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\rm
 
{\bf CLOSED FORM (pun intended!)}
 
{\it Doron Zeilberger}
\footnote{*}{
Department of Mathematics, Temple University,
Philadelphia, PA19122. Supported in part by NSF grant DMS8800663.
}

{\bf Appeared in}: Contemporary Mathematics 143 (1993), 579-608. 

\it
In fond memory of Emil Grosswald, Great Number Theorist and
Wonderful Mensch.
 
\rm
 
{\bf 0. Introduction}
 
Mathematics is infinitely wide, 
while the language that describes it
is finite. It follows from
the pigeonhole principle that there exist distinct concepts that are
referred to by the same name. 
Mathematics is also infinitely deep and 
sometimes entirely different concepts
turn out to be intimately and profoundly
related. 
When the two phenomena coincide, one has a
{\it mathematical pun}. 
 
The phrase {\it closed form} has at least two distinct meanings.
The first meaning is that of "explicit", "nice", or "in finite
terms". For example the definite sum of the binomial coefficients
 
$$
  \sum_{k}  (  { n \atop k }  ) 
$$
 
can be expressed in closed form, namely $2^n $, as can
the sum of their squares
 
$$
  \sum_{k}  (  { n \atop k }  )^{2 \, \,,} 
$$
 
which is known to be $(2n)!/ n!^2 $. On the other hand,
the sum of the cubes
 
$$
  \sum_{k}  (  { n \atop k }  )^{3 \, \,,} 
$$
 
is not expressible in closed form. Similarly, 
$\int x  exp( x^2 ) dx $ can be expressed in closed form while
$\int exp ( x^2 ) dx $ cannot.
 
The other meaning of {\it closed form} is that of \it
closed
(exterior) differential form\rm
is closed if it is annihilated by the exterior derivative $d$.
I will define a natural discrete-continuous analog of $d$, also
to be denoted by $d$, and show that any "nice" identity
 
$$
  \sum_{{\bf k}} \int F( {\bf n} , {\bf k} , {\bf x} , {\bf y} ) d {\bf y}  
\,= \, 1 \, \, \,, 
$$
 
where {\bf n\rm
,\bf
k} are discrete multi-variables and
{\bf x\rm
, \bf
y} are continuous multi-variables, owes its existence
to the fact that the integrand-summand
$F(  n ,  k ,  x ,  y ) d  y \delta  k $ 
is one term of a  so-called "closed holonomic exterior
differential-difference form", and if in luck, that form will have
all its other components nice too, in which case we have what I will
call a {\it WZ form}.
Conversely, I will show that
any WZ form gives rise to several
"Closed Form" (in the previous sense of the phrase)  identities. 
 
The theory of WZ forms is a direct and natural generalization of
that of "WZ pair" ([WZ1],[WZ2]). The pair
$(F(n,k),G(n,k))$ is a  WZ pair iff 
$F(n,k) \delta k + G(n,k) \delta n$ is a WZ form. I am very grateful
to W for discovering the notion of WZ pair, and for countless
conversations.
 
A cheap way to manufacture closed differential
(-recurrence) forms of degree $r$ is to take a form of 
degree $r-1$, and
apply $d$ to it, getting the class of so-called {\it exact forms}.
As we all know, since $d^2 =0$, every exact form is closed, and
thus we have a quick way of generating many (in fact an infinite
number of) "nice" identities. Luckily, this embarrassment of riches
is deceptive, since I will show that identities obtained in this way
are trivial (I will explain what I mean by {\it trivial} later on). The 
{\it nice and interesting} identities are precisely those
that come from closed forms that are not exact. The problem of
classifying all "nice and interesting" identities thus boils down to
computing what I will call the "WZ cohomology".
 
{\bf 1. A Short Review Of The Calculus of Exterior Differential Forms}
 
The calculus of exterior differential forms is of fundamental
importance in Differential Geometry and elsewhere. 
Both Flanders's book[F] and Spivak's book [S]
are excellent introductions. Let
$( x_1 , ... , x_n )$ be continuous variables and
let $( d x_1 , ... , d x_n )$ satisfy the (anti-) commutation
relations : $ d x_i d x_j  \,= \, - d x_j d x_i $. 
An (exterior) {\it differential form} is a linear combination,
with coefficients that are functions (or distributions, or whatever
you can differentiate) of "words" in the alphabet 
{$d  x_1 , ... , d x_n $}. For a subset 
$I= \{ i_1 , ... , i_k \} $ of
{$1, 2 , ... , n$}, let 
 
$$
  d x_I := d x_{i_1} ... d x_{i_k} \, \, \,. 
$$
 
A general 
differential form in $( x_1 , ... , x_n )$  can be written
 
$$
 \omega = \sum_{I c \{ 1 , 2 , ... , n \}}   f_I d x_{I \, \, \,,} 
\eqno(1.1)
$$         
 
where the coefficients $f_I$ are functions in
$( x_1 , ... , x_n )$.
 
A {\it homogeneous r-form} is an expression of the form (1.1)
where all the subsets $I$ have the same cardinality $r$. Of course
a $0-form$ is just a scalar function.
 
Recall that the {\it exterior derivative} $d$ of a form $\omega$ is
defined as follows. For a $0-form$ $f$,
 
$$
 df := \sum_{i=1}^{n} {{\partial f}\over{\partial { x_i }}}  d x_{i \, \,,} 
\eqno(1.2)
$$         
 
and for a general form $\omega$, given by (1.1), 
 
$$
 d \omega = \sum_{I c \{ 1 , 2 , ... , n \}}  ( d f_I ) d  x_{I \, \, \,}. 
\eqno(1.3)
$$         
 
For example
 
$$
  d ( f  d x + g d y + h dz ) = ( g_x - f_y ) dx dy + ( h_x - f_z ) dx dz +( h_y - g_z ) dy dz \, \,, 
$$
 
(the coefficients of which are components of {\it curl} $(f,g,h)$, and
 
$$
  d( f dy dz + g dz dx + h dx dy ) = ( f_x + g_y + h_z ) dx dy dz, 
$$
 
producing the divergence.
 
Recall the fundamental Stokes's Theorem, that for any form $\omega$,
and any oriented manifold $\Omega$, which for our purposes 
can be taken to be a subset of $R^n$, one has
 
$$
 \int_{ \partial \Omega } \omega = \int_{\Omega} d \omega \, \, \,. 
\eqno(STOKES)
$$          
 
It is well known, and easily checked, that $d^2 =0$ and
$d( \omega \theta ) = ( d \omega ) \theta + (-1)^r \omega ( d \theta )$,
where $\omega$ is homogeneous of degree $r$.
 
A differential form $\omega$ is said to be a {\it closed form} if
$d \omega = 0$, and it is said to be {\it exact} if it can be written
as $\omega = d \theta $, for some form $\theta$. Of course every exact
form is automatically closed, and the quotient space of
closed modulo exact, for a given manifold, is the celebrated
{\it de Rham cohomology}. It is well known (e.g. [F]) 
that for "flat" space
$R^n$, the de Rham cohomology is trivial, i.e. every closed form
is exact.
 
{\bf 2. The Calculus of Exterior Difference Forms}
 
The discrete analog of the calculus of differential forms is much
simpler than the continuous case, since we don't have to fuss about
such analytical notions as convergence. We consider functions
$F( m_1 , ... , m_n )$ defined on the rectangular lattice
$Z^n$. The analogs of partial derivatives are  partial
forward difference operators:
 
$$
 \Delta_i F( m_1 , ... , m_i , ... , m_n ):= F( m_1 , ... , m_i +1  , ... , m_n ) - F( m_1 , ... , m_i , ... , m_n ) \, \, \,. 
\eqno(2.1)
$$        
 
A unit r-cube in $Z^n$ is a set of points
of the following form: for some 
subset $I$ of {$1 , 2 , 3 , ... , n $},
of cardinality $r$, $m_i$ is fixed (say $a_i$) if
$i$ is not in $I$, and can take one of two consecutive values
(say $ a_i$ and $a_i +1$) if $i$ belongs to $I$. Of course
a unit $r-cube$ has $2^r$ points. The {\it interior} of a unit
$r-$cube is its "lowest, leftmost" corner,
$( a_1 , ... , a_n )$. The boundary of an r-cube consists of
the union of the $2r$ $ \,(r-1)$-cubes obtained by freezing one of the
$m_i$, for $i$ in $I$, to be $a_i$ or $a_i +1$. 
 
The beauty of the notation for integration is that the variable
with respect to which things are being integrated is explicitly
indicated:
 
$$
  \int f d x 
$$
 
means that the function $f$ is being integrated with respect to the
variable $x$, regardless of whatever other variables or parameters
$f$ may depend upon. For a definite integral, we write
 
$$
  \int_{a}^{b} f d x  \, \,, 
$$
 
rather than
 
$$
  \int_{x=a}^{b} f   \, \,. 
$$
 
This is unlike the corresponding convention for
summation in which the summation variable has to be indicated below
the $SIGMA$, and does not appear within the sum itself. To correct
this inequity, Knuth introduced a useful notation (e.g. [GKP], p.48)
which we shall adopt. In that notation
the sum
 
$$
  \sum_{k=a}^{b} f(k) 
$$
 
is re-expressed as
 
$$
  \sum_{a}^{b} f(k) \delta k \, \, \,. 
$$
 
Similarly, a multiple sum
 
$$
  \sum_{ k_1 , ... , k_r } f( k_1 , ... , k_r )  
$$
 
is written 
 
$$
  \sum f( k_1 , ... , k_r ) \delta k_1 ... \delta k_r  \, \, \, \,. 
$$
 
The mark of a good notation is that it leads to new theory. For
example, matrices started out as shorthand for writing 
systems of linear
equations, and what emerged was linear algebra. In our case, what
comes out of Knuth's notation is the discrete analog of the
calculus of differential forms.
 
Let
$( m_1 , ... , m_n )$ be discrete variables and
let $\delta m_1 , ... , \delta m_n$ be "indeterminates" satisfying
$ \delta m_i \delta m_j  \,= \, - \delta m_j \delta m_i $. 
An (exterior) {\it difference form} is a linear combination,
with coefficients that are discrete functions,
of "words" in the alphabet 
{$\delta  m_1 , ... , \delta m_n $}. For a subset 
$I= \{ i_1 , ... , i_k \} $ of
{$1, 2 , ... , n$}, let 
 
$$
  \delta  m_I := \delta m_{i_1} ... \delta m_{i_k} \, \, \,. 
$$
 
A general 
difference form in $( m_1 , ... , m_n )$  can be written
 
$$
 \omega = \sum_{I c \{ 1 , 2 , ... , n \}}   f_I  \delta m_{I \, \, \,,} 
\eqno(2.2)
$$         
 
where the coefficients $f_I$ are functions of
$( m_1 , ... , m_n )$.
 
{\it An oriented unit cube} is a unit cube with a choice of sign
$\pm$.
An {\it oriented r-manifold} in $Z^n$ is any union of
oriented unit r-cubes. Every $r-manifold$ can be considered as
an oriented one by taking all the signs of its constituent cubes
to be $+$.
The {\it boundary} of an oriented 
unit r-cube $B$,
denoted by $\partial B$, is the union of all the $2r$ $ \,(r-1)$-faces,
with the sign of each face determined in the usual way
([S], p. 98, fig. 4-4): the faces
adjacent to the lower-left corner alternate in sign and opposite
faces have opposite signs. The boundary of an oriented discrete
manifold is the "sum"of all the boundaries of its constituent unit
cubes. (Of course for a connected manifold, of a fixed sign,
all interior walls cancel each other in that sum, and only external
walls survive.)
 
We are now ready to define "integration" of an $r-form$ over
an oriented $r-$manifold.
By additivity, it is enough to define what is
 
$$
 \sum_{B} f_I \delta m_I  \, \, \,, 
\eqno(2.3)
$$            
 
where $B$ is an oriented 
unit $r-$cube whose lower left corner is a typical
point $(m_1 , ... , m_n )$. If the subset of "active"
coordinates of $B$ coincides with $I$, we define the above to be
$\pm  f( m_1 , ... , m_n )$, where we assume that
$i_1 < ... < i_r$, and $\pm$  is the sign of $B$. If the set
of active coordinates of $B$ does not coincide with $I$ then (2.3) is
defined to be $0$.
 
The discrete analog of the exterior derivative $d$, the \it
exterior 
difference\rm
, $\delta$, is defined in analogy with (1.1).
For a $0-form$ $f$,
 
$$
 \delta f := \sum_{i=1}^{n}  ( \Delta_i f ) \delta m_{i \, \,,} 
\eqno(2.4)
$$          
 
and for a general form $\omega$ of (2.2) :
 
$$
 \delta \omega = \sum_{I c \{ 1 , 2 , ... , n \}} ( \delta f_I )  \delta  m_{I \, \, \,}. 
\eqno(2.5)
$$         
 
Let $\omega$ be an $r-form$, and $\Omega$ any oriented discrete
$(r+1)-manifold$, i.e. a union of oriented unit $(r+1)-cubes$. 
We have
 
{\bf The DISCRETE STOKES Theorem}
$$
  \sum_{  \, \partial \Omega } \omega  \,= \, \sum_{\Omega}  \, \delta \omega \, \, \,. 
$$
 
The proof is even more trivial
than its continuous namesake (see [S], p. 104). By additivity, it is
enough to prove it for forms with one term and manifolds $\Omega$
consisting of a single $r-cube$, and this last task is immediate.
 
{\bf 3. Closed Difference Forms}
 
As an immediate corollary of the DISCRETE STOKES theorem, we have
 
{\bf Theorem 1}: Let $\omega$ be a discrete $closed$ difference
$r-form$:
 
$$
 \omega  \,= \, \sum_{|I| \,= \,r}   f_I \delta m_{I \, \, \,,} 
\eqno(3.1)
$$         
 
and assume that for some subset $I_0$ of cardinality $r$ of
{$1,2, ... , n $}, its complement $J_0$ has the property that
all the coefficients of $\omega$ have finite support as functions
of the variables $m_{I_0}$, for any fixed choice of the variables
$m_{J_0}$. The function
 
$$
  g( m_{J_0} ) :=  \sum  f_{I_0} ( m_{I_0} , m_{J_0} ) \delta m_{I_0} 
$$
 
is identically constant in $m_{J_0}$.
 
{\bf Proof:} Let $\Omega$ be the region between two parallel
hyperplanes $m_{J_0}  \,= \, a_{J_0}$ and 
$m_{J_0}  \,= \, a'_{J_0}$. The
difference $g( a_{J_0} ) - g( a'_{J_0} )$ is the integral of
$\omega$ over the boundary of $\Omega$, since the contribution is zero
from the other terms, thanks to the assumption that $\omega$ vanishes
when at least one of the $m_{I_0}$ coordinates is infinite. By the
DISCRETE STOKES theorem, that sum is the sum over
$\Omega$ of $\delta \omega$, but the latter is $0$, since $\omega$ is a
closed form.
 
{\bf 4. CLOSED FORM}
 
Theorem 1 by itself is almost a tautology. It is easy to see that
the "de Rham cohomology" for arbitrary discrete functions on 
$Z^n$ is trivial, i.e. every closed form is exact, and so to
obtain all closed forms of a certain degree,
all one has to do is take all forms of 
degree one less and apply the exterior difference operator 
$\delta$ to them.
 
However, we are not interested in {\it all} identities, only in
the {\it nice\rm
 ones! For us "nice" will mean \rm
Closed Form}.
 
{\bf Definition:} A function $f( m_1 , ... , m_n )$ is
{\it Closed Form} (CF) if for each of its variables $m_i$
($i \,= \,1, ... , n$), the quotient
 
$$
 {{f( m_1 , ... , m_{i-1} , m_i +1 , m_{i+1} , ... , m_n )} \over {f( m_1 , ... , m_{i-1} , m_i , m_{i+1} , ... , m_n )}} 
\eqno(4.1)
$$        
 
is a rational function of $m_1 , ... , m_n$, i.e. a quotient
of two polynomials in $m_1 , ... , m_n$.
 
A typical example of a Closed Form function is the factorial expression
$(c+ a_1 m_1 + a_2 m_2 + ... + a_n m_n )!$, where
$a_1 , ... , a_n$ are (positive or negative) integers, and
$c$ is any constant or indeterminate. Similarly, the reciprocal of
such an expression is CF. Since the product of CF functions is again
CF, any quotient of products of such expression, times powers
$x_1^{m sub 1} \cdot \cdot \cdot x_n^{m sub n}$ is CF.
 
Next we define formally what we mean by a {\it CF identity}
 
{\bf Definition:} A multi-sum identity is CF if both its summand
and sum are CF.
 
The general form of a CF identity is
 
$$
 \sum_{ m_1 , ... , m_a } f( m_1 , ... ,  m_n )  \,= \, g( m_{a+1} , ... , m_n ) 
\eqno(4.2)
$$         
 
where $f$ is CF in 
$m_1 , ... ,  m_n$, and $g$ is
$CF$ in $m_{a+1} , ... , m_n$. Since the quotient of  CF
functions is again CF, every CF identity is equivalent to one in
which the right side is $1$, simply divide through by the right
side! This "innocent observation" of Herb Wilf was the starting
point of the theory of WZ pairs and consequently WZ forms.
 
We would like to study "all" CF identities
 
$$
 \sum  f( m_1 , ... , m_a , m_{a+1} , ... , m_n ) \delta m_1 ... \delta m_a  \,= \, 1 \, \, \, \,. 
\eqno(4.3)
$$        
 
The big surprise is that the converse of the tautological theorem 1
is true, for CF identities!, and not in a tautological manner!
Whenever we have a CF identity, it is so because
the summand is one term in a closed difference form, that can be
algorithmically constructed, and all whose terms can be described
explicitly. Furthermore, in most cases in practice, all the terms of
that closed difference form are CF themselves. 
At any rate, at the very worst, they
belong to the class of so-called holonomic functions, and can be actually
exhibited. In the WZ-pairs miracle [WZ1], we set out to prove
one identity and we also got the
discovery and proof of another one, often more interesting, as a
bonus. Now, whenever we have an identity like (4.3), we can always
reconstruct the whole closed difference form $\omega$ of which the
summand 
$f( m_1 , ... , m_a , m_{a+1} , ... , m_n ) $
$\delta m_1 ... \delta m_a $ is just one term, with all its
${ {n} \choose {a} }$ terms, and we get
{\it free of charge} ${ {n} \choose {a} }  -1$ identities
as bonuses, complete with proofs! In fact the same proof that demonstrated
the original identity, works for all the bonuses simultaneously,
and merely consists of the purely mechanical task of verifying that
the proposed form $\omega$ is indeed closed. I will also describe
an algorithm for constructing $\omega$.
 
{\bf 5. Holonomic Functions Identities in $ {Z^n}$}
 
Holonomic functions
are  natural generalizations of CF functions. Another way of saying
that a function $f$ is CF is to say that it satisfies {\it homogeneous}
ordinary linear recurrences with polynomial coefficients
of the {\it first order} in each of its
variables: 
 
$$
  \, \, \, \, \, 
$$
$$
  P^{(i)} ( m_1 , ... , m_n )  f( m_1 , ... , m_i + 1 , ... , m_n ) + Q^{(i)} ( m_1 , ... , m_n )  f( m_1 , ... , m_n )  \,= \,0 \, \, \,,i \,= \,1, ... , n 
\eqno(5.1)
$$       
 
for some polynomials $P^{(i)}$, $Q^{(i)}$, ($i \,= \,1, ... , n$).
A {\it holonomic function}, roughly speaking, is a function on
$Z^n$ that satisfies ordinary recurrence equations, not
necessarily of the first order, in each of its variables:
 
$$
 \sum_{k=0}^{L_i} P_k^{(i)} f( m_1 , ... , m_i + k , ... , m_n )  \,= \,0 \,. 
\eqno(5.2)
$$        
 
To guarantee that $f$ is genuinely holonomic we must insist that the
leading coefficients ${P_{L_i}}^{(i)}$ do not have
common zeroes. There are also holonomic functions that do not
have the above form: for example the Kronecker $\delta_{m,n}$. 
The full definition of holonomicity is that the set of linear
difference equations with polynomial coefficients 
satisfied by the function forms a so-called
"maximally over-determined system" [Z1]. 
 
The ring of linear partial difference operators with polynomial
coefficients in $Z^n$ is generated by 
$m_1 , ... , m_n$ and 
$\Delta_1 , ... , \Delta_n$. However, it is
more convenient and congenial to take the generators to be
$m_1 , ... , m_n$ and 
$E_1 , ... , E_n$, where $E_i$ are the 
{\it shift operators}:
 
$$
 E_i f( m_1 , ... , m_i , ... , m_n ) := f( m_1 , ... , m_i +1 , ... , m_n )  \, \,. 
\eqno(5.3)
$$         
 
Of course, we have
 
$$
 \Delta_i  \,= \, E_i - 1 \, \,, 
\eqno(5.4)
$$          
 
where "$1$" is the identity operator.
 
Any linear difference operator with polynomial coefficients can be
expressed as $P( m_1 , ... , m_n , E_1 , ... , E_n )$,
where $P$ is a polynomial in the indeterminates
$( m_1 , ... , m_n , E_1 , ... , E_n )$. It is
readily seen that they satisfy the "commutation relations":
 
$$
 E_i m_j  \,= \, m_j E_{i \, \,i} \neq  j \,; \, \, \, E_i E_j  \,= \, E_j E_i  \,, \, \, \, m_i m_j  \,= \, m_j m_i  \, \, \,, 
\eqno(5.5)
$$        
 
and
 
$$
 E_i m_i  \,= \, m_i E_i + E_{i \, \, \,}. 
\eqno(5.6)
$$          
 
The following is a useful mnemonic. The m's are men and the
E's are women, $m_i$ is married to $E_i$. All men pass each other
with no problem and all women pass each other without a trace,
as do a man and a woman that are not married to each other.
However whenever a woman wants to pass to the
right of her husband, a new baby girl is born, that is a clone of
her mother.
 
In [Z1] I showed how to adjust Sylvester's dialytic elimination to
the algebra of linear difference operators. Given two operators
$P( m_1 , ... , m_n , E_1 , ... , E_n )$ and
$Q( m_1 , ... , m_n , E_1 , ... , E_n )$ it is
possible, in general, to eliminate one of the variables, say
$m_1$. This means that one can find linear difference operators
with polynomial coefficients
$A( m_1 , ... , m_n , E_1 , ... , E_n )$ and
$B( m_1 , ... , m_n , E_1 , ... , E_n )$ such that
$AP+BQ$ does not involve $m_1$. Of course this is only possible
when $P$ and $Q$ are "independent" in a certain technical sense.
For example if both of them are left multiples of $m_1$ then
every combination $AP+BQ$ will still have that property, and there is no
getting rid of $m_1$. More generally, if $P_1$, $P_2$,
$... , P_r$ are operators, it is possible, if all goes well, to
eliminate $r-1$ out of the $2n$ indeterminates
$ m_1 , ... , m_n , E_1 , ... , E_n $. 
 
Given a discrete function $f$, the set of linear difference operators
with polynomial coefficients
annihilating it is a left ideal in the whole ring, and is denoted by
$I(f)$. It is a left ideal since if $P$ annihilates $f$, so does
obviously $AP$, for {\it any} operator $A$. 
If $f$ is holonomic, then the ideal $I(f)$ is
"as big as possible" (which means that the Hilbert dimension
of the quotient ring 
$K< m_1 , ... , m_n , E_1 , ... , E_n >/I(f)$
viewed as a left $K< m_1 , ... , m_n , E_1 , ... , E_n >$
module,
is $n$, which is the least that it can possibly be (thanks to 
the celebrated Bernstein inequality [Ber], [Bj]). It follows from the
general theory of holonomic systems ([Ber],[Bj], see also [Z1]) that
if $f$ is holonomic, then $I(f)$ contains an operator that does not
involve $m_2 , ... , m_n$. Let that operator be
$A( m_1 , E_1 , E_2 , ... , E_n )$. Now write
 
$$
 A( m_1 , E_1 , ... , E_n )  \,= \, A_1 ( m_1 , E_1 )  \,- \, ( E_2 -1) A_2  \,- \, ...  \,- \, (E_n -1) A_{n \, \, \,}. 
\eqno(5.7)
$$        
 
Applying both sides of (5.7) on our holonomic $f$, the left side is
zero and
we have  proved the following
 
{\bf Theorem 2}: Let $f( m_1 , ... , m_n )$ be holonomic
in all its variables. For each of its variables $m_i$, there
exists an operator $A^{(i)} ( m_i , E_i )$,
and operators
$A_j^{(i)} $ ($j  \,= \, 1, ... ,i-1,i+1, ... ,n $) 
such that
 
$$
 A^{(i)} ( m_i , E_i ) f  \,= \, \sum_{{{1 <= j <= n} \atop {j \neq  i }} }  
\Delta_j A_j^{(i)} f  \, \, \, \,. 
\eqno(5.8)
$$        
 
Note that we have proved {\it more} than the theorem. We have proved
that the operators $A_j^{(i)}$ can be taken to be
independent of $m_2 , ... , m_n$, which is an
unnecessary extravagance. However, to prove the mere existence of such
operators we can be big spenders, since window-shopping does not cost
anything.  Once we know that our shopping expedition is guaranteed
to succeed
we should do a better job of shopping, and get the cheapest possible
operators.
 
Now define
 
$$
 G_j^{(i)} ( m_1 , ... , m_n ):= A_j^{(i)}   f( m_1 , ... , m_n ) 
\eqno(5.9)
$$         
 
$G_j^{(i)}$ are obviously holonomic, since any linear
difference operator with polynomial coefficients applied to any
holonomic function yields a holonomic function. Furthermore if
$f(- , m_i , - )$ has finite support in 
$( m_1 , ... , m_{i-1} , m_{i+1} , m_n )$, for any fixed 
$m_i$, so do the $G_j^{(i)}$. We thus have
 
{\bf Theorem 3:} For any holonomic $f ( m_1 , ... , m_n )$,
and any of its variables $ m_i$, there exists a non-zero
operator $A^{(i)} ( m_i , E_i )$, and
holonomic $G_j^{(i)}$ (that have the form (5.9)) such that
 
$$
 A^{(i)} ( m_i , E_i ) f( m_1 , ... , m_n )  \,= \, 
\sum_{ {1 <= j <= n \atop j \neq  i }}  \Delta_j G_j^{(i) \, \, \, \,}. 
\eqno(5.10)
$$        
 
Furthermore, if $f(-, m_i , -)$ has finite support for any fixed
$m_i$, so do the $G_j^{(i)}$.
 
An immediate consequence is
 
{\bf Corolary 4}: Let $f( m_1 , ... , m_n )$ be holonomic,
and let $m_i$ be a variable such that for any specific $m_i  \,= \,c$,
there are only finitely many 
$( m_1 , ... , m_{i-1} , m_{i+1} , ... , m_n )$ for
which $f( m_1 , ... , m_{i-1} , c, m_{i+1} , ... , m_n )$
is non-zero. Let
 
$$
 a_i ( m_i ) := \sum_{ m_1 , ... , m_{i-1} , m_{i+1} , ... , m_n } 
f(m_1 , ... , m_n ) \, \, \,, 
\eqno(5.11)
$$        
 
then $a_i ( m_i )$ satisfies an ordinary linear recurrence
 
$$
  A^{(i)} ( m_i , E_i ) a_i ( m_i )  \,= \,0 \, \, \,. 
$$
 
{\bf Proof}: Sum (5.10) w.r.t to 
{$ m_1 , ... , m_{i-1} , m_{i+1} , ... , m_n $}.
The sum on the right is a sum of $n-1$ telescoping series,
each of finite support.
 
Let's consider now identities of the form
 
$$
  \sum_{ m_1 , ... , m_{i-1} , m_{i+1} , ... , m_n } f( m_1 , ... , m_n )  
\, \equiv \, 1 \, \, \,. 
$$
 
How to prove them? Let's call the left side 
$a_i ( m_i )$. Corollary 4 manufactures an operator
$A^{(i)} ( m_i , E_i )$ which annihilates
$a_i ( m_i )$. To prove that $a_i ( m_i )$ is indeed
identically $1$, we must show that $1$ is also annihilated by same,
and that $a_i ( m_i )  \,= \, 1$ for the first $L_i$ values
of $m_i$, where $L_i$ is the order (i.e. degree in $E_i$)
of $A^{(i)}$. Alternatively, Euclid-dividing $A^{(i)}$ by 
$\Delta_i$ (from the right!) gives
 
$$
  A^{(i)} ( m_i , E_i )  \,= \,  \bar{A}^{(i)} ( m_i , E_i ) \Delta_i + b ( m_i ) \, \, \,, 
$$
 
for some operator $\bar{A}^{(i)}$, and 
for some function $b( m_i )$ that must be identically zero,
since $A^{(i)} 1  \,== \, 0$, so we can write
 
$$
 A^{(i)} ( m_i , E_i )  \,= \,  \bar{A}^{(i)} ( m_i , E_i ) \Delta_i  \, \, \,. 
\eqno(5.12)
$$         
 
Now for each $1 <= j <= n , j \neq  i  \,$, let's find new functions
$\bar{G}_j^{(i)}$ such that
 
$$
 \bar{A}^{(i)} ( m_i , E_i ) \bar{G}_j^{(i)}  \,= \,  \,- \, G_j^{(i)}  \, \, \,, 
\eqno(5.13)
$$         
 
subject to appropriate initial conditions to be specified 
momentarily. It is well known, and easy to see, that this is
always possible, although the $\bar{G}_j^{(i)}$ usually
don't inherit the "finite support" property of the 
$G_j^{(i)}$. Since $G_j^{(i)}$ are holonomic,
so are the $\bar{G}_j^{(i)}$.
 
Substituting (5.12) and (5.13) into (5.10), and extracting 
$\bar{A}_i ( m_i , E_i )$ out, we get
 
$$
 \bar{A}^{(i)} ( m_i , E_i )  \,  ( \, \Delta_i f( m_1 , ... , m_n ) + \sum_{1<= j <= n , j \neq  i } \Delta_j \bar{G}_j^{(i)}   \,  )  \,= \,0 \, \, \, \,. 
\eqno(5.14)
$$       
 
Now we are ready to specify the initial conditions promised earlier,
that determine the solutions of (5.13): we choose them in such a way that
 
$$
 \Delta_i f( m_1 , ... , m_n ) + \sum_{1<= j <= n , j \neq  i } \Delta_j \bar{G}_j^{(i)}  =0 \, \, \, \,, for \, m_i =0, 1 , ... , L_i -1 \, \, \,. 
\eqno(5.15)
$$        
 
It is easy to see that this is
always possible, and for this choice of $\bar{G}_j^{(i)}$, it
follows from (5.14) that
 
$$
 \Delta_i f( m_1 , ... , m_n )  \,+ \, \sum_{1<= j <= n , j \neq  i } \Delta_j \bar{G}_j^{(i)}  =0 \, \, \, \,,for \,all \,m_i  \, \,. 
\eqno(5.16)
$$        
 
We have just proved (renaming $\bar{G}_j^{(i)} = H_j  \,$ ,
$f= H_i$, in (5.16))
 
{\bf Theorem 5:} 
Let $f( m_1 , ... , m_n )$ be holonomic.
 
$$
 \sum f( m_1 , ... , m_n ) \delta m_1 ... {\delta \hat{m_i}} ... \delta m_n == constant 
\eqno(5.17)
$$         
 
if and only if there exists a {\it closed difference} form
$\omega$:
 
$$
 \omega := \sum_{j=1}^{n}  \, H_j  \,\delta m_1 ... {\delta \hat{m_j}} ... \delta m_{n \, \, \,,} 
\eqno(5.18)
$$        
 
such that the components $H_j$ are all holonomic, and $H_i =f$.
 
A similar analysis holds when the number of variables that are
being summed over is bigger than one. We have
 
{\bf Theorem 6}:
Let $f( m_1 , ... , m_n )$ be holonomic. Let $I$ be
any subset of {$1,2, ... , n$}. An identity of the form
 
$$
 \sum f( m_1 , ... , m_n ) \delta m_I  == constant \, \, \,, 
\eqno(5.19)
$$         
 
where the constant is independent of the variables in the
complement of $I$, is possible
if and only if there exists a {\it closed difference} form
$\omega$, of degree $k:=|I|$, 
 
$$
 \omega := \sum_{J c \{1, ... , n \}, |J|=k}  \, H_J  \, \delta m_J 
\eqno(5.20)
$$         
 
such that the components $H_J$ are all holonomic, and 
$H_I =f$.
 
We remark that every closed holonomic form is exact, since it is
always possible to take "anti-differences" in the class of
holonomic functions, and the proof is analogous to the proof
that the de Rham cohomology in flat space is trivial.
It follows that the totality of identities of the form (5.19) is
obtained by taking arbitrary holonomic forms of degree $k-1$ and
applying the exterior difference operator $\delta$. This 
is not very exciting, since most identities produced that
way are very boring. We are interested in "nice" identities, in which
$f( m_1 , ... , m_n )$ is a Closed Form function rather than
just a plain holonomic function.
 
{\bf 6. Closed Form Identities and WZ forms}
 
Let's return to our main object of interest: Closed Form identities.
Since any Closed Form function 
$f( m_1 , ... , m_n )$ is holonomic, theorem 6 tells us
that whenever an identity of the from (5.19) holds, it is so because
$f \delta m_I $ is one term of a closed holonomic form. However, more
is true in that case. The $G's$ of (5.9) have special form now! It is
easy to see that if $f$ is CF then 
applying to it
any linear difference operator with polynomial coefficients
yields a multiple of $f$ by a rational function. Indeed,
by iterating (4.1),
 
$$
 E_1^{i_1} ... E_n^{i_n } f /f = f( m_1 + i_1 , ... , m_n + i_n ) /f 
\eqno(6.1)
$$         
 
is a rational function, and since for any linear difference operator
$P( E_1 , ... , E_n , m_1 , ... , m_n )$,
$Pf/f$ is a linear combination, with polynomial coefficients, of
such expressions, $Pf/f$ itself is a rational function.
In general, there is no guarantee that the operator $A^{(i)}$ of
(5.10) is $\Delta_i$, so the terms of the closed form
$\omega$ do not all have to be CF. In practice, however, it happens,
{\it very often}, that this is the case, and then the form
$\omega$ has all CF components, and moreover, they
are all multiples of $f$ by rational functions.
By taking common denominator, we are lead to the following definition of
a {\it WZ form}.
 
{\bf Definition:} A WZ form of degree $k$ is a \it
closed difference
form\rm
that looks as follows: for some CF function 
$f( m_1 , ... , m_n )$, and polynomials 
$P_I ( m_1 , ... , m_n )$:
 
$$
 \omega  \,= \, f \,\cdot \,  [  \, \, \, \sum_{ I c \{ 1,2, ... , n \}, |I|=k} P_I \delta m_I   \, \, \,]  \, \, \,. 
\eqno(6.2)
$$         
 
{\bf 7. Examples of WZ forms}
 
I have already mentioned that any WZ pair $(F(n,k),G(n,k))$ gives
rise to the WZ form $F(n,k) \delta k + G(n,k) \delta n$. So we
can take all the examples of [WZ1].
 
The way to obtain non-trivial WZ pairs is to start with a well
known identity, or a specialization thereof:
 
$$
 \sum_{k} F(n,k)  \,= \, 1 \, \,, 
\eqno(7.1)
$$          
 
and apply Gosper's [G] algorithm for \it
indefinite hypergeometric 
summation\rm
, w.r.t to $k$, to $\Delta_n F(n,k)$, thus hopefully
getting a CF $G(n,k)$ such that
 
$$
 \Delta_n F(n,k) = \Delta_k G(n,k) \, \, \, \,. 
\eqno(7.2)
$$          
 
As was narrated above, one is always guaranteed, thanks to the
theory of holonomic systems, to find a CF $G(n,k)$ and an operator
$A( N , n )$ such that
 
$$
 A(N,n) \Delta_n F(n,k) = \Delta_k G(n,k) \, \, \,, 
\eqno(7.3)
$$          
 
which at any rate "certifies", i.e. proves, the identity. Being lucky
means that the operator $A(N,n)$ turns out to be the identity operator,
and the good news is that we are lucky in the vast majority of
cases.
 
{\bf Brilliant Discovery Of Herb Wilf:} Even in the "unlucky" cases,
there is still hope of getting a WZ pair. If there exists an
operator $B(N,n)$ such that 
$A(N,n) \Delta_n = \Delta_n B(N,n)$, then 
$(B(N,n)F(n,k),G(n,k))$ is a WZ pair.
 
Many times an identity has several auxiliary parameters, and
hitherto we just picked one of them, called it $n$ and left the other
passive. That was arbitrary and unfair, so let's rectify is.
Suppose we have a known identity:
 
$$
 \sum_{k} f( n_1 , ... , n_r , k ) =1 \, \, \,. 
\eqno(7.4)
$$          
 
For each of the parameters that is not being summed over,
$n_1 , ... , n_{r \,}$, do the above, getting, hopefully, CF
$G_i$ s.t
 
$$
 \Delta_{n_i} F( n_1 , ... , n_r , k ) = \Delta_k G_i ( n_1 , ... , n_r , k ), \, \, \, \, \,i=1, ..., r   \, \,. 
\eqno(7.5)
$$         
 
The above $r$  identities are equivalent to the single statement
that
 
$$
 \omega  \,:=  \, F \delta k  \,+ \, \sum_{i=1}^{r} G_i  \, \delta n_i 
\eqno(7.6)
$$         
 
is a closed difference form, and hence a WZ form. It follows that we
have $r$ "companion" identities
 
$$
 \sum_{n_i} G_i ( n_1 , ... , n_r , k)  \,= \, constant \, \, \,, 
\eqno(7.7)
$$          
 
whenever the sum converges. In practice they would usually
not converge,
but one can perform the operation of "shadowing" described
in section 4 of [WZ1], by which 
for any given $i=1, ... , r$, we can always find 
a shadow $\tilde{\omega}^{(i)} $ of $\omega$, such that the coefficient
$\tilde{G}_i$ of $\delta n_i$ in $\tilde{\omega}^{(i)}$ has
the property that for any fixed $ n_1 , ... , n_{i-1} , $
$n_{i+1} , ... , n_r ,k$, there are only finitely many 
$n_i$ for which $\tilde{G}_i$ is non-zero.
 
Using the above procedure for Dixon's identity (compare [WZ1], p. 153)
 
$$
 \sum_{k}   \,(-1)^k {{(a+b)! (a+c)! (b+c)! a! b! c! }\over{(a+k)!(a-k)!(b+k)!(b-k)!(c+k)!(c-k)!(a+b+c)!}} \, \, \,=1 
\eqno(7.8)
$$      
 
we get the following WZ form
 
$$
  \,\omega_{DIXON \,} := (-1)^k {{(a+b)! (a+c)! (b+c)! a! b! c! }\over{2(a+k)!(a-k+1)!(b+k)!(b-k+1)!(c+k)!(c-k+1)!(a+b+c+1)!}} \cdot  
\eqno(7.9)
$$     
 
$$
   [ 2(a-k+1)(b-k+1)(c-k+1)(a+b+c+1) \delta k  \,+ \,  (b-k+1)(b+k)(c-k+1)(c+k) \delta a  \,+ \, 
$$
 
$$
  (a-k+1)(a+k)(c-k+1)(c+k) \delta b  \,+ \, (a-k+1)(a+k)(b-k+1)(b+k) \delta c  ]  \, \, \,. 
$$
 
Saalschu\*:tz's identity
 
$$
 \sum_{k} {{(m-r+s)!(n+r-s)!(r+k)!m!n!(r-m)!(s-n)!}\over{k!(m-r+s-k)!(n-k)!(r-s+k)!(m+n)!(r+k-m-n)!r!s!}} \,= \,1 \, \, \,, 
\eqno(7.10)
$$       
 
leads to, and is proved by, the following WZ form
 
$$
 \omega_{SAALSCHUTZ} := {{(m-r+s)!(n+r-s)!(r+k)!m!n!(r-m)!(s-n)!}\over{k!(m-r+s-k)!(n-k)!(r-s+k)!(m+n)!(r+k-m-n)!r!s!}} \, \cdot 
\eqno(7.11)
$$       
 
$$
    [  \delta k  \,+ \, {{-k(r-s+k)(-r-k+m+n)}\over{(n-k+1)(m+n+1)(n-s)}} \delta n  \,+ \, {{-2r-m+1-s+k)k}\over{(r+1)(-m+r-s)}} \delta r  \,+ \, 
$$
 
$$
  {{-k(-r+s-k)((r+k-m-n)}\over{(m-r+s-k+1)(s+1)(-n-r+s)}} \delta s  \,+ \,  \,{{-k(r-s+k)(-r-k+m+n)}\over{(m-r+s-k+1)(m+n+1)(m-r)}} \delta m   \,   ] \, \,. 
$$
 
Moving right along, Vandermonde's identity, involving two free parameters,
after some shadowing, yields the WZ form
 
$$
 \omega_{VANDERMONDE}  \,:= \, 
\eqno(7.12)
$$          
 
$$
  {{n!^2 k!^2 a!^2}\over{(a+k+1)! (n+k+1)! (n+a+1 )! }}  [ (n+a+1) \delta k  \,+ \, (k+a+1) \delta n  \,+ \, (n+k+1) \delta a  ] \, \,. 
$$
 
So far all our examples were 1-forms, having been obtained from
single sum identities. To find higher degree forms, we must start
with a well known multi-sum CF identity, and we would need
a multivariate analog of Gosper's algorithm. I am presently working
on developing such an algorithm, but until I succeed, all I can present
is the $r$-form arising out of the multinomial identity
 
$$
 \sum  \, {{n!}\over{k_1 ! ... k_r ! (n- k_1 - ... - k_r )! (r+1)^n }} \delta k_1 ... \delta k_{r \,=1 \, \, \,,}  
\eqno(7.13)
$$        
 
which produces
 
$$
 \omega_{MULTINOMIAL} := {{n!}\over{k_1 ! ... k_r ! (n- k_1 - ... - k_r +1 )! (r+1)^n }} \cdot 
\eqno(7.14)
$$        
 
$$
   [ ( n- k_1 - ... - k_r  \,+ \, 1 ) \delta k_1 ... \delta k_r  \,+ \,  \sum_{i=1}^{r} k_i  \, \delta n \delta k_1 ... {\delta  \hat{k_i}} ... \delta k_r   ] \, \, \, \,. 
$$
 
{\bf 8. WZ COHOMOLOGY}
 
Since every holonomic closed form is exact, it follows that every WZ
$r-form$ $\omega$ can be written as $\delta theta$, for some
{\it holonomic} $(r-1)-form$ $theta$. However, this is not an
effective way of cranking out WZ forms, since there is no way of
knowing beforehand which holonomic $(r-1)$-forms 
$theta$ are such that $\delta theta$ are WZ
forms, i.e. all their components are CF. Of course, if $theta$ is CF
to begin with, then $\delta theta$ is a WZ-form: it is closed, and
all its components are CF, as can be easily verified. We will call
such WZ forms {\it exact}. I will now explain why exact WZ forms lead to
trivial identities, but before we must define {\it trivial}.
 
Let's look at the following two definite integrals
 
$$
 \int_{0}^{\infty} x exp(- x^2 ) dx  \,= \, 1/2,  \, \, \, \, \, \int_{0}^{\infty}  exp(- x^2 ) dx  \,= \, \sqrt {\pi} /2 \, \, \,. 
\eqno(8.1)
$$         
 
The first of these is trivial, since the indefinite integral
$\int^{y} x exp(- x^2 ) dx $ can be expressed in finite
terms (in the sense of Liouville, see [DST]), and equals
$ (-1/2)exp( - y^2 ) +C$, from which the 
definite integral can be obtained
by plugging in $y=0$ and $y \,= \, \infty$. On the other hand, the indefinite
integral $\int_{0}^{y}  exp(- x^2 ) dx $ 
cannot be expressed in finite
terms (so a special name, {\it "erf"}, had to be coined for it),
and hence the definite integral has a nice formula for a
deeper reason.  Since according to  Lord Kelvin, the second  definite
integral above was to Liouville
what $2+2=4$ is to an ordinary mortal,
the first integral must have been as clear to him as $0+0=0$
is to the man in the street.
 
Going back to sums, a definite sum
 
$$
 \sum_{ k= - \infty }^{\infty}  F(n,k)  \,= \, 1 \, \,, 
\eqno(8.2)
$$          
 
with  $F(n,k)$ of CF, is {\it trivial} if the indefinite sum
 
$$
 \Phi (n,m):= \sum_{ k= - \infty }^{m}  F(n,k)  \, \,, 
\eqno(8.3)
$$          
 
is CF in $m$ and $n$, so that the definite sum follows from it
by taking $ \Phi ( n, \infty )$ and checking that it is indeed $1$.
As we saw above, the way we proved an identity like (8.2), was
to apply Gosper's algorithm to $\Delta_n F(n,k)$, but if
the identity (8.2) follows from the indefinite version (8.3), then
Gosper's algorithm is already successful on $F(n,k)$ itself. In other
words, there exists a CF function $PHI (n,k)$ such that
 
$$
 F(n,k) \,= \, \Delta_k \Phi (n,k) \, \, \,. 
\eqno(8.4)
$$          
 
Let
 
$$
 G(n,k) \,= \, \Delta_n \Phi (n,k) \, \, \,. 
\eqno(8.5)
$$          
 
It follows that $G(n,k)$ is the WZ-mate of $F(n,k)$ and that
 
$$
 \delta \Phi (n,k)  \,= \, F(n,k) \delta k  \,+ \, G(n,k) \delta n \, \, \,, 
\eqno(8.6)
$$          
 
and the  WZ form $F(n,k) \delta k  \,+ \, G(n,k) \delta n$ is thus an
exact form.
 
Similar considerations hold for identities with $r$ free parameters,
leading to $1-forms$ with $r+1$ variables, and to multi-sum
identities.
The interesting identities are precisely those arising out of
WZ forms that are not exact, so loosely speaking, finding all
interesting identities amounts to computing the {\it WZ cohomology}.
we must be careful to take the word "cohomology" with a grain of salt,
since the set of WZ forms is not a vector space. One way out is
to consider the vector space of linear combinations, but a better way
is as follows. 
 
{\bf Research Problem}: Characterize those CF functions $f$
such that there exist (closed), non-exact, WZ forms 
$f \omega_P$, with $\omega_P$ a polynomial difference
form, and for those successful $f$, compute the cohomology of
closed modulo exact.
 
{\bf 9. Continuous WZ forms}
 
The theory of holonomic systems makes sense, and in fact 
was initiated([Ber]),
in $R^n$. The notion of CF is defined naturally as follows.
(See [AZ], where it is called "hyperexponential".)
 
{\bf Definition:} A function $f( x_1 , ... , x_n )$ is
CF if all its logarithmic partial derivatives
$ \,{{\partial f}\over{ \partial x_i }} /f  \, \,$ are rational functions of
$x_1 , ... , x_n$.
 
In analogy with the discrete case, we define
 
{\bf Definition}: A continuous WZ form on $R^n$ is 
a closed differential form that is the product
a CF function by a polynomial differential form.
 
It is very easy to manufacture new continuous WZ forms out of old
ones. If $\omega$ is any WZ form, so is $\omega (g)$, for any rational
transformation $g$, and if $\omega$ and $theta$ are WZ forms, so is
their wedge product $\omega theta$. But let's not get too exited:
I don't know of any continuous closed WZ form that is not exact!
and in fact I am almost sure that the following conjecture is true:
 
{\bf Conjecture:} The continuous WZ cohomology is trivial, i.e.
every closed continuous WZ form is exact.
 
Although I do know of a few CF identities
 
$$
 \int_{ - \infty }^{\infty} F(x,y) dy  \,= \, 1 \, \, \,, 
\eqno(9.1)
$$          
 
with $F(x,y)$ CF (see [AZ]), none of them come from WZ forms.
What we do have in this case, in analogy to the discrete case, is
a CF $G(x,y)$, and an operator $A( D_x , x )$
such that
 
$$
 A( D_x , x ) D_x F(x,y)  \,= \, D_y G(x,y) \, \, \,, 
\eqno(9.2)
$$          
 
but  $A( D_x , x )$ has never, in my experience, 
turned out to be the identity operator!
 
It seems that the reason that the discrete case leads to so many more
interesting things is that finite differences are not derivations,
and you can't compose with a transformation, so WZ forms are hard
to come by, and hence are more interesting. However we should not
write off the continuous realm altogether, all we have to do is
interface it with the discrete, which brings us to the next section.
 
\bf
10. Holonomic and WZ forms in
$  {R^r \times Z^s }$\rm
 
The full beauty and power of the theory of holonomic functions,
as developed in [Z1], is on functions of several discrete and
continuous variables. Everything we said before extends naturally.
Since "continuous came first" we will denote the exterior
derivative-difference by $d$. It is defined on scalar functions by:
 
$$
 d f ( x_1 , ... , x_r , m_1 , ... , m_s )  \, := \, \sum_{i=1}^{r} {{\partial f}\over{\partial x_i }} d x_i + \sum_{j=1}^{s} ( \Delta_{m_j} f  ) \delta m_j  \, \, \,, 
\eqno(10.1)
$$      
 
and on general forms as before, where all the "letters"
{$d x_1 , ... , d x_n , \delta m_1 , ... , \delta m_n$}
anti-commute. Once again we can raise the question of WZ cohomology,
and trying to find "all of them". The Macdonald conjectures (see [H]
for a nice review and references), 
offer many examples of possible WZ forms, and
for $A_2$ and $G_2$ they were found by Shalosh B. Ekhad
[E].
 
{\bf 11. Examples of Discrete-Continuous WZ forms}
 
The customary proof of Euler's integral
 
$$
 \int_{0}^{\infty} e^{-x} x^k  \,= \, k! 
\eqno(11.1)
$$          
 
is by integration by parts. This proof can be recast by saying that
the form
 
$$
 \omega_{"\Gamma"} := { e^{-x} x^k } over (k+1)!  [ (k+1) dx  \,+ \, x \delta k   \, ] 
\eqno(11.2)
$$         
 
is closed, as is easily verified. The "companion identity" is
obtained by summing $\omega_{"GAMMA"}$ w.r.t to $k$, and, surprise!,
it turns out to be good old
 
$$
  e^x  \,= \, \sum_{k=0}^{\infty} {{x^k}\over{k!}} \, \, \,. 
$$
 
So Euler's integral for the Gamma function could have been discovered
by starting with the series expansion for $e^x$ and finding
its companion identity.
 
Similarly, Euler's beta integral could have been discovered by
taking the companion identity of the binomial theorem for
$(1-y)^{-m}$. Euler's Beta integral gives rise to the
following WZ form, whose (easily verified) closedness proves it.
 
$$
 \omega_{BETA}  \,:=  \, 
{{y^n (1-y)^m (n+m+1)!}\over{(m+1)!(n+1)!}} \,  [ \, ((n+1)(m+1))dy-(y(1-y)(m+1)) \delta n -  (y(1-y)(n+1)) \delta m  ] \, \, \,. 
\eqno(11.4)
$$
 
It is possible to get many more non-trivial examples as follows.
Take a known evaluable $" "_2 F_1 (x)$, (many of which
were found by Gosper and Gessel \& Stanton, 
([PBM], pp. 491-497,[GS]),
convert them to an integral formula, via the well known
integral formula (e.g. [Ba] (1.5),(1), p. 4)
 
$$
 {}_2 F_1 (a,b;c;z) \,= \, {{\Gamma (c) }\over{ \Gamma (b) \Gamma (c-b) }} \int_{0}^{1} t^{b-1} (1-t)^{c-b-1} (1-tz)^{-a} dt \, \, \,, 
\eqno(11.5)
$$        
 
and find the corresponding
WZ proof and form. A MAPLE program, based on the algorithm of
[AZ], that does just that,
is available upon requests, as is
an extensive list of successful outputs. For example, the identity
([GS],(5.23))
 
$$
 {}_2 F_1 (-2n-1/3 \,, \,-n \,; \,2/3 \,; \,-8) \,= \, (-27)^{n \, \, \,,} 
\eqno(11.6)
$$          
 
yields the following continuous-discrete WZ form:
 
$$
{ {y^{-n-1} (1-y)^{n-1/3} (1+8y)^{2n+1/3}  n! } \over 
{ 27^n  \,(5/3)_n  }}
  \,\cdot \,  [ (27n+18) dy \,+ \,(4y-1)(1-y)(1+8y)\delta n  ] \, \, \, \,.  
\eqno(11.7)
$$      
 
\bf
12. A WZ approach 
to Hypergeometric Convergence Acceleration Formulas\rm
 
Some WZ forms can be useful for convergence acceleration. If $\omega$
is a WZ $r-form$ , then its integral is zero over any r-manifold 
that is a boundary of some $(r+1)-$manifold.
By partitioning such a manifold into two subsets, we get that two
different summations are equal. If one of them converges faster then
the other we have a convergence acceleration formula. 
 
Let's specialize to two variables $(n,k)$. Let 
$\omega := F(n,k) \delta k  \,+ \, G(n,k) \delta n$ be a WZ form, and let's
sum it over the discrete contour 
 
$$
\partial \Omega :=  \{ ((n,0) \leftarrow (n+1,0); n>= 0 \}  \,\cup \, 
\{ (n,n) \rightarrow (n+1,n) \rightarrow  (n+1,n+1) \,; \, n >= 0 \}  \, \cup  
$$
$$
\, \{ ( \infty ,k+1 ) \rightarrow ( \infty ,  k ); k >= 0 \}, 
$$
 
which is the "boundary" of the region $\Omega =$  {$(n,k); n>=k$ }.
Since
 
$$
 0 = \sum_{\partial \Omega }  \, \omega = \sum_{n=0}^{\infty} G(n,0) - \sum_{n=1}^{\infty} (F(n,n-1)+G(n-1,n-1)) \, \, \,, 
\eqno(12.2)
$$        
 
we have
 
{\bf Theorem 7:} For any WZ pair $(F,G)$
 
$$
 \sum_{n=0}^{\infty} G(n,0)  \,= \, \sum_{n=1}^{\infty} (F(n,n-1)+G(n-1,n-1)), 
\eqno(12.3)
$$         
 
whenever both sums converge.
 
We will give three applications of theorem 7. The WZ form
 
$$
 \omega_{log(1+x)} := {{(-1)^k n! k!  }\over{(n+k+1)! x^k (1+x)^{n+1} }}  [  \, (1+x)  \delta k  \,+ \, x \delta n  ] 
\eqno(12.4)
$$        
 
leads to the acceleration formula
 
$$
  \sum_{n=1}^{\infty} {{1}\over{n (1+x)^n }} ( \,= \, log((1+x)/x)) \,= \, (1+2x) \sum_{n=1}^{\infty} {{(-1)^{n-1}} \over {n  (  {2n \atop n }  ) ((1+x)x)^n }} \, \, \,. 
\eqno(12.5)
$$       
 
The WZ form
 
$$
 \omega_{ \zeta (2) }  \,:=  \, {{(-1)^{(n+k)} k!^2 (n-k-1)!}\over{ (n+k+1)!}}   [ \delta k  \,+ \, (2(n-k)/(n+1)) \delta n  \,  ] 
\eqno(12.6)
$$        
 
leads to the well known formula for $\zeta (2)$( [P]):
$$
 2 \sum_{n=0}^{\infty} {{(-1)^n}\over{(n+1)^2}} =  
\,( \, \zeta (2)  \,, \, (why?) \,) \,= 
\, 3 \sum_{n=1}^{\infty} {{1} \over {  (  { {2n} \choose {n} }   ) n^2}} \, \, \,, 
\eqno(12.7)
$$       
 
while the WZ form
 
$$
 \omega_{ \zeta (3) } := {{(-1)^k k!^2 (n-k-1)!}\over{(n+k+1)!}}  [ \, (1/(2(k+1))) \delta k  \,+ \, ( (n-k)/ (n+1)^2 )  \delta n  \, ] 
\eqno(12.8)
$$      
 
leads to the acceleration formula ([P])
 
$$
 \sum_{n=1}^{\infty} {{1}\over {n^3}} = (5/2) 
\sum_{n=1}^{\infty} {{(-1)^{n-1}} \over { { {2n} \choose {n} } n^3} } 
\, \, \,, 
\eqno(12.9)
$$      
 
which was the starting point of Ap\'ery's wonderful proof,
which we will discuss and redo later.
 
The notion of WZ pairs and forms thus gives a unified setting for
proving and discovering convergence acceleration formulas. To discover
such new formulas we take any known identity, or \it
a specialization
thereof\rm
, find the corresponding WZ pair, take appropriate 
shadow ([WZ1], sect. 4),
and cross our fingers that one of the sums 
in (12.3) converge slowly to a well
known number, while the other sum converges fast.
 
The same reasoning can be applied to $1-forms$ in more variables
and also to higher degree forms. We will only state it for
$1-forms$ of 3 variables, since the general statement, though
straightforward, is rather cumbersome.
we have
 
{\bf Theorem 8:} For any WZ 1-form of three variables,
 
$$
 \omega := F (n,k,a) \delta k  \,+ \, G(n,k,a) \delta n  \,+ \, H(n,k,a) \delta a  \, \, \,,we \,have \, \,, 
\eqno(12.10)
$$         
 
$$
 \sum_{a=0}^{\infty} H(0,0,a)  \,= \, \sum_{n=1}^{\infty} (H(n,n,n-1)+F(n,n-1,n-1)+G(n-1,n-1,n-1)), 
\eqno(12.11)
$$         
 
provided both sides converge. 
 
Applying theorem  8 to 
$\omega_{VANDERMONDE}$ of (7.12) we get an
even better acceleration for $\zeta (2)$:
 
$$
  \sum_{n=1}^{\infty} {{1}\over {n^2}} = \sum_{n=1}^{\infty} {{n!^6 (21n+13)}\over{8(2n+1)!^3}} \, \, \,. 
$$
 
{\bf 13. A Conceptual Frame To Ap\'ery's Magic}
 
Ap\'ery ([A1][A2][P][R][M]) used the fact that if 
$\alpha$ is a rational number,
then for any sequence $a_n / b_n ( \neq  \alpha )$ of rational numbers
tending to 
$\alpha$, $| \alpha - a_n / b_n | >=  C/| b_n |$
(if $\alpha = c/d$ then 
$| \alpha - a_n / b_n | > (1/ d)/ b_n$). Thus if
one can exhibit a sequence $a_n / b_n ( \neq  \alpha )$, of rational
numbers, with $a_n , b_n$ integers, such that
$| \alpha - a_n / b_n | < C/ | b_n |^{ 1 + \delta }$,
$\delta > 0$, then one has proven the irationality of $\alpha$.
For $\alpha$ defined by a hypergeometric series
 
$$
 \alpha = \sum_{n=0}^{\infty} f(n) \, \, \,, ( \, f(n+1)/f(n)   \,a \,rational \,function \,in \,n) \, \,, 
\eqno(13.1)
$$         
 
the natural temptation is to take the approximating sequence
to be the sequence of partial sums
 
$$
 \sum_{k=0}^{n} f(k) \, \,, 
\eqno(13.2)
$$          
 
and this works for $e$, for example. However, for $\zeta (n)$ the
convergence is far too slow, although the denominators are not
too bad. To rectify the slow convergence, one may try to find
a convergence acceleration formula, like (12.7) for $\zeta (2)$ and
(12.9) for $\zeta (3)$. Alas now the denominators of the
partial sums grow too fast, and
the improved numerical convergence does not make up for it.
 
Ap\'ery's breakthrough ([P][R][M]) consisted in finding
a doubly-indexed sequence $c(n,k)$ 
(defined for $n >= k >= 0$) that tends to the number
of interest $\alpha$ (in his case $\zeta (3)$ and $\zeta (2)$),
no matter how you go to $( \infty , \infty )$ in the region
$ n >= k >= 0$. The sequence $c(n,0)$ turned out to be
the sequence of partial sums of the defining series (whose
denominators were decent, but whose convergence rate was too slow),
while the diagonal sequence $c(n,n)$ turned out to be
the sequence of partial sums of the accelerated series
(12.7) and (12.9), whose convergence rate was good, but
whose denominators grew too large. Somehow it was necessary
to have a tradeoff. This was achieved by introducing
a new approximating sequence, by taking
a weighted average of the $c(n,k) , k= 0 , ... , n$,
with explicitly defined weights $b(n,k)$:
 
$$
 d_n  \,= \, {{a(n)}\over {b(n)}} := {{ \sum_{k=0}^{n} c(n,k) b(n,k) } \over { \sum_{k=0}^{n} b(n,k) }}             \, \, \,. 
\eqno(13.3)
$$        
 
(Please be warned that the $c(n,k)$ are rational numbers,
so the numerator $a(n)$ in the above expression 
is not an integer.)
It turned out that for a judicious choice of $c(n,k)$ and
weights $b(n,k)$, things 
worked out for Ap\'ery. To wit, for
$\zeta (2)$ he gave
 
$$
 c(n,k):= 2 \sum_{m=1}^{n} {{(-1)^{m-1}} \over{ m^2 }} + 
\sum_{m=1}^{k} {{(-1)^{n+m-1}} \over { m^2  (  { n \atop m }  )  
          (  { n+m \atop m }  ) }}  
\,, \, \, \, \, b(n,k)=  (  { n \atop k }  )^2
                 (  { n+k \atop k }  ) \, \, \,, 
\eqno(13.4)
$$  
 
while for $\zeta (3)$ 
Ap\'ery gave
 
$$
 c(n,k):=  \sum_{m=1}^{n} {{1}\over{ m^3 }} + \sum_{m=1}^{k} {{(-1)^{m-1}} \over { 2 m^3  (  { n \atop m }  )            (  { n+m \atop m }  ) }}  \,, \, \, \, \, b(n,k)=  (  { n \atop k }  )^2                 (  { n+k \atop k }  )^2  \, \, \,. 
\eqno(13.5)
$$  
 
The way Ap\'ery 
proved that $d_n$ had the desired properties
was to show that both the top 
($a(n)$) and bottom ($b(n)$) of (13.3) were solutions
of a certain three term linear recurrence with polynomial
coefficients. In the case of $\zeta (2)$ it turned out
that both $x_n = a(n)$ and $x_n = b(n)$ were solutions
of
 
$$
 n^2 x_n - (11 n^2 - 11 n +3) x_{n-1} - (n-1)^2 x_{n-2} =0 \, \, \, \,, 
\eqno(13.6)
$$         
 
while in the case of $\zeta (3) $ the recurrence was
 
$$
 n^3 x_n - (34 n^3 - 51 n^2 +27n-5) x_{n-1} + (n-1)^3 x_{n-2} =0 \, \, \, \,. 
\eqno(13.7)
$$         
 
In Alf van der Poorten's delightful account of Ap\'ery's proof([P])
(with details that were filled in by H. Cohen), everything
is presented as magic, in tune with Ap\'ery's own personality
and style ([A1],[M]). 
Subsequently, Beukers[Beu] presented a new proof that
took away the magic but retained the charm of the original proof.
Beukers's elegant proof was extended and generalized by himself,
Allady and Robinson[AR], and others, while Ap\'ery's original proof
seemed ad-hoc and ungeneralizable.
 
Ken Wilson once said that today's tricks are
tomorrow's theory. I will now demystify Ap\'ery's original proof by
placing it in the context of
holonomic functions and WZ theory.
Hopefully this will
open the door for further applications and generalizations. I will also
show how all the "hairy" steps in Ap\'ery's proof are now
purely routine, and provable by computer, using the fast algorithm
[Z3] that is based on Gosper's [G] algorithm. See the appendix for
an example.
 
Let's make two key
observations about Ap\'ery's proofs:
 
{\bf Observation 1}:
The doubly-infinite sequence $c(n,k)$ is the "potential
function" of a WZ form! Namely it is the "contour sum"
of a WZ form $\omega = F \delta k + G \delta n$ from $(0,0)$ to
$(n,k)$, and since $\omega$ is closed, the contour is immaterial.
In other words, $\omega = \delta c$. Since the holonomic cohomology
is trivial, we know that every closed 1-form $\omega$, qua holonomic
form, can be expressed as $\delta c$, for some holonomic 0-form
(i.e. function) $c(n,k)$. Note that $c(n,k)$ is not CF, or else
everything would be trivial. The $c(n,k)$ for $ \zeta (2)$, (13.4),
is the potential function of the WZ form $\omega_{\zeta (2)}$ given in
(12.6), and the $c(n,k)$ for $\zeta (3)$, (13.5), is the potential
function for the WZ form $\omega_{\zeta (3)}$ given in (12.8).
(The way $c(n,k)$ is presented in (13.4) and (13.5) is the "contour sum"
of  $\omega_{\zeta (2)}$ and $\omega_{\zeta (3)}$ over the
contour
$(0,0) \rightarrow (1,0) \rightarrow ... \rightarrow (n,0) \rightarrow (n,1) \rightarrow ... \rightarrow (n,k)$.)
 
{\bf Observation 2}: The "weighting" function $b(n,k)$ is CF.
 
Let's recall the "hairy steps" in Ap\'ery's proof ([P],[R]). First
a recurrence operator $P(N,n)$ is pulled out of the hat, and it
is claimed that both $a(n)$ and $b(n)$ are annihilated by it.
The way these claims are proved is to manufacture a $B(n,k)$,
again out of the blue, such that
 
$$
  P(N,n) b(n,k) = B(n,k)  \,- \, B(n,k-1) \, \, \,, 
$$
 
and a $D(n,k)$ such that
 
$$
  P(N,n) (b(n,k)c(n,k))= B(n,k)c(n,k) \,- \,B(n,k-1)c(n,k-1)+         D(n,k)-D(n,k-1) \, \, \,. 
$$
 
Furthermore, the $B(n,k)$ and $D(n,k)$ turned out to be CF.
 
I will now show that for any $c(n,k)$ that is a potential function
of a WZ 1-form $\omega$, and any CF function $b(n,k)$
there exists such
$P(N,n)$, $B(n,k)$, and $D(n,k)$.
In other words:
 
{\bf 1)\rm
 We have the "meta-theorem" that for \it
every} $c(n,k)$ such
that $\delta c$ is WZ, and for {\it every} CF $b(n,k)$, we
have an Ap\'ery-style proof.
 
{\bf 2)} The form of the proof is always the same.
 
I will also show how my computer (or your computer, if you have 
MAPLE and ask
for my program), can generate the proof, in every given case,
out of the input $((F(n,k),G(n,k)),b(n,k))$, 
(where $\delta c = F \delta k + G \delta n$, and the latter is WZ.)
The proof consists in presenting the recurrence operator
$P(N,n)$ and the two "certificates" $B(n,k) \,,D(n,k)$.
 
Unfortunately, the recurrence $P(N,n)$ is usually of higher order,
and usually does not yield irrationality,
and so far I am unable to prove irrationality of new interesting
numbers.
 
After this long introduction, let's go to business.
 
{\bf Theorem 9}: Let $c(n,k)$ be the potential function of
a WZ 1-form $F(n,k) \delta k + G(n,k) \delta n $ 
in the two variables $(n,k)$. In other words,
 
$$
  F(n,k)= c(n,k+1)-c(n,k) \,, \, \, \,G(n,k)=c(n+1,k)-c(n,k) \, \,, 
$$
 
and let $b(n,k)$ be CF. Let $a(n)$ and $b(n)$ be the
top and bottom of (13.3):
 
$$
  a(n):= \sum_{k=0}^{n} c(n,k) b(n,k)  \,, \, \, \, \, \, \, b(n):= \sum_{k=0}^{n} b(n,k)     \, \,. 
$$
 
 There exist
(rapidly exhibitable) linear recurrence operators
with polynomial coefficients $R (N,n)$ and
$S (N,n)$ such that
 
$$
 R (N,n) b(n)  \,= \,0 \,, \, \, \, \, \, S (N,n) R (N,n) a(n) \,= \,0 \, \, \, \,. 
\eqno(13.9)
$$         
 
Furthermore, there exist rapidly exhibitable
CF "certificates" $B(n,k)$ and $D(n,k)$ such that
the following routinely verifiable identities are true:
 
$$
 R (N,n) b(n,k) = B(n,k)-B(n,k-1) \,, 
\eqno(13.10)
$$          
 
$$
  S (N,n) R (N,n) ( b(n,k) c(n,k) ) = S (N,n) ( c(n,k)B(n,k) - c(n,k-1)B(n,k-1)) + D(n,k)-D(n,k-1) \, \, \,. 
$$
 
In addition, $B(n,k)/b(n,k)$ and $D(n,k)/(b(n,k)F(n,k))$ are
both rational functions.
 
Before proving Theorem 9, let's make a few remarks. Since
$R (N,n) b(n)  \,= \,0$ obviously implies 
$S (N,n) R (N,n) b(n)  \,= \,0$, it follows that both
$a(n)$ and $b(n)$ are annihilated by the same operator
$P(N,n) = S (N,n) R (N,n)$, but the latter is 
in general {\it not} the minimal order operator that annihilates
$b(n)$. $P(N,n)$ is minimal for both $a(n)$ and $b(n)$ whenever
$S (N,n)$ is the identity operator. This was what
happened in Ap\'ery's cases, and explained why he was successful,
since both $a(n)$ and $b(n)$ satisfied \it
the same minimal second order
linear recurrence\rm
$S$ is the identity operator 
an {\it Ap\'ery pair}.
 
Note also that (13.10) immediately imply (13.9), by summing w.r.t
$k$,
and that (13.10) is routinely verifiable, as we will see.
 
{\bf Proof of Theorem 9:} The first parts of (13.9) and (13.10)
were proved in [Z3], where an algorithm for constructing 
$R (N,n)$ and $B(n,k)$ was given, and it was shown that
$B(n,k)/b(n,k)$ is a rational function of $(n,k)$.  Now let's write
 
$$
 R (N,n) = \sum_{i=0}^{L} r_i (n) N^i 
\eqno(13.11)
$$          
 
and consider $R (N,n)  \,( \,b(n,k) c(n,k) \,) \,$.  We have
 
$$
 R (N,n)  \,( \,b(n,k) c(n,k) \,) \, - c(n,k) R (N,n) b(n,k) = 
\eqno(13.12)
$$          
 
$$
  ( \sum_{i=0}^{L} r_i (n) N^i  ) b(n,k) c(n,k) \,- \, c(n,k) ( \sum_{i=0}^{L} r_i (n) N^i ) b(n,k) 
$$
 
$$
   \,= \,\sum_{i=0}^{L} r_i (n) b (n+i,k) [ c(n+i,k)-c(n,k)] \,= \, \sum_{i=0}^{L} r_i (n) b(n+i,k)  \,[ \, \sum_{j=0}^{i-1} G(n+j,k)  \,] \, \, \,. 
$$
 
It follows from this and the first part of (13.10) that
 
$$
 R (N,n) ( \,b(n,k) c(n,k) \,)  \,= \, 
\eqno(13.13)
$$          
 
$$
  c(n,k) ( B(n,k) - B(n,k-1) )  \,+ \, \sum_{i=0}^{L} r_i (n) b(n+i,k)   \,[ \, \, \sum_{j=0}^{i-1} G(n+j,k)  \, \,] \, \, \,. 
$$
 
But
 
$$
 c(n,k) ( B(n,k) - B(n,k-1) ) = 
\eqno(13.14)
$$          
 
$$
  c(n,k)B(n,k) - c(n,k-1) B(n,k-1) - (c(n,k)-c(n,k-1)) B(n,k-1)= 
$$
 
$$
  c(n,k)B(n,k) - c(n,k-1) B(n,k-1) - F(n,k-1) B(n,k-1) \, \, \,. 
$$
 
It follows that
 
$$
 R (N,n)  \,( \,b(n,k) c(n,k)  \,) = c(n,k) B(n,k) - c(n,k-1) B(n,k-1) + H(n,k) \, \, \, \,, 
\eqno(13.15)
$$         
 
where $H(n,k)$ is given by the expression:
 
$$
 H(n,k) =   \,\sum_{i=0}^{L} r_i (n)  \, b(n+i,k) \,  [ \, \, \sum_{j=0}^{i-1} G(n+j,k)  \, \,] \,-F(n,k-1) B(n,k-1) \, \, \,. 
\eqno(13.16)
$$        
 
The pleasant surprise it that $H(n,k)$ is CF on its own right!
Indeed we can write
 
$$
 H(n,k)  \,= \,   \{ \sum_{i=0}^{L} r_i (n) (b(n+i,k) /b(n,k)) [ \, \sum_{j=0}^{i-1}  ( \, {{G(n+j,k)}\over {F(n+j,k)}}  \,)  \,( \, {{F(n+j,k)}\over {F(n,k)}}  \,) \, ] 
\eqno(13.17)
$$       
 
$$
   \,-( {{{F(n,k-1)}\over {F(n,k)}}} {{B(n,k-1)}\over {b(n,k-1)}}  {{b(n,k-1)}\over {b(n,k)}}   \} b(n,k) F(n,k) \, \, \,. 
$$
 
Since $b(n,k)$ and $F(n,k)$ are CF, and $B(n,k)/b(n,k)$ and
$G(n,k)/F(n,k)$ are rational functions, it follows that the
expression inside the braces in (13.17) is a rational function,
since it is a sum of products of them. Obviously
$b(n,k) F(n,k)$ is CF, and it follows that $H(n,k)$ is CF.
 
It follows once again from [Z3] that there exists an operator
$S (N,n)$ and a CF function $D(n,k)$ that is a
multiple of $H(n,k)$ by a rational function, such that
 
$$
 S (N,n) H(n,k) = D(n,k) - D(n,k-1) \, \, \,. 
\eqno(13.18)
$$          
 
Applying $S (N,n)$ to (13.15), and using (13.18), we get
 
$$
 S (N,n) R (N,n) ( \, b(n,k) c(n,k) \,)  \,= \, 
\eqno(13.19)
$$          
 
$$
  S (N,n) [ c(n,k) B(n,k) - c(n,k-1) B(n,k-1) ] + D(n,k) - D(n,k-1)  \, \, \, \,, 
$$
 
which is the second half of (13.10), which we had to prove, and that
immediately implies (13.9).   QED
 
One of the rewards of a general theory is that it points the way to
fruitful future generalizations. Instead of a two-variable
$c(n,k)$, we can consider a multi-variate
$c(n, k_1 , ... , k_r )$ that is a potential function
of a WZ 1-form 
 
$$
  \omega  \,= \, F(n, k_1 , ... , k_r ) \delta n + \sum_{i=1}^{r} G_i ( n, k_1 , ... , k_r )  \delta k_i  \, \, \,. 
$$
 
Now we consider weighted
averages
 
$$
 d_n ={{ \sum_{ k_1 , ... , k_r } b(n, k_1 , ... , k_r ) c(n, k_1 , ... , k_r )} \over { \sum_{ k_1 , ... , k_r } b(n, k_1 , ... , k_r )}} \, \, \,. 
\eqno(13.20)
$$      
 
It is possible to show that (13.9) still holds, while (13.10) is
replaced by the existence of 
$B_i ( n , k_1 , ... , k_r )$, 
$D_i ( n , k_1 , ... , k_r )$, $i=1, ... , r$, all CF,
such that
 
$$
 R (N,n) b(n, k_1 , ... , k_r ) \,= \, \sum_{i=1}^{r} \Delta_{k_i} B_i ( n , k_1 , ... , k_r ) \, \, \,, 
\eqno(13.21)
$$        
 
$$
 S (N,n) R (N,n) (  \,b \, c \, )= S (N,n)  (  \sum_{i=1}^{r} \Delta_{k_i} ( c B_i )  )  \,+ \, \sum_{i=1}^{r} \Delta_{k_i} D_i  \, \, \, \,. 
\eqno(13.22)
$$       
 
Furthermore, the $B_i$'s are all multiples of
$b(n,k)$ by rational function, and the $D_i$ are all multiples
of $b(n,k) F(n,k)$ by  rational 
functions. The only thing different is that
at present there is no fast algorithm for finding the
recurrence operators $R$, $S$ and the certificates
$B_i$, $D_i$, and the slow algorithm is too slow. I hope
that soon I or someone else will find such a fast algorithm.
\eject
{\bf Concluding comments}
 
Another approach to the insight behind the Ap\'ery recurrence
for $b(n)$ (but not for $a(n)$) was given by Askey and Wilson[AW].
Askey and Wilson derive recurrences for much more general sequences.
It would be interesting to find the analog of $c(n,k)$
and $a(n)$ for these more general sequences, since perhaps they
can be used to prove the irrationality of new numbers.
 
The referee has pointed out that so far we didn't show how the
WZ-form is affected by specialization, save for doing it
empirically. Herb Wilf and I, succeeded in doing just this, and this
will appear in a forthcoming paper. The referee has also pointed out
that our definition of "closed form" excludes sequences such as
$f(n)=0, n \, odd, f(n)= (n/2)!, n \, even$. It would be interesting to
generalize the present theory to this more general class.
 
\bf
APPENDIX: AN EXAMPLE OF AN Ap\'ery-STYLE COMPUTER-GENERATED PROOF:
The Irrationality of log(2)\rm
 
My program is available upon request. Here is the input file
for the irrationality of log(2). (TOPb, BOTb, bPol, bARGn, bARGk)
describe b: b is bPol times $(bARGn)^n (bARGk)^k$ times
the product of the factorials in the list TOPb divided by the
product of the factorials in the list BOTb. Ditto,
regarding $F$, for
(TOPF, BOTF, FPol, FARGn, FARGk), GoF is the rational function that
is $G/F$, $SEDER$ is the conjectured order of $R (N,n)$,
while SEDER1 is the conjectured order of $S (N,n)$.
 
\#Begin Input File
read `full\_path\_name\_of\_program\_file` :
TOPb:=[n+k]:
BOTb:=[k,k,n-k]:
bPol:=1:
 
bARGn:=1:
bARGk:=1:
TOPF:=[n,k]:
BOTF:=[n+k+1]:
FPol:=1:
FARGn:=1/2:
 
FARGk:=-1:
GoF:=1/2:
SEDER:=2;
SEDER1:=0:
NAME:=log(2):
 
apery(TOPb, BOTb, bPol, bARGn, bARGk, TOPF, BOTF, FPol, FARGn,
FARGk, GoF, NOTATION, SEDER, SEDER1, NAME):
quit;
\#End of Input File
 
Placing the input file in a file called, say, inlog2, I instructed
my computer, Shalosh B. Ekhad (that runs under UNIX):
 
                          maple $<$ inlog2
 
After a few seconds, came the output:
 
AN APERY PAIR THAT IMPLIES THE IRRATIONALITY OF   ln(2)
 
by SHALOSH B. EKHAD, 85 Wilson Road, Princeton, NJ 08540
 
Theorem : Let W(n,k):=
 
$$
\, \, \,{{(-1)^k (1/2)^n n! k!}\over{(n+k+1)!}} 
$$
 
let $Z(n,k):=(1/2)W(n,k)$ and let $b(n,k):=$
 
$$
\, \, \, \, {{\,(n+k)!}\over{k!^2 (n-k)! }} 
$$
 
then ((W(n,k),Z(n,k)),b(n,k)) is an Ap\'ery pair and both
b(n):=Sum\_k b(n,k) and a(n):=Sum\_k b(n,k)c(n,k) are solutions
of the linear recurrence equation, (c(n,k) is the 
potential function of (W,Z))
 
$$
        (n N^{-1} - 6 n - 3 + (n + 1) N) u(n) = 0 \, \, \,.  
\eqno(1)
$$         
PROOF: We cleverly construct $B(n,k):=$
 
$$
                            {{(- 4 n - 2) (n + k)! }\over{k!^2 (n - k)!}} 
$$
 
with the motive that
 
$$
 (n N^{-1} - 6 n - 3 + (n + 1) N) b(n, k) = B(n, k) - B(n, k - 1),  
\eqno(2)
$$          
 
(check!), which upon summing w.r.t to k shows that b(n) is a solution
of (1).
To establish that a(n) also satisfies (1), we cleverly
construct
 
$$
 A(n,k):=  B(n, k) c(n, k) + {{ (-1)^k  (1/2)^n n! }\over{k! (n - k + 1)! }} 
\eqno(3)
$$     
 
with the motive that
 
$$
      (n N^{-1} - 6 n - 3 + (n + 1) N) b(n, k) c(n, k) =  A(n, k) - A(n, k - 1) 
$$
 
(To get (3) use (2) and the expression for differences of
c(n,k) in terms of W and Z)
and the result follows upon summing w.r.t to k.
 
My program also reproduced within seconds the irrationality proofs
of $\zeta (2)$ and $\zeta (3)$.
 
{\bf REFERENCES}
 
[A1] Ap\'ery, R., 
\it
Irrationalite\*` 
de $\zeta (2)$ et $\zeta (3)$\rm
, Asterisque {\bf 61}(1979), 11-13.
 
[A2] Ap\'ery, R., \it
Interpolation de fractions continues 
et irrationalite\*' de certaines constantes\rm
, Bulletin de la
section de sciences du C.T.H.S. \#3, 37-53.
 
[AR] Allady, K., and Robinson, M. L., \it
Legendre polynomials and
irrationality\rm
, Crelle's Journal {\bf 318}(1980), 137-155.
 
[AW] Askey, R.,and Wilson, J., \it
A recurrence relation generalizing
those of Ap\'ery\rm
, J. Aust. Math. Soc. {\bf 36}(1984), 267-278.
 
[AZ] Almkvist, G., And Zeilberger, D., \it
The method of differentiating 
under the integral sign\rm
, J. Symbolic Computation, to appear.
 
[Ba] Bailey, W. N. , "{\it Generalized Hypergeometric Series}", Cambridge
Math. Tracts {\bf 32}, Cambridge University Press, London, 1935.
(Reprinted: Hafner, New York, 1964 .)
 
[Ber] Bernstein, I. N. , \it
Modules over a ring of
differential operators, study of the fundamental solutions of
equations with constant coefficients\rm
, Functional Analysis and Its
Applications, Vol {\bf 5}, No. 2(1971), Russian original :1-16,
English translation:89-101.
 
[Beu] Beukers, F., \it
A note on the irrationality of
$\zeta (2)$ and $\zeta (3) $, Bull. London Math. Soc. {\bf 11}
(1979), 268-272.
 
[Bj] Bjo\*:rk, J. -E., "{\it Rings Of Differential Operators}",
North-Holland, Amsterdam, 1979.
 
[DST] Davenport, J. H., Siret, Y., Tournier, E., 
"{\it Computer Algebra}", Academic Press, London, 1988.
 
[E] Ekahd, S. B., \it
A one-line proof of the Habsieger-Zeilberger
$G_2$ constant term identity\rm
, J. Computational and Applied
Mathematics, to appear.
 
[F] Flanders, H., 
"{\it Differential Forms: With Applications to the Physical Sciences}",
Academic Press, New York,  1963.
 
[G] Gosper, R. W., Jr., \it
Decision Procedure for indefinite 
hypergeometric summation\rm
,
Proc. Natl. Acad. Sci. USA\rm
 {\bf 75}(1978), 40-42.
 
[GKP] Graham, R., Knuth, D.E., and Patashnik, O., "\it
Concrete
Mathematics\rm
", Addison Wesley, Reading, 1989.
 
[H] Habsieger, L., {\it Macdonald conjectures and the Selberg integral},
in: "q-Series and Partitions", edited by D. Stanton, IMA volumes
{\bf 18}, Springer, New York, 1989.
 
[M] Mende\*`s France, M., {\it Roger Ap\'ery et l'irrationel},
La Recherche, {\bf 10}(1979), 170-172.
 
[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.
 
[PBM] Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I.,
{\it "Integrals and Series"}, vol. 3., Translated from the Russian
by G. G. Gould, Gordon and Breach, New York, 1990.
 
[R] Reyssat, E., \it
Irrationalite\*' 
de $\zeta (3)$ 
selon Ap\'ery\rm
,
Seminaire DELANGE-PISOT-POITU (Theorie des nombres)
{\bf 20}(1978/1979), \#6, 6 pp.
 
[S] Spivak, M., {\it "Calculus On Manifolds"}, Addison-Wesley,
Reading, Mass, 1965.
 
[WZ1] Wilf, H.S., and Zeilberger, D.,
\it
Rational functions certify combinatorial
identities\rm
, J. Amer. Math. Soc. {\bf 3}(1990), 147-158.
ied Math., to appear.
 
[Z2] Zeilberger, D.,
\it
A Fast Algorithm for proving terminating hypergeometric
identities\rm
, Discrete Math {\bf 80}(1990), 207-211.
 
[Z3] Zeilberger, D.,
{\it The method of creative telescoping}, J. Symbolic Computation, 
to appear.
\end
 
