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\leftheadtext{George E. Andrews}


\def\marco{\text{Marco Sch\"utzenberger}}
\def\c{\cite}
\def\fr{\frac}
\def\call{\Cal L}


\topmatter
\title
	Some Debts I Owe
\endtitle
\author
	by \\
	\ \  \\
	George E. Andrews$^{(1)}$
\endauthor
\abstract
The primary objects of this paper are:

(1) \ to acknowledge my debts to a number of important mathematicians
	who have passed away, and

(2) \ to describe some of the beginnings of several themes in my
	research.
\endabstract
\endtopmatter


\document
\footnote"{}"{$^{(1)}$Partially supported by National Science Foundation
Grant DMS-9206993 and by the University of Strasbourg.}
%\baselineskip 20pt
\subhead
1. \ Introduction
\endsubhead
It is impossible for me to express adequately my gratitude to all who
were involved with the 42$^{\text{nd}}$ Seminaire Lotharingien held in
Maratea, Italy during September 1--5, 1998.  I am especially indebted
to Dominique Foata who really made the trip possible for me and
managed so many aspects of the conference so well.  Also I extend
special thanks to Peter Paule who originally suggested the role I
would play.  The surprise of being asked to participate in a
conference to commemorate your 60$^{\text{th}}$ birthday is quite
overwhelming.  It is impossible to feel worthy of such an event.
However, I have always told others that such events were important for
the cohesiveness of the mathematical community, and so it would hardly
be consistent for me to back out when this event was proposed.  After
some thought, it occurred to me that, I might use my participation to
honor some of those who are no longer with us and who influenced me and
many others.  I will follow that same general outline in this paper.

After a belated tribute to \marco, I will describe the
beginnings of the paths that eventually led to: \ (1) \ Ramanujan's
Lost Notebook, (2) \ Bailey Chains, (3) \ Determinant Evaluations, and (4)
\ Partition Analysis.  I will conclude with a look at one of my current
paths which I have called the Liouville Mystery.

\subhead
2. \ \marco
\endsubhead

Sch\"utzenberger was a grand master of mathematics and a powerful
and vivid presence.  I had the good fortune to meet him at a number
of European conferences, mostly at Oberwolfach.  He had great style
and great insight.

Let me elaborate with slight variations on two of the stories that
began Dick Askey's tribute to Marco.

Dick started off by describing his first meeting with Marco.  I
was there with a different viewpoint.  I knew both of them fairly
well prior to this meeting.  Each was a strong personality with 
passionately held views resulting from years of devoted research
and study.  I knew Dick also to be a committed non-smoker, and I 
knew of Marco's chain smoking.  I also knew that I had arranged 
for Dick's invitation to the meeting:  I believed that one of the
world's premier workers in special functions would have valuable
interactions with the world of enumerative combinatorics.  So I
watched this meeting with some anxiety.  Within minutes of the
beginning of the conversation Marco introduced a discussion of $X$
(a famous European mathematician) whom he described as ``the 
cross I bear!''  Unknown to Marco, Dick had had a very serious
confrontation with $X$ over fundamental principles dear to his
heart.  From that moment on, my anxiety vanished.  I am convinced that 
(if asked) Dick would have happily lit Marco's cigarettes for the
rest of the afternoon.  They agreed on much more than $X$, but 
$X$ was the perfect starting point.

In Dick's second story concerning a flawed combinatorial proof
of a famous result, I remember the conversation at the end a bit
differently.  Dick, Marco and I were seated near each other.  At
the conclusion of the talk Marco asked what we thought.  I replied
that I would reserve judgement.  He responded:  ``You reserve
judgement on THAT?  You make me doubt that the result is even
true.  When I return to Paris I will check it on the computer.''

Most meaningful were the many ways Marco attempted both to educate
and encourage many people (including me) over many years.  I recall
his gracious letter enthusiastically welcoming my edited
publication of MacMahon's collected papers.  I remember how he
began a conversation with me on my long memoir devoted to a 
topic I had named ``partition ideals.''  He said ``You do not
write well.''  He went on to explain exactly what he meant, and
by the end of the evening had convinced me of the flaws in my
exposition.

I vividly remember a letter from him which he sent to me at a
low point in my career.  I know I carefully saved the letter.
In fact, I so carefully saved it that I have hunted for it for
two years without success.  So I will paraphrase it from memory.
The meaning is preserved but his eloquence is gone.  It went
something like this:

{\it Do not be too concerned about your current disappointment.
The best that any mathematician can really hope for is to prove
some first class theorems and have them understood and appreciated 
by a few good mathematicians.  Everything else is politics.}

His advice in this letter is something I have thought about at
numerous times and in numerous settings.

We will all miss his wisdom.

\subhead
3. \ The Path to Ramanujan's Lost Notebook
\endsubhead

One of the most remarkable events in my career was the discovery of
Ramanujan's Lost Notebook in the Wren Library of Trinity College,
Cambridge.  I have provided lengthy accounts of that event elsewhere
\c{18}, \c{21; pp. 5--6}.  Prior to 1976, the only information 
anyone had about Ramanujan's work in 1919--1920 (the last year of his
life) was contained in a letter on a new topic, mock theta functions,
which Ramanujan sent to Hardy early in 1920.  The letter was published
as the last two pages in Ramanujan's Collected Papers \c{50;
pp. 354--355}.  Suffice it to say that what I needed at the time of the
discovery was a deep knowledge of Ramanujan's mock theta functions.
It was the extensive list of formulas for these functions in the Lost
Notebook that made clear to me that I had found the lost discoveries
made by Ramanujan during the last year of his life.

As incredible as it may seem, I learned all about the mock theta
functions in graduate school at the University of Pennsylvania.
My thesis advisor was Hans Rademacher, the famous German number
theorist, who had emigrated to the U. S. in the 1930's.  His work 
on partitions and modular forms had led him to Ramanujan,  and
he had asked his student, Leila Dragonette, to study the third
order mock theta functions for her Ph.D. thesis \c{41}.  
Rademacher told me how pleased he was with her work, but he
felt that it could be greatly improved.  So he set me to work 
to improve it \c6.

Simultaneously, Nathan Fine gave a course entitled: Basic
Hypergeometric Functions.  I signed up for it mistakenly 
thinking that ``basic'' meant ``beginning'' or ``elementary.''
I soon found out that ``basic'' meant ``$q$''; however, Fine
was sufficiently mesmerizing that I never regretted my error
\c{25}.

Rademacher's assigned project for me consisted of:  (1) 
determining the behavior of the third order mock theta functions
under the transformations of the modular group, and (2) applying
the celebrated circle method to obtain asymptotic series for the
power series coefficient of the third order mock theta functions.  
He believed that the combination of these two projects would
yield improvements on Dragonette's results, and this turned out
to be the case \c6.  This project was immensely interesting
to me, but it did not provide me with a real feel for the 
inner workings of the mock theta functions.  This was to be
provided by Fine's course.  His course was based on a manuscript
he had been perfecting for a decade; it eventually became a book
\c{42}.  He covered the first chapters of his book.  In this
material, he discussed the third order mock theta functions at length.
Realizing that Rademacher had assigned me a thesis project on mock
theta functions, Fine asked me to present an account of the fifth
order mock theta functions from Watsons second paper \c{57}.  So I
plunged in.  Of this paper, Watson had once said: ``The basic
hypergeometric series which has been used hitherto is of no avail for
these functions, and other means must be sought to establish
Ramanujan's relations which connect functions of order 5.  After
spending a fortnight on fruitless attempts, I proceeded to attack the
problem by the most elementary methods available, namely applications
of Euler's formulae mingled with rearrangements of repeated series;
and within the day I had proved not only the five relations set out by
Ramanujan but also five other relations whose existence he had merely
stated.  My proofs of these relations are all so long that I took the
trouble to analyse one of the longest in the hope of being able to say
that it involved ``thirty-nine steps''; it was, however, disappointing
to a student of John Buchan to find that a moderately liberal count
revealed only twenty-four."  I suspect I was one of the first to read
this paper carefully.  I was able to extend Watson's methods in \c3,
\c4 and \c5.  The upshot of this exercise was that I emerged from
graduate school with an intimate familiarity with mock theta functions
on an almost individual basis.

A little over a decade later in 1976, I was invited to participate in
the conference, Combinatoire et Repres\'entation du Groupe
Sym\'etrique, a Table Ronde organized by Dominique Foata at the
University of Strasbourg.  The then current manifestation of airline
ticket fare irrationality was that if you stayed in Europe for at least
three weeks, your fare was miniscule.  Because of this financial
incentive, the University of Wisconsin (where I was visiting for the
academic year thanks to Richard Askey) allowed me to undertake a 3
week European itinerary which included several days in Cambridge.  I
went there as the guest (and at the suggestion) of Lucy Slater.  She
had told me that many of Watson's papers (G. N. Watson died in 1965)
had been deposited in the Trinity College Library.  So I went with
minimal expectations to examine Watson' papers.  In one box was a
manuscript of nearly one hundred pages written in Ramanujan's
inimitable hand.  Perusing it I saw many of the formulas from Watson's
second paper as well as other formulas which Watson had suggested
couldn't exist.  The manuscript had few words mostly formulas.
However when I saw series like
$$
	1 + \fr{q}{1 + q} + \fr{q^4}{(1 + q)(1 + q^2)} + \cdots
$$
or
$$
	q + q^3(1 + q) + q^6 (1 + q)(1 + q^2) + \cdots\,,
$$
I recognized immediately my old friends the fifth order mock
theta functions.

Of course, the Lost Notebook \c{51} was a gold mine.  I have 
spent a significant portion of the last two decades studying
this marvellous collection of formulas that Ramanujan stated
without proof.

The Lost Notebook has many amazing formulae; so I will conclude
with two formulas that quite surprised me \c{22}.  Consider
Euler's function
$$
	S(q) = \prod_{n=1}^{\infty} (1 + q^n) = \prod_{n=1}^{\infty}
	\fr1{1 - q^{2n-1}}
$$
and define
$$
	R(q) = 1 + \sum_{n=1}^{\infty} 
	\fr{q^{n(n+1)/2}}{(1 + q)(1 + q^2) \cdots (1 + q^n)}\;,
$$
and
$$
	D(q) = - \fr12 + \sum_{n=1}^{\infty} \fr{q^n}{1 - q^n}\;.
$$
Then Ramanujan \c{51; p. 14} asserts:
$$
	\sum_{n=0}^{\infty} (S(q) - (1 + q)(1 + q^2) \cdots
	(1 + q^n)) = S(q) D(q) + \fr12 R(q)
$$
and
$$
	\sum_{n=0}^{\infty} \left(S(q) - \fr1{(1 - q)(1 - q^3) \cdots
	(1 - q^{2n+1})}\right) = S(q) D(q^2) + \fr12 R(q)\,.
$$

This study subsequently led to the discovery by Freeman Dyson,
Dean Hickerson and me \c{33} that most of the power series 
coefficients of $R(q)$ are zero; however every integer appears
infinitely often as a coefficient.

\subhead
4. \ The $q$-Series Path to Bailey Chains
\endsubhead

In this topic, my experiences in graduate school played a very
important role as well.  Here too both Rademacher and Fine had
important things to say.

Rademacher loved Schur's paper \c{53} which contains Schur's
independent discovery of the Rogers-Ramanujan identities, and
Rademacher used Schur's unique statement \c{53; p. 303 translated}:

\proclaim
{Theorem}  We define the infinite determinant
$$
	D(x_1,x_2,x_3,\dots) = \vmatrix
		1 & x_1  & 0 & 0 & \hdots  \\
		-1 & 1  & x_2 & 0 & \hdots  \\
		0 & -1  & 1 & x_3 & \hdots  \\
		0 &  0  & -1 & 1 & \hdots  \\
		\vdots & \vdots  & \vdots &  \vdots  & 
	\endvmatrix
\tag4.1
$$
and we let
$$
	D_{\mu}(q) = D(q^{\mu},q^{\mu+1}, q^{\mu+2},\dots)\,,
\tag4.2
$$
then for $|q| < 1$
$$
	D_1(q) = \prod_{n=1}^{\infty} \fr1{(1 - q^{5n-4})(1 - q^{5n-1})}
$$
and
$$
	D_2(q) = \prod_{n=1}^{\infty} \fr1{(1 - q^{5n-3})
	(1 - q^{5n-2})}\;.
\tag4.3
$$
\endproclaim

Still following Schur, we let
$$
	D(x_1,x_2,\dots,x_m) = \vmatrix 
		1 & x_1 & 0 \; & \hdotsfor2 & 0  \\
		-1 & 1 & x_2 \; & \hdotsfor2 & 0  \\
		0 & -1 & 1  \; & \hdotsfor2 & 0 \\
		\vdots & \vdots & \vdots \quad & {} & {} & \vdots \\
		0 & 0 & 0 & -1  & 1 & x_m   \\
		0 & 0 & 0 & 0 & -1 & 1  
	\endvmatrix
\tag4.4
$$

The proof of the standard form of the Rogers-Ramanujan identities
then proceeds as follows.  If
$$
	\Delta(z,q) = D(z,zq,zq^2,zq^3,\dots)\,,
\tag4.5
$$
then expansion of $\Delta(z,q)$ along its top row yields
$$
	\Delta(z,q) = \Delta(zq,q) + z \Delta(zq^2,q)\,,
\tag4.6
$$
and substituting a power series expansion for $\Delta(z,q)$
into (4.6) and comparing coefficients of $z^n$, we find directly
that
$$
	\Delta(z,q) = 1 + \sum_{n=1}^{\infty}
	\fr{q^{n^2 - n}z^n}{(1 - q)(1 - q^2) \cdots (1 - q^n)}\;,
\tag4.7
$$
which means
$$
	D_{\mu}(q) = \Delta (q^{\mu},q) = 1 + \sum_{n=1}^{\infty}
	\fr{q^{n^2 + (\mu -1)n}}{(1 - q)(1-q^2) \dots (1 - q^n)}\;.
\tag4.8
$$

Finally it is possible to deduce (4.3) from letting $n \to \infty$
in the following polynomial identities
$$
	D(q,q^2,\dots,q^n) = \sum_{\lambda = -\infty}^{\infty}
	(-1)^{\lambda} q^{\lambda(5\lambda + 1)/2}
	\bmatrix n + 1 \\ \left\lfloor \fr{n+1-5\lambda}{2}\right\rfloor
	\endbmatrix
\tag4.9
$$
and
$$
	D(q^2,q^3,\dots,q^n) = \sum_{\lambda = -\infty}^{\infty}
	(-1)^{\lambda} q^{\lambda(5\lambda - 3)/2}
	\bmatrix n + 1 \\ \left\lfloor \fr{n+1-5\lambda}{2}\right\rfloor
	+ 1
	\endbmatrix
\tag4.10
$$

A careful study of the several aspects of Schur's work suggests
the beginning both of much further development of $q$-difference
equations \c7 (of which (4.6) is an example) and of a study of
polynomial identities (such as (4.9) and (4.10)) which lead to
the theory of Bailey chains \c{20}.

Indeed the observation that $\Delta(z,q)$ yields (4.6) by
expansion along the top row immediately suggests that other
$q$-difference equations such as Atle Selberg's \c{54}
generalization of (4.6) can be translated back into infinite
determinants.  From there the determinants can be truncated
(just as (4.4) truncates (4.1)).  Then if one is lucky one can
expand the truncations along their last column (as Schur did for
(4.4)), and, with some luck, it will be possible to read off what
partitions are being generated.  This is, in fact, the genesis of
\c7, and the work there led to the sequence of papers 
\c8 \c{11} \c{12} \c{14} culminating in \c{17}.
In recent years, these studies have led to collaboration with
J. Olsson and C. Bessenrodt \c{24}, \c{30}, \c{34}.

As noted in \c{20; p. 279}, (4.9) and (4.10) are instances of
Bailey pairs.  However the distance between Schur's paper and
Bailey pairs in general is great.  Indeed it was the study of
$q$-series alluded to in the last paragraph that eventually 
led to my collaboration with Richard Askey \c{27}.  In trying
to link that paper with the work of Bailey \c{38}, \c{39}, I
was led to \c{20}.

The ``aha!'' moment for Bailey chains came in the summer of 1982 in
Toronto.  I had agreed to give a paper in a special session on the
work of Gabor Szeg\"o run by Richard Askey.  I settled on trying to
say something about the Rogers-Szeg\"o polynomials.  At the last
minute, as I tried to fit these recalcitrant objects into the format of
earlier work, I saw the power of Bailey's lemma \c{20; p. 270}
unfolding in front of my eyes.  This was quickly written up in \c{20}.

The basic idea is a sequence of pairs of rational functions 
$(\alpha_n,\beta_n)_{n\geqq 0}$ which is called a Bailey pair
provided that for each $n \geqq 0$
$$
	\beta_n = \sum_{j=0}^n \fr{\alpha_j}
	{\overset{n-j}\to{\underset{h=1}\to\prod}
	(1 - q^h)\overset{n+j}\to{\underset{k=1}\to\prod}(1 - aq^k)}\;.
$$

The Bailey Lemma \c{20}, \c{39} may succinctly be stated as
follows:

{\bf Bailey's Lemma.}  If $(\alpha_n,\beta_n)_{n\geqq 0}$ is a
Bailey pair, then so is $(\alpha_n',\beta_n')$ where
$$
	\alpha_n' = \fr{(\rho_1;q)_n (\rho_2;q)_n \left(
	\fr{aq}{\rho_1 \rho_2}\right)^n \alpha_n}
	{\left( \fr{aq}{\rho_1};q\right)_n \left(\fr{aq}{\rho_2};
	q\right)_n}\;,
$$
and
$$
	\beta_n' = \overset{n}\to{\underset{j=0}\to\sum}
	\fr{(\rho_1;q)_j (\rho_2;q)_j \left(
	\fr{aq}{\rho_1 \rho_2};q\right)_{n-j} 
	\left(\fr{aq}{\rho_1\rho_2}\right)^j \beta_j}
	{(q;q)_{n-j}\left( \fr{aq}{\rho_1};q\right)_n \left(\fr{aq}{\rho_2};
	q\right)_n}\;.
$$

Bailey never wrote this result down in this form, and consequently
he missed the power of this result to produce infinite chains 
(Bailey Chains) of Bailey pairs.  The applications of \c{20}
continue to this day.  One of the subsequent highlights was 
the use of Bailey chains by F. Dyson, D. Hickerson and me \c{33}
to prove that
$$
\aligned
	1 + \sum_{n=1}^{\infty} & \fr{q^{n(n+1)/2}}{(1 + q)(1 + q^2)
	\cdots(1 + q^n)}   \\
	& = \sum_{\Sb n \geqq 0 \\ |j| \leqq n\endSb}
		(-1)^{n+j} q^{n(3n+1)/2-j^2} (1 - q^{2n+1})\,.
\endaligned
$$

Suppose $\Delta_e(n)$ (resp. $\Delta_0(n)$) is the number of
partitions of $n$ into distinct parts with even (resp. odd)
rank.  The rank is the largest part minus the number of parts.

Identity (4.1) implies that
$$
	\Delta_e(n) - \Delta_0(n)
$$
is almost always $0$ but that it also takes any integral value
infinitely many times.

This is the result alluded to at the end of Section 3.

\subhead
5. \ Determinant Evaluations
\endsubhead

The problems that led to my study of determinant evaluation have
already appeared in the previous section.  Namely, when learning
about Schur's work from Rademacher, I noticed that setting $q = 1$
in (4.9) yielded a new representation of the Fibonacci numbers,
namely
$$
	F_{n+1} = \sum_{\lambda = -\infty}^{\infty} (-1)^{\lambda}
	\binom{n}{\left\lfloor\fr{n-5\lambda}{2}\right\rfloor}\;.
\tag5.1
$$
Furthermore this formula implies immediately that if $p$ is a
prime congruent to 1 modulo 5 (so necessarily $p = 10 j + 1$),
then
$$
	F_{p+1} \equiv (-1)^{2j} \equiv 1 \pmod{p}\,.
\tag5.2
$$
This is the only proof of (5.2) I know which does not rely on
the Binet formula \c{55; p. 15} for $F_n$.

I did consider very early \c{13; Received by the editors, March
16, 1966} a full generalization of (4.9) \c{13; p. 302}
$$
	\Delta_{k,n} = \sum_{\lambda=-\infty}^{\infty}
	(-1)^{\lambda} q^{\lambda((2k+1)\lambda+1)/2}
	\bmatrix n \\ \left\lfloor\fr{n-(2k+1)\lambda}{2}\right\rfloor
	\endbmatrix\;.
\tag5.3
$$
However, it took a number of years before I stumbled on the 
successive ranks theorem \c{15}, \c{16}, and a number of years
after that before these polynomials became important in 
statistical mechanics \c{21; Ch. 8}, \c{28}.

However, I did discover a number of interesting properties of
$$
	F_{k,n} = \sum_{\lambda = -\infty}^{\infty} (-1)^{\lambda}
	\pmatrix n \\ \left\lfloor\fr{n-(2k+1)\lambda}{2}\right\rfloor
	\endpmatrix\;.
\tag5.4
$$
These were recorded in a paper \c{10} whose results are 
somewhat overshadowed by the fact that the name ``Einstein''
replaced ``Eisenstein'' throughout, and my middle initial
``E'' was replaced by ``H.''  I can only say in my defense 
that these mistakes were not in my original manuscript, and
I was not sent any sort of page or galley proofs to correct.

In any event, I showed in \c{10} that the roots of the auxiliary
polynomial for the minimal recurrence for $F_{k,n}$ do, in fact,
define the maximal real subfield of $Q(e^{2\pi i/(2k+1)})$.  Among my
unpublished discoveries was the fact that
$$
	\delta_k = \det (F_{k,i+j-1})_{1 \leqq i,j \leqq k} = 
	\det \left(\pmatrix i+j - 1 \\ \left\lfloor \fr{i+j-1}{2}
	\right\rfloor\endpmatrix\right)_{1 \leqq i,j\leqq k}
\tag5.5
$$
satisfied
$$
	\delta_k = (-1)^{\lfloor k/2\rfloor}
\tag5.6
$$

I found this assertion nearly impossible until I happened upon
the following two evaluations in Muir's famous book \c{49; pp.
435--436}
$$
	\det \left(\pmatrix 2i + 2j - 1  \\ i + j-2\endpmatrix 
		\right)_{1 \leqq i,j \leqq n} = 1\,,
\tag5.7
$$
and
$$
	\det \left( \fr1{2j+2i-1}\pmatrix 2i + 2j - 1  \\ i + j-2\endpmatrix 
		\right)_{1 \leqq i,j \leqq n} = 1\,.
\tag5.8
$$

While (5.7) and (5.8) only involve the odd central binomial 
coefficients whereas (5.5) concerns all central binomial 
coefficients, it was clear that Muir's method was precisely
what the doctor ordered to establish (5.6).

Namely, Muir multiplies the determinant in (5.7) on the left by
$$
	\det \left(\fr{(-1)^{i+j-1}(2i-1)}{(2j-1)}\pmatrix i + j - 1  
		\\ 2j - 2\endpmatrix 
		\right)_{1 \leqq i,j \leqq n} 
\tag5.9
$$
to produce a lower triangular determinant with 1's on the main
diagonal. To treat (5.8) he multiplies that determinant on the left by
$$
	\det \left( (-1)^{i+j} \binom{i+j-2}{2j-2}\right)_{1 \leqq
	i,j \leqq n}\;.
\tag5.10
$$
In each instance, Muir states the underlying binomial coefficient
identity necessary to prove that the result of multiplication is
an upper triangular determinant with ones on the main diagonal.
In fact, each identity is an instance of the Pfaff-Saalsch\"utz
$_3 F_2$ summation \c{35; p. 9}, a fact not realized by Muir nor
by me at the time.

Following up on this idea was quite easy.  One defines a lower
triangular determinant $\det (c_{ij})_{1\leqq i,j \leqq k}$ with 
1's on the main diagonal. Then the undetermined $c_{ij}$ can 
easily be found to force
$$
	\det (c_{ij})_{1\leqq i,j \leq k} \cdot \delta_k
	= \epsilon_k
\tag5.11
$$
where $\epsilon_k$ is upper triangular.  The $i - 1$ entries
$c_{i,j}$, $1 \leqq j \leqq i - 1$, must fulfill the $i -1$
linear equation
$$
	\sum_{k=1}^{i-1}  c_{i,k} \binom{k+j-1}{\left\lfloor
	\fr{k+j-1}{2}\right\rfloor} + \binom{i+j-1}{\left\lfloor
	\fr{i+j-1}{2}\right\rfloor} = 0\,.
\tag5.12
$$

From (5.12) we can empirically produce as many of the $c_{ij}$ as
we want.  It is then an easy matter to guess that if
$$
	a_{ij} = \fr{(-1)^{i+j-1}(2j-1)}{(2i-1)} 
	\binom{i+j-1}{2i-2}
\tag5.13
$$
and
$$
	b_{ij} = (-1)^{i+j}\binom{i+j-1}{2i-1}\,,
\tag5.14
$$
then
$$
	c_{i,j} = \cases
		a_{h,m} \qquad & \text{ if } i = 2h-1,\;j=2m-1  \\
		0 \qquad & \text{ if } i = 2h-1,\;j=2m  \\
		b_{h,m} \qquad & \text{ if } i = 2h,\;j=2m  \\
		-2 b_{h,m} \qquad & \text{ if } i = 2h,\;j=2m-1  \\
	\endcases
\tag5.15
$$

Two further binomial coefficient identities are required and
again each is an instance of the Pfaff-Saalsch\"utz summation.

There were several ideas I took away from this.  First, 
something like this method ought to work on any determinant
of binomial coefficients.  This faith led to all of my work 
on plane partitions \c{19}, \c{23}, \c{24}, \c{31}.  Second
the complexity of the rules defining $c_{ij}$ in (5.15) 
suggests that there will be a mixture of summation theorems
required to finish off a given result. 

In recent years, C. Krattenthaler has built a number of 
powerful methods for determinant evaluation (see for example \c{43}). 
These methods go well beyond the technique described here.

\subhead
6. \ Partition Analysis and P. A. MacMahon
\endsubhead

Elsewhere in this volume, Peter Paule, Axel Riese, and I present an
account of one aspect of our work to implement the method of partition
analysis in Mathematica.

So I shall abbreviate this section down to a short acknowledgement
of P. A. MacMahon's influence on much of my work.  Thanks to
a 1971 invitation by Gian-Carlo Rota to edit MacMahon's 
Collected Papers \c{47}, \c{48}; I became aware of many rich 
and little explored areas of combinatorics and partitions.

MacMahon's account of partition analysis makes clear that it
is a powerful method \c{46; Section VIII}.  However, his failure
to refine it adequately to evaluate the generating functions
for plane partitions caused the method to fall into disuse.
In the following 75 years only Richard Stanley \c{54} made
significant use of it (in his proof of the Anand-Dumir-Gupta
conjecture \c1).

Once it was realized that the method is in fact algorithmic, the task
of implementing it in computer algebra became an important project.

\subhead
7. \ The Liouville Mystery
\endsubhead

The last topic in this collection of vignettes is a subject
that has been on my mind for the last four years.  In the
summer of 1994, David Crippa, Klaus Simon and I collaborated
on the application of $q$-series to certain problems in
random graphs \c{32}.  Among the results we required was the
following, old, often rediscovered chestnut
$$
	\sum_{n=1}^\infty \fr{(-1)^n q^{n(n+1)/2}}
		{(1-q)(1 - q^2) \cdots (1 - q^{n-1})(1 - q^n)^2}
	= \sum_{m,n \geqq 1} q^{mn}\,.
\tag7.1
$$

Somewhat related were some papers by W. N. Bailey \c{36}, \c{37}
which had been inspired by Bell's proof \c{40} of ``Liouville's
Last Theorem.''  Indeed, Bell's paper begins:

``In the usual notation,
$$
	N = N[n = wx + xy + yz + zu,; w,x,z,u > 0,\;y \geqq 0]
$$
denotes the number of sets $(w,x,y,z,u)$ of integers, subject
to the conditions indicated, satisfying the stated equations 
in which $n$ is a arbitrary constant integer $> 0$.  Then
$$
	N = D_2(n) - n\;D_0(n)\,.
\tag7.2
$$
This curious result is the only one of the numerous theorems 
on quadratic forms stated by Liouville for which (apparently)
no proof has been published.''

Bell devoted a significant portion of his career (cf. \c{51}) to an
explication of Liouville's work in number theory.  To gain some idea
of the mystery and controversy surrounding Liouville's original work
on this topic, let us refer to L\"utzen's biography \c{45; pp. 228 and
229}.  ``Liouville had begun publishing on quadratic forms in 1856.  In
1860, he inserted more than a dozen notes on this question in his {\it
Journal}, and in 1861, he ran amuck, publishing more than 30 notes of
one or two pages each, all with the same structure: a theorem stating
that numbers of a particular form, $a + b\mu$ ($a,b$ are specified
numbers, $\mu$ a variable), can be written in a given number of ways,
by way of a particular quadratic form, for example, $Ax^2 + By^2 +
Cz^2 + Dt^2$ ($A,\;B,\;C,\;D$ are specified numbers) The theorems were
not proved, but merely illustrated with a particular value of $\mu$.
Thus, not only did Liouville keep the proofs of his theorems about
number-theoretical functions to himself, he also hid how they could be
applied to quadratic forms. . . .

``If Liouville hid his methods, like renaissance mathematicians, 
in order to impress his colleagues with his results, he did not
entirely succeed.  This can be seen in a letter from Hermite to
Catalan, probably written shortly after 1865, when Catalan had
moved to Li\`ege:
\block
	For a long time, I have shared your sentiments of regret
	concerning Liouville's last arithmetical publications.
	The secret behind his numerous theorems has not been 
	long in becoming known (P. P\'epin has proved them).
	He would have gained much by showing his principles
	and his methods at once instead of keeping them to
	himself; his meager and monotonous verifications make
	one smile a little.''
\endblock

While Pepin, Humbert and other solved some of Liouville's
riddles, Bell became famous as their master.  For this he
received the B\^ocher Prize in 1924 \c{52; p. 201}.

To my great surprise, Liouville's Last Theorem and related
results involving rather messy convolutions of sums of powers
of divisors, are in fact closely related to (7.1).

To understand the relationship, we define
$$
	\call_k(q) = \sum_{n_1,n_2,\dots,n_{k+1} \geqq 1}
	q^{n_1 n_2 + n_2 n_3 + n_3 n_4 + \dots + n_k n_{k+1}}\,,
\tag7.3
$$
and
$$
	M_k(q) = \sum_{n=1}^{\infty} \fr{(-1)^n q^{n(n+1)/2}}
		{(1 - q)(1 - q^2) \cdots (1 - q^{n-1})(1 - q^n)^{k+1}}\;.
\tag7.4
$$
Then (7.1) asserts
$$
	M_1 (q) = \call_1(q)\,.
\tag7.5
$$

Using the results of \c{32}, it is easy to prove
$$
	M_2(q) = \call_1(q) + \call_2(q) + \call_3(q)\,,
\tag7.6
$$
and with a lot of effort and several results like Liouville's
Last Theorem one can prove \c{26}
$$
	M_3(q) = \call_1(q) + 2 \call_2(q) + 3 \call_3(q) 
	+ 2 \call_4(q) + \call_5(q)\,.
\tag7.7
$$

Given that the coefficients of the $\call_i(q)$ in (7.5)--(7.7)
form the table
$$
\gathered
	1  \\
	1\;1\;1  \\
	\;1\;2\;3\;2\;1  \,,
\endgathered
$$
one expects that we are considering the famous table of
trinomial coefficients.  If this is true, the next line should be
$$
	1\;3\;6\;7\;6\;3\;1
$$

But
$$
\align
	M_4(q) & = \call_1(q) + 3\call_2(q) + 6\call_3(q) 
		+ 7 \call_4(q)  \tag7.8    \\
	& \quad + 6\call_5(q) + 3 \call_6(q) + \call_7(q)   \\
	& + (q^7 + 2q^8 + 6q^9 + 11 q^{10} + 22 q^{11} + 33q^{12}
	\\
	& \qquad + 57q^{13} + 83q^{14} + \cdots )
\endalign
$$

Thus the beautiful pattern of (7.5)--(7.7) almost holds up,
but not quite.  And so, Liouville's Last Theorem becomes linked
with a new mystery:  What is going on in (7.8), and what is the
correct continuation of the pattern beginning with (7.5)--(7.7)?

\subhead
8. \ Conclusion
\endsubhead

It is impossible within the confines of a survey paper to provide
anything like an accounting of my many debts.  To all who helped
organize the Maratea conference and to all who participated you have
my deep thanks.

\Refs

\ref
  \no 1
  \by H. Anand, V. C. Dumir and H. Gupta
  \paper A combinatorial distribution problem
  \jour Duke Math. J.
  \vol 33
  \yr 1966
  \pages 757--770
\endref

\ref
  \no 2
  \manyby G. E. Andrews
  \paper A simple proof of Jacobi's triple product identity
  \jour Proc. Amer. Math. Soc.
  \vol 16
  \yr 1965
  \pages 333--334
\endref

\ref
  \no 3
  \bysame  
  \paper On basic hypergeometric series, mock theta functions,
	and partitions, I
  \jour Quart. J. Math.
  \vol 17
  \yr 1966
  \pages 64--80
\endref

\ref
  \no 4
  \bysame
  \paper On basic hypergeometric series, mock theta functions,
	and partitions, II
  \jour Quart. J. Math.
  \vol 17
  \yr 1966
  \pages 132--143
\endref

\ref
  \no 5
  \bysame
  \paper $q$-identities of Auluck, Carlitz, and Rogers
  \jour Duke Math. J.
  \vol 33
  \yr 1966
  \pages 575--582
\endref

\ref
  \no 6
  \bysame
  \paper On the theorems of Watson and Dragonette for Ramanujan's
	mock theta functions
  \jour Amer. J. Math.
  \vol 88
  \yr 1966
  \pages 454--490
\endref

\ref
  \no 7
  \bysame
  \paper An analytical proof of the Rogers-Ramanujan-Gordon identities
  \jour Amer. J. Math.
  \vol 88
  \yr 1966
  \pages 844--846
\endref

\ref
  \no 8
  \bysame
  \paper On Schur's second partition theorem
  \jour Glasgow Math. J.
  \vol 9
  \yr 1967
  \pages 127--132
\endref

\ref
  \no 9
  \bysame
  \paper On partition functions related to Schur's second
	partition theorem
  \jour Proc. Amer. Math. Soc.
  \vol 18
  \yr 1968
  \pages 441--444
\endref

\ref
  \no 10
  \bysame
  \paper Some new formulae for the Fibonacci sequence with
	generalizations
  \jour Fibonacci Quart.
  \vol 7
  \yr 1969
  \pages 113--130
\endref

\ref
  \no 11
  \bysame
  \paper A general theorem on partitions with difference conditions
  \jour Amer. J. Math.
  \vol 91
  \yr 1969
  \pages 18--24
\endref

\ref
  \no 12
  \bysame
  \paper A new generalization of Schur's second partition theorem
  \jour Acta Arith. 
  \vol 14
  \yr 1968
  \pages 429--434
\endref

\ref
  \no 13
  \bysame
  \paper A polynomial identity which implies the Rogers-Ramanujan
	identities\hfil\break
  \jour Scripta Math.
  \vol 28
  \yr 1970
  \pages 297--305
\endref

\ref
  \no 14
  \bysame
  \paper A generalization of the classical partition theorems
  \jour Trans. Amer. Math. Soc.
  \vol 145
  \yr 1968Q
  \pages 205--221
\endref

\ref
  \no 15
  \bysame
  \paper Sieves for theorems of Euler, Rogers and Ramanujan
  \jour Lecture Notes in Mathematics \#251
  \paperinfo Springer-Verlag, NY, pp. 1--20 (1972)
\endref

\ref
  \no 16
  \bysame
  \paper Sieves in the theory of partitions
  \jour Amer. J. Math.
  \vol 94
  \yr 1972
  \pages 1214--1230
\endref

\ref
  \no 17
  \bysame
  \paper On the general Rogers-Ramanujan theorem
  \jour Memoir. Amer. Math. Soc.
  \vol 152
  \yr 1974
  \pages 86
\endref

\ref
  \no 18
  \bysame
  \paper An introduction to Ramanujan's ``lost'' notebook
  \jour Amer. Math. Monthly
  \vol 86  (2)
  \yr 1979
  \pages 89--108
\endref

\ref
  \no 19
  \bysame
  \paper Plane partitions, I: The MacMahon conjecture
  \jour Adv. in Math. Suppl. Studies
  \vol 1
  \yr 1979
  \pages 131--150
\endref

\ref
  \no 20
  \bysame
  \paper Multiple series Rogers-Ramanujan type identities
  \jour Pac. J. Math.
  \vol 114 (2)
  \yr 1984
  \pages 267--283
\endref

\ref
  \no 21
  \bysame
  \paper $q$-Series:  Their Development and Application in Analysis,
	Number Theory, Combinatorics, Physics and Computer Algebra
  \paperinfo C.B.M.S. Regional Conference Series in Math, No. 66,
	American Math. Soc. Providence (1986)
\endref

\ref
  \no 22
  \bysame
  \paper Ramanujan's ``lost'' notebook, V:  Euler's partition
	identity
  \jour Advances in Math.
  \vol 61 (2)
  \yr 1986
  \pages 156--164
\endref

\ref
  \no 23
  \bysame
  \paper Plane partitions, IV: A conjecture of Mills-Robbins-Rumsey
  \jour Aequationes Mathematicae
  \vol 33
  \yr 1987
  \pages 230--250
\endref

\ref
  \no 24
  \bysame 
  \paper Plane partitions, V: The T.S.S.C.P.P. conjecture
  \jour J. Combinatorial Th. (A)
  \vol 66
  \yr 1994
  \pages 28--39
\endref

\ref
  \no 25
  \bysame
  \paper Nathan Fine
  \jour Notices of the Amer. Math. Soc.
  \vol 42
  \yr 1995
  \pages 678--679
\endref

\ref
  \no 26
  \bysame
  \paper Stacked lattice boxes
  \paperinfo Annals of Combinatorics, (to appear)
\endref

\ref
  \no 27
  \bysame (with R. Askey)
  \paper Enumeration of partitions:  The role of Eulerian series
	 and $q$-orthogonal polynomials.
  \paperinfo {\bf Higher Combinatorics}, M. Aigner, ed.,
	Reidell Publ. Co., Dordrecht, Holland, pp. 3--26 (1977)
\endref

\ref
  \no 28
  \bysame (with R. J. Baxter and P. J. Forrester)
  \paper Eight-vertex SOS model and generalized Rogers-Ramanujan-type
	identities
  \jour J. of Stat. Phys.
  \vol 35
  \yr 1984
  \pages 193--266
\endref

\ref
  \no 29
  \bysame (with C. Bessenrodt and J. B. Olsson)
  \paper Partition identities and labels for some modular characters
  \jour Trans. Amer. Math. Soc.
  \vol 344
  \yr 1994
  \pages 597--615
\endref

\ref
  \no 30
  \bysame (with C. Bessenrodt and J. B. Olsson)
  \paper A refinement of a partition identity and blocks of some
	modular characters
  \jour Arch. Math.
  \vol 66
  \yr 1996
  \pages 101--113
\endref

\ref
  \no 31
  \bysame (with W. H. Burge)
  \paper Determinant identities
  \jour Pac. J. Math.
  \vol 158
  \yr 1993
  \pages 1--14
\endref

\ref
  \no 32
  \bysame (with D. Crippa and K. Simon)
  \paper $q$-series arising from the study of random graphs
  \jour S.I.A.M. J. Discrete Math.
  \vol 10 (1)
  \yr 1997
  \pages 41--56
\endref

\ref
  \no 33
  \bysame (with F. J. Dyson and D. R. Hickerson)
  \paper Partitions and indefinite quadratic forms
  \jour Invent. Math.
  \vol 91
  \yr 1988
  \pages 391--407
\endref

\ref
  \no 34
  \bysame (with J. B. Olsson)
  \paper Partition identities with an application to group
	representation theory
  \jour J. Reine Angew. Math.
  \vol 413
  \yr 1991
  \pages 198--212
\endref

\ref
  \no 35
  \manyby W. N. Bailey
  \paper Generalized Hypergeometric Series
  \paperinfo Cambridge Math. Tract No. 32, Cambridge University
	Press, London and New York, 1935. (Reprinted: Hafner,
	New York, 1964.)
\endref

\ref
  \no 36
  \bysame
  \paper An algebraic identity
  \jour J. London Math. Soc.
  \vol 11
  \yr 1936
  \pages 156--160
\endref

\ref
  \no 37
  \bysame
  \paper Some identities connected with representations of
	numbers
  \jour J. London Math. Soc.
  \vol 11
  \yr 1936
  \pages 286--289
\endref

\ref 
  \no 38
  \bysame
  \paper Some identities in combinatory analysis
  \jour Proc. London Math. Soc. (2)
  \vol 49
  \yr 1947
  \pages 421--435
\endref

\ref 
  \no 39
  \bysame
  \paper Identities of the Rogers-Ramanujan type
  \jour Proc. London Math. Soc. (2)
  \vol 50
  \yr 1949
  \pages 1--10
\endref

\ref 
  \no 40
  \by E. T. Bell
  \paper The form $wx + xy + yz + zu$
  \jour Bull. Amer. Math. Soc.
  \vol 42
  \yr 1936
  \pages 377--380
\endref

\ref 
  \no 41
  \by L. A. Dragonette
  \paper Some asymptotic formulae for the mock theta series of
	Ramanujan
  \jour Trans. Amer. Math. Soc.
  \vol 72
  \yr 1952
  \pages 474--500
\endref

\ref 
  \no 42
  \by N. J. Fine
  \paper Basic Hypergeometric Series and Applications
  \paperinfo Math. Surveys and Monographs, No. 27, Amer. Math.
	Soc., Providence, 1988
\endref

\ref 
  \no 43
  \by C. Krattenthaler
  \paper The major counting  of noninteresting lattice paths and
	generating functions for tableaux
  \jour Memoirs of the Amer. Math. Soc., No. 552
  \vol 115
  \yr 1995
  \pages vi+ 109
\endref

\ref 
  \no 44
  \by J. Liouville
  \paper $x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5$
  \jour Comptes Rendus
  \vol 62
  \yr 1866
  \pages 714
  \finalinfo and Journal de Mathematiques (2), 12 (1867), 47--48
\endref

\ref 
  \no 45
  \by J. Lutzen
  \paper
  \jour
  \vol
  \yr
  \pages
\endref

\ref 
  \no 46
  \manyby P. A. MacMahon
  \paper Combinatory Analysis
  \paperinfo Volume 2, Cambridge University Press, Cambridge, 1916
\endref

\ref 
  \no 47
  \bysame  
  \paper Collected Papers
  \paperinfo Vol.~1, G. E. Andrews, ed., MIT Press, Cambridge,1978
\endref

\ref 
  \no 48
  \bysame  
  \paper Collected Papers
  \paperinfo Vol.~2, G. E. Andrews, ed., MIT Press, Cambridge,1986
\endref

\ref 
  \no 49
  \by T. Muir
  \paper A Treatise on the Theory of Determinant
  \paperinfo Dover, 1960
\endref

\ref 
  \no 50
  \by S. Ramanujan 
  \paper Collected Papers
  \paperinfo Cambridge University Press, London and New York,
	1927.  (Reprinted:  Chelsea, New York, 1962).
\endref

\ref 
  \no 51
  \by S. Ramanujan
  \paper The Lost Notebook and Other Unpublished Papers
  \paperinfo (introduction by G. E. Andrews), Narosa, New
	Delhi, 1988
\endref

\ref 
  \no 52
  \by C. Reid
  \paper The Search for E. T. Bell
  \paperinfo Math. Assoc. of America, Washington, 1993
\endref

\ref 
  \no 53
  \by I. Schur
  \paper Ein Beitrag zur additiven Zahlentheorie und zur Theorie
	der Kettenbr\"uche
  \paperinfo S.-B. Preuss Akad. Wiss Phys.-Math. Kl., 1917,
	pp. 302--321 (Reprinted in I. Schur, Gesammelte
	Abhandlungen, Vol. 2, pp. 117--136, Springer, Berlin,
	1973)
\endref

\ref 
  \no 54
  \by A. Selberg
  \paper \"Uber einige arithmetische Identit\"aten
  \jour Avhl. Norske Vid.
  \vol 8
  \yr 1936
  \pages 23
\endref

\ref 
  \no 55
  \by R. Stanley
  \paper Linear homogeneous Diophantine equations and magic
	labelings of graphs
  \jour Duke Math. J.
  \vol 40 
  \yr 1973
  \pages 607--632
\endref

\ref 
  \no 56
  \by N. N. Vorobyov
  \paper The Fibonacci Numbers
  \paperinfo D. C. Heath, Boston, 1963
\endref

\ref 
  \no 57
  \by G. N. Watson
  \paper The mock theta functions (2)
  \jour Proc. London Math. Soc. (2)
  \vol 42
  \yr 1937
  \pages 274--304
\endref

\endRefs

\vskip .5in
\baselineskip 12pt
\noindent Department of Mathematics  \newline
The Pennsylvania State University  \newline
University Park, PA  16802  USA  \newline
andrews\@math.psu.edu

\enddocument
