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\rm

{\bf Identities}

\it 
Doron Zeilberger*, Department of Mathematics, Drexel University,
Philadelphia, PA 19104\rm 

Current Address (1990-): 
Department of Mathematics, Temple University,
Philadelphia, PA 19122; zeilberg@math.temple.edu
\rm 

{\bf Appeared in: q-Series and Partitions, D. Stanton, ed.,
 IMA seris 18, 67-75 (1989) } 

{\bf Abstract}: "Professor Littlewood, when he makes use of an
algebraic identity, always saves himself the trouble of proving it;
he maintains that an identity, if true, can be verified in a few
lines by anybody obtuse enough to feel the need of verification."
(Dyson [Dy]).  

Nowadays obtuse mathematicians can use computer algebra. 
Littlewood's dictum is obvious for finite algebraic identities like 
$(a+b)^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3$. 
In this paper,
identities are classified into three classes: Real time-Littlewoodian,
Zillion years-Littlewoodian, and Non-Littlewoodian.  I suggest a
research program whose purpose will be to classify identities, and
families of identities, into these classes, and to try and enlarge
the family of real-time Littlewoodian identities by devising
efficient algorithms.

*Supported in part by NSF grant DMS-8800663
\medskip

{\bf 1. Introduction}

David Blackwell once said that in order for a
result to be interesting, it is not enough that it be new. For
example, if one takes any two ten-digit numbers and multiplies them 
together, then the resulting identity is very probably a new result,
but it is uninteresting.  In any case, it is unnecessary nowadays to
prove a result like "$123 \times 234 = 28782$", because it clearly
belongs to a class of identities for which there is a well known
verification algorithm.  This algorithm of multiplication of two
integers written in decimal notation is known to any four-grader 
is programmed into every calculator ( for integers whose
product has up to ten digits ).  Furthermore, in most computer
algebra systems, one can multiply integers with an arbitrary number
of digits, where the limit depends on the available memory of the
computer.  

So let us make the following definition:

{\bf Definition\rm 
: An identity is \it 
Littlewoodian} if it clearly
belongs to a class of identities for which there is a known  and 
programmable algorithm for determining the truth of the identity.

For example any identity like "$123 \times 234 = 28782$" is obviously
Littlewoodian, as are finite algebraic identities like

$$
  (a+b)^4 = a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^{4}. 
$$

Of course being Littlewoodian is time-dependent, and it is not
enough that there {\it is} an algorithm, we must know how to perform it.
It is conceivable that one can prove that a class of
identities is intrinsically non-Littlewoodian, in analogy with
Matijasevic's
solution of Hilbert's tenth problem.

Consider the identity

$$
  ( 100^{( 100^{100} )} + 1 ) \times ( 100^{(100^{100} )} - 1 ) = 100^{ 2 ( 100^{100} )} -1, 
$$

where it is assumed that the integers are spelled out in decimal
notation, with $2 ( 100^{100} ) $ digits.  Although this identity is 
certainly Littlewoodian, it will take a zillion years, and zillion
bytes of memory to perform.  Such an identity I call
{\it Zillion years Littlewoodian}.  Of course any moderately smart
human or artificially smart machine will recognize the above
identity as a special case of the algebraic identity
$(a+1)(a-1)= a^2 -1$, which in itself is Real-time Littlewoodian,
but this is not allowed, because finding the right generalization is
not a purely mechanical operation in general.

The notion of "Real time-Littlewoodian" is also time dependent,
because any day there may be a better algorithm.  However it is
conceivable that a class of identities be declared intrinsically
zillion years-Littlewoodian, in analogy 
with the notion of intractableness in computer science.

Once a class of identities has been declared Littlewoodian, the
human mathematician is far from superseded.  We still need human
mathematicians to conjecture nice identities, relate them to other
branches of mathematics, and give proofs with
insight, that explain {\it why} the identity is true.  
This insight will often
show us how to generalize it to an identity that is non-Littlewoodian.  
The purpose of the computer is to save  humans the dreadful
task of devising proofs that Hardy called scornfully "essentially
verifications", and concentrate on trying to find insightful proofs.

{\bf 2 Quiz}

For each of the following identities, state whether the identity is 
Real time-Littlewoodian, Zillion years-Littlewoodian, or
Non-Littlewoodian.  Answers to this quiz are given in the next
section.

{\bf 1}   (Euler) 

$2^{ 2^5 } + 1= 641 \times 670041$.

{\bf 2}  (Lander and Parkin [L-P])

$$
  27^5 + 84^5 + 110^5 + 133^5 = 144^{5}. 
$$

{\bf 3}

a)(Leonardo of Pisa)

$$
  ( a^2 + b^2 ) \times ( c^2 + d^2 ) = (ac+bd)^2 + (ad-bc)^{2}. 
$$

b.(Euler)

$$
  ( a^2 + b^2 + c^2 + d^2 ) \times ( A^2 + B^2 + C^2 + D^2 ) =  
$$
$$
  (aA+bB+cC+dD)^2 + (aB-bA-cD+dC)^2 + (aC+bD-cA-dB)^2 + (aD-bC+cB-dA)^{2}. 
$$

{\bf 4} 

$$
  \sum_{i=1}^{N} n^3 = (N(N+1))^2 /4. 
$$

{\bf 5}

$sin(x+a)=sin x cos a + cos x sin a $.

{\bf 6} (Cassini) 

Let $F_n$ be the Fibonacci numbers defined by
$F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$,
then

$$
   F_{n+1} F_{n-1} - F_n^2= (-1)^{n}. 
$$

{\bf 7}

 Let $b(n)$ be the number of complete binary trees with $n$ leaves,
and $t(n)$ be the number of ordered trees with $n$ vertices, then
$b(n)=t(n)$.

{\bf 8}

Consider the lattice

$$
  L(n,2):=\{( a_1 , ...  , a_n ); 2 \geq a_1 \geq ... \geq a_n \geq 0 \}. 
$$

The following chains ($ 0 \leq  2i \leq  n $) constitute a symmetric
chain decomposition of $L(n,2)$.

$$
  C_i^{(n)} := 2^i \rightarrow 2^i 1 \rightarrow ... \rightarrow 2^i  1^{n-2i}  \rightarrow 2^{i+1} 1^{n-2i-1} ... \rightarrow 2^{n-i}. 
$$

{\bf 9}

a)For all square matrices $A$,$B$, of size $n \leq  9$, 
$det(AB)=det(A)det(B)$.

b)For all square matrices $A$,$B$, of size $n = 9000$, 
$det(AB)=det(A)det(B)$.

c) For all square matrices of arbitrary size $n$,
$det(AB)=det(A)det(B)$.

{\bf 10} (Amitsur-Levitski)

For an arbitrary $n$, let
$A_1 , ... , A_{2n} $  be $2n$ square $n \times n$ matrices,
then:

$$
  \sum_{ \pi  \epsilon  S_{2n} } sgn ( \pi ) A_{ \pi (1) } ... A_{ \pi (2n) } \equiv 0. 
$$

{\bf 11} (Dyson)

 a) The constant term of

$$
  [(1-x/y)(1-y/x)(1-x/z)(1-z/x)(1-y/z)(1-z/y)]^a 
$$

is  $(3a)!/ a!^3 $.

b)  The constant term of

$$
  \prod_{ 1 \leq  i \neq  j \leq  1000 } ( 1- {{x_i}\over {x_j}} )^a  
$$

is $(1000a)!/ a!^{1000} $.

c) ( Dyson's conjecture, see [An])
the constant term of

$$
  \prod_{ 1 \leq  i \neq  j \leq  n } ( 1- {{x_i}\over {x_j}} )^a  
$$

is $(na)!/ a!^n $.

{\bf 12} ( Omar Qayam )

a)

$$
  (x+y)^3 = x^3 + 3 x^2 y + 3 x y^2 + y^{3}. 
$$

b)
$$
  (x+y)^n = \sum_{i=0}^{n} n! x^k y^{n-k} / (k!(n-k)!). 
$$

c)

$$
  ( x_1 + ... x_m )^n = \sum_{ k_1 + ... + k_m = n } n! x_1^{k_1} ... x_m^{k_m} / ( k_1 ! ... k_m ! ). 
$$

{\bf 13}

 a) ([Po])

$$
  \sum_{n=1}^{N} {{1}\over{ n^3 }}  -  { {{5}\over {2}} } \sum_{n=1}^{N}  {{ (-1)^{n-1} } \over { n^3   (   { 2n  \atop n }  ) }} = \sum_{k=1}^{N} {{ (-1)^k } \over { 2 k^3  (   { N+k  \atop k }  )  (   { N  \atop k }  ) }}. 
$$

b) 

$$
  \sum_{n=1}^{\infty} {{1}\over{ n^3 }}  =  { {{5}\over {2}} } \sum_{n=1}^{\infty} {{ (-1)^{n-1} } \over { n^3   (   { 2n  \atop n }  ) }}. 
$$

{\bf 14} (Rogers-Ramanujan)

 a)

$$
  \sum_{n=0}^{\infty} {{ q^{ n^2 } }\over{ (1-q) ... ( 1- q^n ) }} = \prod_{i=0}^{\infty} 1 over { ( 1- q^{5i+1} )  (1- q^{5i+4} ) }. 
$$

b) ([Br])

$$
  \sum_{r=0}^{n} {{ q^r sup 2 }\over{ (q)_r (q)_{n-r} }} = 1 + \sum_{r=1}^{n} {{ (-1)^r q^{(5 r^2 - r )/2} (1+ q^r )} \over { (q)_{n-r} (q)_{n+r} }}. 
$$

{\bf  3 Answers to the Quiz }

Real time-Littlewoodian : 1-8,9a, 11a, 12a, 12b, 13a, 14b.

Zillion years-Littlewoodian : 9b, 11b.

Non-Littlewoodian : 9c,10,11c,12c,13b,14a.

{\bf  4 Discussion }

{\bf 1\rm 
 and \bf 
2} :

These are examples of interesting
numerical identities. 1) gives a refutation of Fermat's conjecture that
$ 2^{ 2^n } + 1$ is always a prime, and 2) refutes Euler's
conjecture that it is impossible to express a fifth power as a sum
of four perfect fifth powers.  A formal proof of these identities
will involve writing all the integers as a combination of powers of 
$10$, invoking the associative and distributive laws, and then using the
$100$ "lemmas" that comprise the multiplication table and the 
$100$ lemmas that constitute the addition table.  Since arithmetic with
specific integers is so much part of our culture, it would not occur to
anyone to give a proof of a statement like 2), and the proof is
considered routine by a general consensus.

{\bf 3}:

Way back in 1202, when {\it Liber Abaci} first appeared, identity 3a)
was a deep theorem, and there were very few people who understood its
proof.  Nowadays this is a routine exercise that has the same
epistemological stature as $9 \times 11 = 99$.  But although routine,
it is an important identity in number theory, as it implies the
non-trivial fact that the product of two integers that are
expressible as a sum of two squares is itself a sum of two squares.
Similar remarks apply to identity 3b), which implies
that if one can show that every prime is expressible as a  sum
of four squares, then every integer can be so expressed.

Although the proofs of 3a) and 3b) are completely routine, they are
nevertheless elegant identities, and one may want to 
understand {\it why} they
are true, as opposed to {\it whether} they are true.  One such
explanation is that for two complex numbers $z_1 = a + ib$,
and $z_2 = c + id$, 
$| z_1 z_2 | = | z_1 | | z_2 | $.  The analogous
identity in the quarterions gives 3b).  These observations  were perhaps
the motivation to Gauss's discovery of the Gaussian integers and to
Hamilton's discovery of the quarterions.

{\bf 4}:

Any identity of the form

$$
  \sum_{n=1}^{N} p(n) = q(N) , 
$$

with $p$ and $q$ polynomials, is routinely verifiable, since it is
equivalent to

$$
  q(N)-q(N-1)=p(N) . 
$$

Unfortunately, many inane identities of this kind are assigned to
students , expecting them to use mathematical induction.  
Many students who do not understand mathematical induction "prove"
this identity by plugging in a few values.   They are not that
wrong.  For example,
in order to prove identity 4) all we need is check it in 5 different
values, say $N=0,1,2,3,4$, since both sides are polynomials of degree
$4$.

{\bf 5}:

Trigonometric identities are feared and hated by many a high school
student.  Using $cos x = ( e^{ix} + e^{-ix} )/2$ and
$sin x = ( e^{ix} - e^{-ix} )/2i$, they are all routine, and
indeed trigonometric simplifications are built-in into most computer
algebra systems.

{\bf 6}:
Writing the Fibonacci numbers in terms of Binet's formula, identities
like 6) are trivially verifiable, as are all identities involving
sums and products of Fibonacci numbers, or more generally,
solutions of constant coefficients homogeneous linear recurrences.

{\bf 7}:

Thanks to the work of the Lothargian school (Schutzenberger, Foata,
Viennot and their students), 
many combinatorial sequences are known to have
generating functions that are algebraic formal power series. It is
by now a routine matter to find the equation satisfied by such a
generating function and to see whether two 
such generating functions are equivalent.  In fact it should be
possible to find {\it a priori} bounds for the order of the equation
satisfied by the generating function, as well as bounds for the
degrees of the coefficients.  Then it should be possible to find an 
{\it a priori} integer $N$ such that if the two combinatorial sequences
of the problem are equal up to N, then they are identical.  
It follows that many theorems on trees, and on two and three dimensional
lattice walks are either routine or routineable.  Of course it is
nice to have  {\it nice} proofs , preferably bijective, 
to prove such results,
but if the proof is going to be ugly it may just as well be done by
computer.

{\bf 8}:

This statement is not only "routine" in the colloquial sense of
the word, but is also technically routine, since it can be easily encoded
in terms of an identity involving rational functions of 
a {\it fixed} number of variables.

Consider 

$$
  M(n):=\{ ( a_1 , ... , a_m ) ; for somem \geq 0, n \geq a_1 \geq a_2 \geq ... \geq a_m \geq 0 \}. 
$$

(Note that we allow some trailing zeroes).

Introduce the commuting indeterminates $ x_1 , ... , x_n $ and
$t$,
and define the weight

$$
  weight( a_1 , ... , a_m ) = x_{a_1} ... x_{a_m} t^{m,} 
$$

with the convention that $x_0 =1$.  The subset of $M(n)$
of partitions of length $m$ ( allowing zeroes) can be easily 
identified with Young's lattice $L(m,n)$.  It is easily seen that
the total weight of $M(n)$ ( i.e. the sum of the weights is ):

$$
  (1-t)^{-1} ( 1- x_1 t )^{-1} ... ( 1 - x_n t )^{-1}. 
$$

The total weight of the chains $C_i $ , for a fixed $n$ is

$$
  ( x_2^i+ x_2^ix_1 + ... + x_2^{n-2i} + x_2^{i+1} x_1^{n-2i-1} + ... x_2^{n-i} ) t^{n,} 
$$

which is some (simple) rational function, and summing with respect to
$n$ yields geometric series that are easily summed. Thus the
statement that the chains $C_i^{(n)} $ cover $L(2,n)$, for
every n, is equivalent to a simple identity among rational
functions.  Of course it is obvious by inspection that 
$C_i^{(n)}$ are symmetric chains.

By the same token, the verifications of West's[We] constructions of
a symmetric chain
decomposition of L(4,n) and Riess's [Ri] for L(3,n) and L(4,n) are
purely routine and could have been omitted.  This does not take away
from their achievements, because the hard part was to find the
constructions.  But once found, the verification is a purely routine
matter and two pages of the European Journal of Combinatorics could
have been spared.

On the other hand, a symmetric chain decomposition of L(m,n) for
general $m$ {\it and} $n$ is not going to be routinely verifiable,
at least not at present.  Neither is O'Hara's[O'h] recent
magnificent symmetric chain decomposition of the "complete" version
of L(m,n) where any two partitions with different ranks are related.

{\bf 9}:

There are many identities in matrix algebra, of which 9) is one of
the simplest.  Writing the matrices in generic form, the statement
$det(AB)=det(A)det(B)$, for a fixed $n$ is nothing but a finite
algebraic identity, but of course for $n=9000$ we would have 
more than $9000!^2$ terms !  Of course identity 9c), for general 
$n$, is non-Littlewoodian ( at least today).

{\bf 10}:

This is the celebrated Amitsur-Levitski identity, that
was given an elegant proof by Rosset [Ro].  For general $n$ it is
non-Littlewoodian.

{\bf 11}:

Identity 11c) is Dyson's ex-conjecture, that was proved by Wilson
and Gunson and that was given a short and elegant proof by 
Good( see [An] ).
For general $n$ this is certainly non-Littlewoodian, but for $n=3$
(identity 11a) and for $n=1000$ ( 11b) it is real time-Littlewoodian
and zillion years-Littlewoodian respectively.  This follows from my
approach ([Ze]) to special functions identities that is 
based on I.N.Bernstein's theory of holonomic systems.

{\bf 12}:

No one would argue with the assertion that 12a) is real
time-Littlewoodian.  That the binomial theorem is real
time-Littlewoodian follows from my above mentioned approach ([Ze]),
but the multinomial theorem, that involves an $indefinite$ number of 
variables is not covered by this theory, and is thus
non-Littlewoodian at present.

{\bf 13}:

Identity 13b) was the starting point of Apery's incredible proof of
the irrationality of $\zeta (3)$ ([Po]).  At present such identities
that state the equality of two infinite hypergeometric series ( and
whose
summands only depend on the index of summation, and not on an auxiliary
parameter) are non-Littlewoodian.  On the other hand, 13b), which is
the "finite form" of 13a), is nothing but a binomial coefficients identity
and is certainly real time-Littlewoodian.  Note that from a human
point of view, 13b) is a trivial consequence of 13a), upon taking
$n \rightarrow \infty$, and thus 13a) is "deeper" and more general.

{\bf 14}:

This final example illustrates the q-analog of the point made in
the previous example.  14a) is one of the Rogers-Ramanujan identities,
and involves the equality of an infinite q-hypergeometric series and
an infinite q-hypergeometric product.  Since we do not have any
extra parameters to provide elbow room, it is, at present,
non-Littlewoodian.  On the other hand, the finite form of this identity,
14b) ( that implies 14a upon taking $n \rightarrow \infty$ and using the Jacobi
triple product identity ) is nothing but a q-binomial coefficients
identity , and
as such is real time-Littlewoodian ([Ze]). 
Given an infinite q-hypergeometric identity we still need, at present,
a Schur, an Andrews, or a Bressoud to come up with a
{\it conjectured} finite form, that my program can then do.

{\bf 5 An Open Problem}

It would be nice if identities like 13b) and 14a) would be provable
by computer.  Consider an identity

$$
 \sum_{i=1}^{\infty} a(n) = \sum_{i=1}^{\infty} b(n), 
\eqno(*)
$$          

where $a(n)$ and $b(n)$ are  hypergeometric sequences, i.e.,
there exist polynomials $P(n), Q(n), R(n), S(n)$ such that

$$
  a(n+1)/a(n)=P(n)/Q(n), b(n+1)/b(n)=R(n)/Q(n). 
$$

It would be interesting to develop a decision procedure, in the style
of [Go], that will decide whether an identity like (*) is true or
false. 

Consider

$$
         c(N):= \sum_{i=1}^{N} ( a(n)-b(n)). 
$$

The statement (*) is equivalent to the fact that $c(N) \rightarrow 0$ as
$N \rightarrow \infty$.  It is not hard to find a second order linear recurrence
equation (with polynomial coefficients)
satisfied by $a(n)-b(n)$, and hence a third order linear recurrence
equation satisfied by $c(N)$.  The problem of deciding whether
identities of the form (*) are true would follow from the following
more general problem:

{\bf Problem}

\it
Given a homogeneous linear recurrence with polynomial coefficients:

$$
  p_0 (n) a (n) + p_1 (n) a (n-1) + ... + p_i (n) a (n-i) + ... p_K (n) a (n-K) =0, 
$$

and $K$ initial conditions : $a(0), ... , a(K-1)$, decide whether or not
the solution $a(n)$ of the equation , subject to the initial conditions, 
has the property that $a(n) \rightarrow 0$ as $n \rightarrow \infty$.
\rm

The Birkhoff-Trijinski method ( resurrected in [Wi-Ze]) enables us to
find the complete asymptotics of the {\it dominant} solution of a
linear recurrence equation.  What we need is some way to handle the
asymptotics of an arbitrary solution, under prescribed initial
conditions.
On the other hand we are not asking for {\it complete} asymptotics 
but only whether or not it tends to zero.

q-Analogously, one may pose the q-problem, that will enable our machines
to prove Rogers-Ramanujan style identities, and will free us humans
to prove $multi-variate$ extensions, like the multi-variate
extensions of Andrews, Gordon, and Bressoud ( see [An], (3.45),
(3.46)).

{\bf q-Problem}

\it
Given a homogeneous linear recurrence with polynomial coefficients
(in $q$ and $q^n$):

$$
  p_0 ( q^n , q) a (n) + p_1 ( q^n , q) a (n-1) + ... + p_i ( q^n , q) a (n-i) + ... + p_K ( q^n , q)  a (n-K) =0, 
$$

and $K$ initial conditions : $a(0), ... , a(K-1)$,( certain formal
power series 
in $q$)  decide whether or not the solution $a(n)$ of the equation , 
subject to the initial conditions, has the property that $a(n) \rightarrow 0$ 
as $n \rightarrow \infty$ , in the sense of formal power series in $q$.
\rm

I am offering $50$ dollars for a solution of the problem, and an
additional $(1- q^{50} )/(1-q)$ dollars for a solution of the
{\it q-problem} ( where, for this one, I adopt the analyst's custom of
taking $| q | < 1$ ).

{\bf  6 Concluding Remarks }

The computer is here to stay, for better or for worse, and as
Marshal McLuhan has taught us, it will shape our practices, problems,
ways of thinking
and even our tastes.  This was put beautifully by Ruelle[Ro]:

\it
   My guess is that, within fifty or hundred years ( or it might be one 
   hundred and fifty ) computers will successfully compete with the human 
   brain in doing mathematics, and that their mathematical style will be 
   rather different from ours.  Fairly long computational verifications 
   ( numerical or combinatorical ) will not bother them at all, and this 
   should lead not just to different sorts of proofs, but more importantly
   to different sorts of  theorems being proved.
\rm

Most mathematicians nowadays still consider computer-assisted proofs
as "cheating", and a priori "ugly" and "not giving any
insight", as was manifested by the cold and hostile reception of
Appel and Haken's marvelous tour-de-force.  But I am not worried
for Appel and Haken.  In a hundred years, their proof will be
considered as elegant as any human proof, and its only drawback will
be the fact that it involves {\it too much} human effort.  

In the future, given a "conjecture", one would try to embed it into
a class of statements having a given format, and then develop
an effective method for deciding the truth of the statements in this
class, or whether it is "undecidable" or "intractable".

Given an algorithm to decide the truth of such a class of statements,
nobody would care if the 
computer-generated proof, in every given instance, is long and
ugly, and this is just as well.  What is perhaps sad is that nobody
would probably care whether the algorithm itself is elegant or not,
and everything will be judged by its computational complexity.  Be
that as it may, let us enjoy the present exciting transition 
era, where we can
both enjoy the rich human  heritage of the past , as well as witness
the first crude harbingers of the
marvelous computer-mathematics
revolution of the late $21^{th}$ century.  

{\bf References}

[An] Andrews (George),
"q-Series: Their Development and Applications in Analysis,
Number Theory, Combinatorics, Physics, and Computer Algebra",
CBMS series {\bf 66}, Amer. Math. Soc., Providence, 1986.

[Br] Bressoud (David M.) , {\it The Bailey Lattice: An introduction},
in "Ramanujan Revisited", edited by G.E.Andrews et. al., Academic Press,
San Diego, 1988.

[Dy] Dyson (Freeman J.) {\it Some guesses in the theory of partitions},
Eureka (Cambridge ), {\bf 8} (1944), 10-15.

[Go] Gosper (R.William, Jr. ), \it 
Decision procedures for indefinite
hypergeometric summation\rm 
, Proc. Natl. Acad. Sci. USA, {\bf 75} (1978),
40-42.

[L-P] Lander (L.J.), and Parkin (T.R.), \it 
Counterexample to Euler's
conjecture on sums of like powers\rm 
, Bull.Amer.Math.Soc. {\bf 72}
(1966) 1079.

[O'H] O'Hara ( Kathleen), \it 
Unimodality of the Gaussian coefficients:
a constructive proof\rm 
, J.C.T.(A), to appear.

[Po] van der Poorten (Alfred), \it 
A proof that Euler missed ...
Apery's proof of the irrationality of $\zeta (3)$\rm 
, Math. Intell.,
{\bf 1}(1979), 195-203.

[Ri] Riess (W.), {\it Two optimization problems of ordering} ( in German),
Pamphlets of the the institute for Mathematics {\bf 11}(1978), \#5,
Erlangen.

[Ro] Rosset (Shmuel), \it 
A new proof of the Amitsur-Levitski
identity\rm 
, Israel J. Math {\bf 23}(1976), 187-188. 

[Ru] Ruelle (David), \it 
Is our mathematics natural? the case of
equilibrium statistical mechanics\rm 
, Bull. (New Series) Amer. Math.
Soc. {\bf 19} (1988), 259-268.

[We] West, (Douglass B.) \it 
A symmetric chain decomposition of
L(4,n)\rm 
, Europ. J. Comb. {\bf 1} (1980), 379-383.

[W-Z] Wimp (Jet), and Zeilberger (Doron), \it 
Resurrecting the
asymptotics of linear recurrences\rm 
, J. Math. Ana. Appl. {\bf 111}
(1985),162-177.

[Ze] Zeilberger (Doron), \it 
A holonomic systems approach to special
functions identities\rm 
, preprint.
\end
