4n-2/3 \, , \, 4/3 \, , \, 2\delta  )=
(3^3(26 \delta +5))^n
\, \cdot \,
{{(5/12)_n} \over {(5/4)_n}} \quad.
$$
Proof : 
$$
(3710744+18504312n+607916\delta n+
33746032n^2+26726016n^3+7767936n^4+661656\delta n^2-
$$
$$
1315958k\delta n-404352kn^3\delta +533802k^2\delta n+
268020k^2\delta n^2-62100k^3\delta n-1280676k\delta n^2-
$$
$$
3169096k-5589216kn^3-14213784kn^2-11770892kn+1684440k^2n^2+
2829540k^2n-240624k^3n
$$
$$
-60696k^3\delta +260835k^2\delta -
439228k\delta +235872n^3\delta +182668\delta 
$$
$$
+13446k^4-200988k^3+1159494k^2+4941k^4\delta )
\cdot
$$
$$
{{
(1/364122)k(1+3k)
(61\delta -776)
} \over {
(12n+14-3k)(12n+11-3k)(12n+8-3k)(12n+5-3k)(2n-k+1)(2n-k+2)(2\delta -1)
}}  \quad \halmos 
$$


Theorem 28 : Let $\alpha \, = \, 2 \pm \sqrt{3}$.
$$
F(-2n \, , \, -3n-1/4 \, , \, -4n \, , \, 4\alpha  ) \, = \, 
((39/2)\alpha -21/4)^n \, \cdot \,
{{(5/12)_n} \over {(1/4)_n}} \quad.
$$

Proof : 
$$
(171978n+238000n^2+104736n^3-12496k\alpha n^2-
32440k\alpha n+7780k^2\alpha n+17488\alpha n^2+
$$
$$
30900\alpha n+8407k^2\alpha -1216k^3\alpha -18581k\alpha +
39182+13262\alpha 
$$
$$
-81808kn^2+15490k^2-1792k^3-43403k-
122224kn+21112k^2n)
\cdot
$$
$$
{{
(-1/3273)
k(4n-k+1)
(397\alpha -1484)
} \over
{
(12n+13-4k)(12n+9-4k)(12n+5-4k)(2n-k+1)(2n-k+2)}}  
\quad \halmos
$$

Theorem 29 : 
$$
F(-2n \, , \, -3n+1 \, , \, -2n+2 \, , \, -2 )\, = \, (3^3)^n \, \cdot \,
(-2n+1)
\quad .
$$

Proof : 
$$
{{
(-1/9)
k(2n-k-1)
(39n^2+55n-21kn-14k+3k^2+18)
} \over {
(3n+2-k)(3n+1-k)(3n-k)(2n+2-k)}}  \quad \halmos
$$

Theorem 30 : 
$$
F(-2n \, , \, -3n \, , \,  -2n+1/2 \, , \, 1/4 ) \, = \, 
\left ( {{3^3} \over {2^6}} \right )^n
\, \cdot \,
{{(1/3)_n (2/3)_n} \over
{(1/4)_n(3/4)_n }} \quad.
$$

Proof :
$$
{{
( -2/3)(74n^3+168n^2-57kn^2-87kn+124n+12k^2n-32k+30-k^3+9k^2)(4n+1-2k)k
} \over {
(3n+3-k)(3n+2-k)(3n+1-k)(2n+1-k)(2n+2-k)}}  \quad \halmos
$$

Theorem 31 : Let $\eta \, = \, -7 \pm 4 \sqrt{3}$.
$$
F(-2n \, , \,  -3n-1/4 \, , \, -n+3/4 \, , \, \eta  ) \, = \, 
(312\eta +24)^n 
\, \cdot \,
{{(5/12)_n} \over {(1/4)_n}} \quad .
$$

Proof : 
$$
(485694n+611008n^2+248736n^3-5600k^2\eta n+
11504k\eta n^2+23720\eta nk-11880\eta n-
$$
$$
6512\eta n^2+752k^3\eta +22744k^2+11851\eta k-5432\eta k^2-
204544nk-79751k-2416k^3
$$
$$
+28576k^2n-126928kn^2-5299\eta +124358)
\cdot
$$
$$
{{
(1/124368)
(47\eta +809)
(4n+1-4k)k
} \over
{(12n+13-4k)(12n+9-4k)(12n+5-4k)(2n-k+1)(2n-k+2)(\eta -1)}}  \quad \halmos
$$

Theorem 32 : 
$$
F(-2n \, , \, -2n+1/3 \, , \, -n+5/6 \, , \, -1/8  \, )= \,
\left ( {{3^3} \over {2^4} } \right )^n
\, \cdot \,
{{(1/3)_n} \over {(1/6)_n}} \quad .
$$

Proof : 
$$
{{
(-4/27)(132n^2-60nk+202n+74+9k^2-45k)(6n+1-6k)k}
\over
{(6n+5-3k)(6n+2-3k)(2n-k+1)(2n-k+2)}}  \quad \halmos
$$

Theorem 33 : Let $\zeta \, = \, -1 \pm \sqrt{2}$.
$$
F(-2n \, , \, 1/2 \, , \, -4n \, , \, 2\zeta  ) \, = \,
{{(3/8)_n(5/8)_n} \over
{(1/4)_n (3/4)_n }} \quad.
$$

Proof : 
$$
{{
(-1/2)
(3+\zeta )
(6\zeta n-2n+4\zeta -3k^2-k^2\zeta +8kn+6k)(4n-k+1)k
} \over {
(2n-k+2)(2n-k+1)(8n+5)(8n+3)}}  
\quad \halmos
$$

Theorem 34 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$ then
$$
F(-2n \, , \, 2n+1 \, , \, -3n+1/2 \, , \, \phi  ) \, = \,
\left ( {{5^2} \over {3^3}} (4 \phi+3) \right )^n
\, \cdot \,
{{(3/10)_n(7/10)_n} \over
{(1/6)_n(5/6)_n}} \quad .
$$

Proof : 
$$
(1678n+2708n^2+1448n^3+136k^2\phi n-264k\phi n^2-364k\phi n-
118k\phi +324\phi n
$$
$$
+264\phi n^2-26k^3\phi +96k^2\phi +343+96\phi -
267k-19k^2+24k^3-42k^2n-536kn^2-750kn)
\cdot
$$
$$
{{
(2/905)k(6n+1-2k)(-43+16\phi )
} \over {
(2n+1)(2n-k+2)(2n-k+1)(10n+7)(10n+3)}}  \quad \halmos
$$

Theorem 35 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$.
$$
F(-2n \, , \, 2n+1 \, , \, -3n \, , \, \phi  ) \, = \,
\left ( {{5^2} \over {3^3}} (4 \phi+3) \right )^n
\, \cdot \,
{{(2/5)_n(3/5)_n} \over {(1/3)_n(2/3)_n}} .
$$

Proof : 
$$
(1334n+1716n^2+724n^3-270k\phi n+68k^2\phi n-132k\phi n^2-125k\phi +
214\phi n+132\phi n^2
$$
$$
-13k^3\phi +68k^2\phi +82\phi -177k-21k^2+12k^3-
21k^2n-268kn^2-433kn+342)
\cdot
$$
$$
{{
(1/905)k(3n-k+1)
(-43+16\phi )
} \over {
(2n-k+2)(2n-k+1)(5n+3)(5n+2)(n+1)}}  \quad \halmos
$$

Theorem 36 : Let $\zeta_1 \, = \, 1 \pm \sqrt{2}$.
$$
F(-2n \, , \, 2n+1 \, , \, -2n+1/2 \, , \, (1/2)\zeta_1  ) \, = \,
(4+8\zeta_1 )^n \, \cdot \,
{{(3/8)_n(5/8)_n} \over {(1/4)_n(3/4)_n}} \quad .
$$

Proof : 
$$
(56n^2+72n+8\zeta_1 n+24+5\zeta_1 -6k^2-16kn-12k+4k^2\zeta_1 -
8\zeta_1 kn-6\zeta_1 k)
\cdot
$$
$$
{{
(2/7)k(4n+1-2k)
(-4+\zeta_1 )
} \over {
(2n-k+2)(2n-k+1)(8n+5)(8n+3)}}  \halmos
$$

Theorem 37 : Let $\alpha \, = \, 2 \pm \sqrt{3}$.
$$
F(-2n \, , \, 2n+1 \, , \, -n+3/4 \, , \,  (1/4)
\alpha  )=(3-6\alpha )^n \, \cdot \,
{{(5/12)_n} \over {(1/4)_n}} \quad .
$$

Proof : 
$$
{{
(1/3)(8n+5+\alpha -2k\alpha +2k)
(-5+\alpha )(4n+1-4k)k } \over 
{(2n-k+2)(2n-k+1)(12n+5)}}  \quad \halmos
$$

Theorem 38 : 
$$
F(-2n \, , \, 3n+1 \, , \, -2n+2 \, , \, 2/5 ) \, = \,
\left ( {{5^3} \over {3^3}} \right )^n
\, \cdot \, 
(-2n+1)
\quad.
$$

Proof : 
$$
{{
(-1/25)(147n^2+33nk+179n+3k^2+23k+50)(2n-k-1)k} \over
{n(2n+2-k)(3n+2)(3n+1)}}  \quad \halmos
$$

Theorem 39 : Let $\zeta_2 \, = \, 1 \pm \sqrt{3}/3$.

$$
F(-2n \, , \, 4n+2 \, , \, 4/3 \, , \, (2/3)\zeta_2  ) \, = \, 
(-9+18\zeta_2 )^n \, \cdot \,
{{(5/12)_n} \over {(5/4)_n }} \quad .
$$

Proof : 
$$
(352n^2+432n+168\zeta_2 n+116+138\zeta_2 -2k^2+9\zeta_2 k^2-
$$
$$
288\zeta_2 nk+152nk+126k-237\zeta_2 k)
\cdot
$$
$$
{{
(-1/88)
(-16+9\zeta_2 )
(1+3k)k } \over {
(2n-k+2)(2n-k+1)(12n+5)(4n+3)}}  \quad \halmos
$$

Theorem 40 : Let $\zeta_2 \, = \, 1 \pm \sqrt{3}/3$.
$$
F(-2n \, , \, 4n+1 \, , \, 2/3 \, , \, (2/3)\zeta_2  ) \, = \,
(-9+18\zeta_2 )^n \, \cdot \,
{{(7/12)_n} \over {(3/4)_n}} \quad .
$$

Proof : 
$$
(352n^2+264n+264\zeta_2 n+20+174\zeta_2 -2k^2+9\zeta_2 k^2-
288\zeta_2 nk+152nk+102k-195\zeta_2 k)
\cdot
$$
$$
{{
(-1/88)
(-16+9\zeta_2 )
(-1+3k)k
} \over {
(2n-k+2)(2n-k+1)(12n+7)(4n+1)}}  \quad \halmos
$$

\rm
{\bf Acknowledgement:}  Many thanks are due to my beloved
mentor, Doron Zeilberger, for writing the Maple package {\tt twoFone},
and to Mohamud Mohammed for stimulating conversations with
him.

{\bf REFERENCES}

[AAR] G.E. Andrews, R. Askey, and R. Roy, {\it ``Special Functions''},
Cambridge Univ. Press, 1999.

[GS] I. Gessel and D. Stanton, 
{\it Strange evaluations of hypergeometric series},
SIAM J. Math. Anal. {\bf 13}(1982), 295-308.

[PWZ] M. Petkovsek, H. S. Wilf and D. Zeilberger,
{\it ``A=B''}, AK Peters, Wellesley, (1996).
[available on-line from the authors' websites.]

[M] R.S. Maier, {\it A Generalization of Euler's hypergeometric 
transformation},
preprint. (available from {\tt arXiv.org}).

[MZ] M. Mohammed and D. Zeilberger,
{\it Sharp upper bounds for the orders outputted by the Zeilberger and
q-Zeilberger algorithms}, 
to appear in J. Symb. Comp., available from the authors' websites.

[WZ] H.S. Wilf and D. Zeilberger, {\it Rational functions certify combinatorial
identities}, J. Amer. Math.  Soc. {\bf 3} (1990), 77-83.

[Z] D. Zeilberger,
{\it DECONSTRUCTING the ZEILBERGER algorithm},
submitted, [available on-line from the author's website].

\end
















