.S D +4
.EQ
delim $$
.EN
.nr Pi 10
.nr Pt 1

\fBA SHORT, ELEMENTARY, AND EASY, WZ PROOF
OF THE ASKEY-GASPER INEQUALITY THAT WAS USED BY DE BRANGES IN HIS
PROOF OF THE BIEBERBACH CONJECTURE\fR
.SP1
.LP
\fIShalosh B. Ekhad*\fR
.FS*
c/o D. Zeilberger, Department of Mathematics, Temple University,
Philadelphia, PA19122. Supported in part by NSF grants DMS8800663
and DMS8901690.
.FE
.SP1
.S -2
A number of people have lamented the fact that there is not an easier
proof of the [Askey-Gasper] inequality... . If [anybody] wants an
easier proof there are two alternatives. One is to find one. 
The other alternative is ...
.SP1
.DS
                          (Richard Askey and George Gasper,[AG2], p. 21)
.DE
.S +2
.SP1
.P
We will follow the first alternative in the above quotation.
Throughout this paper, $n,j,m,k$ denote non-negative \fIintegers\fR,
while $alpha$ and $t$ denote real numbers (or alternatively,
commuting indeterminates.)  Let, as usual
.SP1
.EQ
(a) sub n := a(a+1) ... (a+n-1),~~~; n!:= (1) sub n ~,~
~1/(-n-1)! := 0~~,
.EN
.SP1
and for any non-negative integers $p$ and $q$,
.SP1
.EQ
"" sub p F sub q left ( cpile {
                                {a sub 1 , ... , a sub p }
                              above
                                 {b sub 1 , ... , b sub q }
                               }
                 ~;~ t right ) ~:=~

sum from k=0 to inf 

      {( a sub 1 ) sub k ... ( a sub p ) sub k }
     over
      {k! ( b sub 1 ) sub k ... ( b sub q ) sub k }

               ^ t sup k ~~~.
.EN
.SP1
We will present a new proof of the following celebrated
inequality, that was used by de Branges[deB] to prove a famous
conjecture of an infamous man.
.SP1
\fBTHE ASKEY-GASPER INEQUALITY\fR([AG1],[AG2])
.SP1
.EQ
"" sub 3 F sub 2 left ( cpile {
                                { -n , n+ alpha + 2 , ( alpha + 1)/2 }
                              above
                                 {( alpha + 3)/2, alpha + 1 }
                               }
                 ~;~ t right ) ~ > ~ 0, ~~,~0 <= t < 1 ~,~alpha > -1
~~.
.EN
.SP1
As in [AG1] [AG2], this follows immediately (and with a bit of effort
also for $alpha > -2$, see[AG2]) from
.SP1
\fBTHE ASKEY-GASPER EQUALITY\fR([AG1],[AG2])
.SP1
.EQ
{( alpha + 2) sub n } over n!
"" sub 3 F sub 2 left ( cpile {
                                { -n , n+ alpha + 2 , ( alpha + 1)/2 }
                              above
                                 {( alpha + 3)/2, alpha + 1 }
                               }
                 ~;~ t right ) ~ =~
sum from j 
{ (1/2) sub j ( alpha /2 +1) sub n-j ( alpha /2 +3/2) sub n-2j 
( alpha +1) sub n-2j }
over
{ j!  ( alpha /2 +3/2) sub n-j ( alpha /2 +1/2) sub n-2j (n-2j)!}
~ times ~
.EN
.SP1
.EQ
left [
   
"" sub 2 F sub 1 left ( cpile {
                                { -n/2 + j ~,~ n/2~+~ alpha /2 +^1/2^-j}
                              above
                                 { alpha /2 +1 }
                               }
                 ~;~ t right ) 

right ] sup 2 ~~~.
.EN
.SP1
.P
Let's recall the [WZ] methodology of presenting proofs of identities
of the form
.SP1
.EQ(*')
sum from k U(m,k) = rhs(m)~~~.
.EN
.SP1
First one divides through by rhs(m), and tries to proves instead,
writing $F(m,k):=U(m,k)/rhs(m)$,
.SP1
.EQ(*)
sum from k F(m,k) = 1 ~~~.
.EN
.SP1
All the prover has to do is present the "certificate" $R(n,k)$,
a certain specific rational function, from which the readers can
reconstruct the proof as follows. They set
.SP1
.EQ
G(m,k):= R(m,k) F(m,k-1),
.EN
.SP1
and then verify 
.SP1
.EQ(WZ)
F(m+1,k)-F(m,k)~=~G(m,k+1)~-~G(m,k).
.EN
.SP1
This is always a purely routine identity, since dividing by $F(m,k)$
results in a specific identity involving sums of rational functions.
Having verified (WZ), the identity (*) follows upon summing 
(WZ) w.r.t $k$, which shows that
$sum from k F(m,k)$ is identically constant. That constant
is shown to be $1$
by checking that plugging in $m=0$ in (*) gives you indeed $1$.
.SP1
.P
The Askey-Gasper identity follows immediately from the
following two lemmas, the first of which is due to Clausen[C],
and which was given a WZ proof in [WZ]. To make this paper
self-contained, we will give it again.
.SP1
\fBLemma 1'\fR([C]):
.SP1
.EQ
left [
   
"" sub 2 F sub 1 left ( cpile {
                                { -n/2 + j ~,~ n/2 ^+^ alpha /2 +1/2-j}
                              above
                                 { alpha /2 +1 }
                               }
                 ~;~ t right ) 

right ] sup 2 ~=~

"" sub 3 F sub 2 left ( cpile {
                                { -n+2j , n-2j + alpha + 1 , 
( alpha + 1)/2 }
                              above
                                 {( alpha + 2)/2, alpha + 1 }
                               }
                 ~;~ t right ) ~~~.
.EN
.SP1
\fBLemma 2':\fR
.SP1
.EQ
{( alpha + 2) sub n } over n!
"" sub 3 F sub 2 left ( cpile {
                                { -n , n+ alpha + 2 , ( alpha + 1)/2 }
                              above
                                 {( alpha + 3)/2, alpha + 1 }
                               }
                 ~;~ t right ) ~ =~
sum from j 
{ (1/2) sub j ( alpha /2 +1) sub n-j ( alpha /2 +3/2) sub n-2j 
( alpha +1) sub n-2j }
over
{ j!  ( alpha /2 +3/2) sub n-j ( alpha /2 +1/2) sub n-2j (n-2j)!}
~ times ~
.EN
.SP1
.EQ
"" sub 3 F sub 2 left ( cpile {
                                { -n+2j , n-2j + alpha + 1 , 
( alpha + 1)/2 }
                              above
                                 {( alpha + 2)/2, alpha + 1 }
                               }
                 ~;~ t right ) ~~~.
.EN
.SP1
.P
By comparing the coefficient of a typical term $t sup m$ on either sides,
it is clear that they are equivalent respectively
to lemmas 1 and 2 below.
.SP1
\fBLemma 1:\fR
.SP1
.EQ
sum from k 
{ (-n/2 +j) sub k ((n+ alpha + 1)/2 -j ) sub k (-n/2 +j) sub m-k
((n+ alpha + 1)/2 -j ) sub m-k}
over
{ k! (m-k)! ( alpha /2 +1) sub k ( alpha /2 +1) sub m-k }
~=~
.EN
.SP1
.EQ
{(-n+2j) sub m (n-2j+ alpha +1) sub m ( ( alpha +1)/2) sub m}
over
{m!^ ( alpha /2 +1) sub m ^( alpha +1) sub m   }~~~.
.EN
.SP1
\fBProof\fR:
.SP1
.EQ
R(m,k):=

{   (2 j - n + 2 k) ( alpha  - 2 j + n + 1 + 2 k) (1 +  alpha  + 3 m - 2 k) }

 over {2 ( alpha  + 2 m + 1) (n - 2 j +  alpha  + m + 1) (- n + 2 j + m)}

~~.~~~"\(sq" 
.EN   

\fBLemma 2:\fR
.SP1
.EQ
{( alpha +2) sub n } over n! {(-n) sub m (n+ alpha +2) sub m 
( ( alpha + 1)/2) sub m }
over
{m!^  (( alpha +3) /2 ) sub m ( alpha +1) sub m} ~=~
.EN
.SP1
.EQ
sum from j
{(1/2) sub j ( alpha / 2 + 1) sub n-j 
( alpha / 2 + 3/2) sub n-2j 
( alpha +1) sub n-2j
 (-n+2j) sub m (n-2j+ alpha +1) sub m (( alpha +1)/2) sub m}
over
{j! ( ( alpha +3)/2) sub n-j (( alpha +1)/2) sub n-2j (n-2j)!
( alpha /2 +1) sub m ( alpha +1) sub m m!}~~.
.EN
.SP1
\fBProof:\fR
.SP1
.EQ
R(m,j):=

{(2 j + 1) (m - n + 2 j + 1) (- n + 2 j + m) ( alpha  + 2 n - 2 j + 1) }

over{ ( alpha  + 1 + 2 n - 4 j) ( alpha  + 2 m + 2) (- n + m) 
(n +  alpha  + 2 + m)}~~.~~"\(sq"
.EN
.SP1
\fBRemarks:\fR The Askey-Gasper identity can also be viewed as
an identity for formal power series, and then the restriction
$0 <= t < 1$ is unnecessary and, in fact, meaningless.
The present proof is very elementary (it only requires
junior high-school algebra (literally!)), very easy (modulo
purely routine algebraic verifications, that can be left
to a machine or a competent high school student), and very
short (most of the paper was spent in introducing notation
and stating the theorem, the proof itself consists of the
proofs of Lemmas 1 and 2, that occupy together two  lines.)
.P
In fairness to the original proof(s) of the Askey-Gasper
\fIinequality\fR we must concede that we would have been
unable to prove the \fIinequality\fR directly.  All we
did was give a new proof of the \fIequality\fR that was
stated (and of course, first proved) by Askey and Gasper.
Of course, once the \fIequality\fR is available, the inequality
immediately follows, but finding the right equality is the
true breakthrough.
As we know from Polya's
principle, finding a stronger statement is often the crucial
step in solving a problem. 
In the case of the Askey-Gasper inequality, the crucial
step was their expression for the quantity of interest as a sum of
squares. Since, at least for us, equalities are much easier to prove
than inequalities, the actual \fIproof\fR of the \fIequality\fR is minor
compared to its conception. In other words: Any fool can
\fIprove\fR an identity, once stated, but only wise men can
conjecture good ones.

.SP1
\fBREFERENCES\fR
.SP1
[AG1] Richard Askey and George Gasper, \fIPositive Jacobi
polynomial sums II, Amer. J. Math. \fB98\fR(1976), 709-737.
.SP1
[AG2] _____, \fIInequalities for polynomials\fR, in:
" `The Bieberbach Conjecture`, Proceedings of the symposium
on the occasion of the proof" ( A. Baernstein et. al, editors), 7-32, 
Math. surveys and monographs \fB21\fR, Amer. Math. Soc., Providence,
1986.
.SP1
[deB] Louis de Branges, \fIA proof of the Bieberbach conjecture\fR,
Acta Math. \fB154\fR(1985), 137-152.
.SP1
[C] Thomas Clausen, \fIUber die Fa\*:lle,wenn dei Reihe von der
Form...ein Quadrat von der Form ... hat, J. Reine Angew. Math.
\fB3\fR (1828), 89-91.
.SP1
[WZ] Herbert Wilf and Doron Zeilberger, \fIRational
functions certify combinatorial identities\fR, J. Amer.
Math. Soc. \fB3\fR (1990), 147-158.
.SP1
April 30, 1991.