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\bf 
GAUSS'S $ {{}_2 F_1 (1)}$ CANNOT BE GENERALIZED
TO $ {{}_2 F_1 (x)}$\rm
 
{\it Doron Zeilberger}
\footnote{*}{
Department of Mathematics, Temple University,
Philadelphia, PA19104. Supported in part by NSF grant DMS8800663.
}
 
{\bf Appeared in:} J. Comp. Appl. Math. {\bf 39} (1992), 379-382.
 
{\bf Abstract:} Using ideas of Jet Wimp and Richard McIntosh, it is
proved that Gauss's explicit evaluation of ${}_2 F_1 (a,b;c;1)$
cannot be generalized to ${}_2 F_1 (a,b;c;x)$, for arbitrary
$a,b,c$, and $x$.  A short proof of Wimp's theorem that asserts that
${}_3 F_2 (a,b,c;d,e;1)$ cannot be expressed in closed form
is also given.
 
{\bf 0 Introduction}
 
As we all know, a {\it geometric series} is a series
 
$$
 \sum_{k=0}^{\infty} A_k 
\eqno(1)
$$          
 
such that the ratio $A_{k+1} / A_k$ of consecutive terms
is identically equal to a {\it constant}, say $r$, and if $r<1$,
such a series
is {\it explicitly summable}, the sum being
$ A_0 /(1-r)$.
 
A {\it hypergeometric series} (e.g. [R], [B]) is a series (1) where
$A_{k+1} / A_k$ is a {\it rational function} of $k$. Writing this
rational function in factored form
 
$$
 {{ x (k+ a_1 ) ( k + a_2 ) ... ( k + a_p )}\over{  (k+1) (k+ b_1 ) ( k + b_2 ) ... ( k + b_q )}} \, \,, 
\eqno(2)
$$        
 
the resulting series (1) (with $A_0 = 1$) is denoted by
 
$$
  {}_p F_q  (  {   { a_1  \, , ... \,, \, a_p } \atop   { b_1  \,, \,... \,, b_q }                                }  \,; \, x                   ) \, \, \,.                   
$$
 
(The extra factor $(k+1)$ in the denominator of (2) is there for 
historical reasons,
of course it is possible to get rid of it by making one of the numerator
parameters $a_i$ equal to $1$.) In terms of the
{\it raising factorial} $(a)_k := a(a+1) ... (a+k-1)$, the
general hypergeometric series can be written, and is usually defined
by
 
$$
  {}_p F_q  (  {   { a_1  \, , \, ... \,, \, a_p } \atop   { b_1  \,, \,... \,, \, b_q }                                } ; x                   ) \,= \, \sum_{k=0}^{\infty} {{ ( a_1 )_k ... ( a_p )_k }\over{ k! ( b_1 )_k ... ( b_q )_k }} x^k  \, \, \,. 
$$
 
There are many cases in which a hypergeometric series can be
summed "explicitly", i.e. in terms of powers and products of
Gamma functions.
The simplest case is the binomial theorem:
 
$$
 {}_1 F_0  (  {   { a \, } \atop   { -  \, }                                }  \,; \, x \,                   ) \,= \, (1-x)^{-a \, \, \,}.  
\eqno(3)
$$    
 
Moving up to ${}_2 F_1 $, we have Gauss's celebrated
formula(e.g. [R], theorem 18, p.49):
 
$$
 {}_2 F_1  (  {   { a,b } \atop   { c }                                } ; 1                   ) \,= \,  {{\Gamma (c) \Gamma (c-a-b) }\over{\Gamma (c-a) \Gamma (c-b)}} \, \, \,. 
\eqno(4)
$$    
 
Despite many attempts, no such formula for $F(a,b;c;x)$, with general
$x$ and general $a,b,c$ was ever found, and the first purpose of the
present paper is to prove that such a formula is impossible. Of course,
for various specializations of $x$ and $a,b,c$ there do exist closed
form formulas([B],[G-S]), the most well known being that of Kummer,
in which $x=-1$ and $c=a-b+1$([R], Theorem 26, p. 68):
 
$$
 {}_2 F_1  (  {   { a,b } \atop   { 1+a-b }                                } \, ; \, -1                   ) \,= \,  {{\Gamma (1+a-b) \Gamma (1+a/2) }\over{\Gamma (1+a/2-b) \Gamma (1+a)}} \, \, \,. 
\eqno(5)
$$    
 
Another natural generalization of Gauss's formula would be
an explicit expression for the general
 
$$
 {}_3 F_2  (  {   { a_1 , a_2 , a_3 } \atop   { b_1 , b_2 }                                }  \,; \, 1                   ) \, \, \,.                   
\eqno(6)
$$    
 
The celebrated formula of Pfaff-Saalschutz sums (6) ([R], p. 87) when
one of the numerator parameters is a negative integer (so
the series is {\it terminating}) and in addition it is
{\it balanced}: $a_1 + a_2 + a_3 + 1= b_1 + b_2$.
This gives a four-parameter formula. The formula of Dixon([R], p. 92)
sums (6) when it is {\it well-poised}: $a_1 +1 = $
$b_1 + a_2 = b_2 + a_3$, giving a 3-parameter
formula. Other 3-parameter formulas are associated with the names
of Watson and Whipple (see [B]). In addition there are many
"strange" 2- and 1-parameter formulas conjectured by Gosper and
proved by Gessel and Stanton[G-S]. The impossibility of a closed
form evaluation of a general ${}_3 F_2 (a,b,c;d,e;1)$
is a remarkable result due to Wimp[W].  Wimp used recurrences 
and an ingenious asymptotic argument. The second purpose
of this paper is to give a short proof of Wimp's theorem. My proof
uses Wimp's beautiful ideas, but the details are much shorter.
I was also very much
influenced by McIntosh's brilliant thesis[M] and his approach to
proving minimality of recurrences.
 
{\bf 1 $ { {}_2 F_1 (a,b;c;x)}$ Is Not Nice}
 
{\bf Theorem}: There is no formula of the form
 
$$
 {}_2 F_1 (a,b;c;x) =  K \lambda^{ A_0 a + B_0 b +  C_0 c } 
{{\prod_{j=1}^{p} \Gamma ( A_j a + B_j b + C_j c + D_j )} 
\over {\prod_{j=1}^{q} \Gamma ( A'_j a + B'_j b + C'_j c + D'_j )}} \, \,, 
\eqno(7)
$$     
 
with $A_0  \,, \, A_1  \,, \, ...  \,, \, A_p  \,,$
$ \, B_0  \,, \, B_1  \,, \, ... ,$
$B_p  \,, \, C_0  \,, \, C_1  \,, \, ...  \,, \, C_{p \,,}$
$ A'_1  \,, \, ...  \,, \, A'_q  \,, \, B'_0  \,, \, B'_1  \,, \, ...  \,, \,$
$B'_q   \,, \, C'_1  \,, \, ...  \,, \, C'_q$ all rational numbers,
and $K$ and $\lambda$ are allowed to depend on $x$.
 
{\bf Proof:} Suppose there is such a formula, then there would be
a formula for 
 
$$
  a(n) := {}_2 F_1 ( -n , n+1 ; 1; -1 ) = \sum_{k=0}^{n}  (  { n+k \atop k }  )  (  { n \atop k }  ) 
\eqno(8)
$$        
 
of the form
 
$$
 a(n) =  K \lambda^{ A_0 n } 
{{\prod_{j=1}^{p} \
\Gamma ( A_j n + D_j )} \over {\prod_{j=1}^{q} \Gamma ( A'_j n + D'_j )}} \, \,, 
\eqno(9)
$$     
 
with $A_0 , A_j $, and $A'_j$ rational numbers. Let
$K$ be the least common denominator of $A_0 , A_j ,$
$A'_j$.Then obviously $a(n+K)/a(n)$ would be a rational
function of $n$, i.e., there would be polynomials $P(n)$ and
$Q(n)$ such that
 
$$
 P(n) a(n+K) - Q(n) a(n) \equiv 0,  \, \, \, \,n=0,1,2 ... \, \, \,. 
\eqno(10)
$$          
 
Note that $P(n)$ and $Q(n)$ can be chosen to have {\it integer}
coefficients, since $a(n)$ are integers, and writing $P(n)$
and $Q(n)$ in generic form, plugging in the first few values
of $n$, and solving the resulting system of linear equations,
gives rational, and hence integer, solutions.
 
A routine application of the method of [K], pp. 66-67, shows
that $a(n+1)/a(n)$ tends to $(1 + \sqrt 2 )^2$, as
$n \rightarrow \infty$ (see also [P], p. 202). This implies
that $a(n+K)/a(n)$ tends to $(1 + \sqrt 2 )^{2K}$ as
$n \rightarrow \infty$. On the other hand if (10) were true, it would follow that
if $a(n+K)/a(n)$ tends to any non-zero limit, that
limit must be a rational number. Now $(1+ \sqrt 2 )^{2K} $
always has the form $a+ b \sqrt 2$, with $a$ and  $b$
integers, and $b \neq  0 $ (use the binomial theorem), and this
must be irrational, thanks to Hippasus of Metapontum (see also [W']).
 
{\bf 2 A Short Proof of Wimp's Theorem}
 
I will now give a short proof of Wimp's theorem[W] that
${}_3 F_2 (a,b,c;d,e;1)$ cannot be expressed in closed form,
in the form analogous to (7). If it were then
 
$$
 b(n) := {}_3 F_2 ( -n  \,, \, -n  \,, \, n+1  \,; \, 1  \,, \, 1 \,; \, 1) = \sum_{k=0}^{n}  (  { n+k \atop k }  )  (  { n \atop k }  )^2 
\eqno(11)
$$        
 
would have a formula of the form (9), which would entail, for
some integer $K$, and polynomials $P(n),Q(n)$ with integer coefficients
that
 
$$
 P(n) b(n+K) - Q(n) b(n) \equiv 0,  \, \, \, \,n=0,1,2 ... \, \, \,. 
\eqno(12)
$$          
 
A routine application of the method of [K], pp. 66-67, shows
that $b(n+1)/b(n)$ tends to $((1 + \sqrt 5 )/2)^5$, as
$n \rightarrow \infty$ (see also [P], p. 200). The rest of the proof
goes ditto, with $\sqrt 2$ replaced by $\sqrt 5$, and
$2K$ replaced by $5K$.
 
{\bf Epilogue}
 
The referee has found a much quicker proof of the main result of
this paper. He pointed out that the specialization
 
$$
  {}_2 F_1 (a,a+1/2;1/2;x)= (1/2) (1+ \sqrt x )^{-2a} + (1/2) (1+ \sqrt x )^{2a} 
$$
 
also immediately implies that Gauss's evaluation of
${}_2 F_1 (1)$ cannot be extended to
${}_2 F_1 (x)$. 
 
In spite of this beautiful proof of the referee, the present paper
still serves a purpose. I believe that, although the referee calls
his or her proof "trivial", it is very sleek, and I am glad that
I can use the present paper to present it. Moreover, trivial or not,
the result stated in the title is interesting, 
and ought to be pointed out.
Finally, the {\it method} described in this paper, that was inspired
by McIntosh and Wimp, can be used for proving non-closed-formness
of many other hypergeometric and combinatorial sums.
 
{\bf REFERENCES}
 
[B] Bailey, W. N. , "Generalized Hypergeometric Series", Cambridge
Math. Tracts {\bf 32}, Cambridge University Press, London, 1935.
(Reprinted: Hafner, New York, 1964 .)
 
[G-S] Gessel, I., and Stanton, D., \it 
Strange evaluations
of hypergeometric series\rm 
, SIAM J. Math. Anal. {\bf 13} (1982),
295-308.
 
[K] Knuth, D. E.,  "Sorting and Searching", \it 
The art of computer
programming\rm 
, vol. 3, Addison-Wesley, Reading, 1973.
 
[M] McIntosh, R. J., \it 
Asymptotic and Arithmetic Properties of
Recurrent Sequences\rm 
, Ph.D. dissertation, UCLA, 1989.
 
[P] van der Poorten, A., \it 
A Proof that Euler missed..., Ap\'ery's
proof of the irrationality of\rm 
 $\zeta (3)$, Math. Intel.
{\bf 1}(1979), 195-203.
 
[Ra] Rainville, E. D., "{\it Special Functions}", Chelsea, Bronx,
1971. Originally published by Macmillan, 1960.
 
[W] Wimp, J., \it 
Irreducible recurrences and representation theorems
for\rm 
 ${}_3 F_2 (1)$, Comp. \& Maths. with Appl. {\bf 9}(1983), 
669-678.
 
[W'] Wimp, J,.{\it The square root of two is irrational},
in: "Against Infinity", an anthology of contemporary mathematical poetry,
(edited by E. Robson and J. Wimp), Primary Press, Parker Ford, PA, p. 79.
 
May 1990.  Revised: Oct, 1991.
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