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\leftheadtext{Richard Askey}
\rightheadtext{The Work of George Andrews}

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\def\fr{\frac}
\def\c{\cite}
\def\golly{G\"ollnitz}
\def\mode{\text{mod }}
\def\ovwed{\overline{\wedge}}
\def\lam{\lambda}

\topmatter
\vglue2cm
\title
	The Work of George Andrews:  \\
	A Madison Perspective
\endtitle
\author
	by  \\
	\ \ \ \\
	Richard Askey
\endauthor
\endtopmatter

\document
%\baselineskip 20pt
\head
{\bf 1. \ Introduction.}
\endhead

In his own contribution to this volume, George Andrews has
touched on a number of themes in his research by looking at
the early influences on him of Bailey, Fine, MacMahon,
Rademacher and Ramanujan.

In this paper, I propose to present a survey of his work organized on
a different theme.  George has often alluded to the fact that his
1975--76 year in Madison was extremely important in his work.  So it
seems a reasonable project to survey his career from a Madison
perspective.  To make this story complete, I must begin in Sections
2--4 with Andrews' work in the late 1960's that led inexorably to our
eventual lengthy collaboration.  The year in Madison set in motion
three seemingly separate strands of research that were fundamental in
much of his subsequent work.  These are described in Sections 5--8.
Section 9 briefly describes his collaboration with Rodney Baxter.
Section 10 describes the discovery of the crank.  Section 11 contains
a few concluding and summarizing comments.

\head
{\bf 2. \ Partition Identities.}
\endhead

{}From the time Rademacher taught him about the magic and mystery of the
Rogers-Ramanujan identities, George was fascinated with such results.
At first, he pursued them purely for their esthetic appeal.
Rademacher \c{79; Lectures 7 and 8, pp. 68--84} presented the
Rogers-Ramanujan identities as follows:
$$
\aligned
	1 + \fr{q}{1 - q} + \fr{q^4}{(1 - q)(1 - q^2)} & +
	\fr{q^9}{(1 - q)(1 - q^2)(1 - q^3)} + \cdots
\\
	& = \prod_{\Sb n > 0 \\ n \equiv \pm 1 \pmod{5} \endSb}
	^{\infty}  \fr1{(1 - q^n)}\;,
\endaligned
\tag2.1
$$
$$
\aligned
	1 + \fr{q^2}{1 - q} + \fr{q^6}{(1 - q)(1 - q^2)} & +
	\fr{q^{12}}{(1 - q)(1 - q^2)(1 - q^3)} + \cdots
\\
	& = \prod_{\Sb n > 0 \\ n \equiv \pm 2 \pmod{5} \endSb}
	^{\infty}  \fr1{(1 - q^n)}\;,
\endaligned
\tag2.2
$$
and he noted following MacMahon and Schur \c{79; pp. 69--72}
that each may be interpreted partition -- theoretically.

\goodbreak
Identity  (2.1) is equivalent to the assertion that the
partitions of $n$ in which parts differ by at least 2 are
equinumerous with the partitions of $n$ into parts congruent
to $\pm 1$ modulo $5$.

Identity (2.2) is equivalent to the assertion that the partitions
of $n$ into parts each $> 1$ in which the parts differ by at
least $2$ are equinumerous with the partitions of $n$ into
parts congruent to $\pm 2$ modulo $5$.

Rademacher \c{79; pp. 72--73} goes on to say
\block
``The unexpected element in all these cases is the association of
partitions of a definite type with divisibility properties.  The
left-side in the identities is trivial.  The deeper part is the right
side.  It can be shown that there can be no corresponding identities
for moludi higher than $5$.  All these appear as wide generalizations
of the old Euler theorem in which the minimal difference between the
summands is, of course, 1.  Euler's theorem is therefore the nucleus
of all such results.''
\endblock

Now while it may be argued that Rademacher was making a narrow
statement here, it was taken by George to mean that there are no
more results of this type.

So in his first year at Penn State much to his surprise, he discovered
a generalization of the Rogers-Ramanujan identities by
studying a paper of Selberg \c{83} which was quoted in a paper
of Dyson \c{60} which in turn was quoted in Hardy and Wright
\c{72}.  Several months after his discovery, he learned that it had
already been found by Basil Gordon \c{67}.

Selberg based his extension of the Ramanujan-Rogers \c{80;
pp. 214--215, 344--346} proofs of (2.1) and (2.2) on the
following function
$$
	C_{ki} (x;q) = \sum_{n=0}^{\infty}
	\fr{(-1)^n x^{kn} q^{\fr12(2k+1)n(n+1)-in}(xq;q)_n
	(1 - x^i q^{(2n+1)i})}{(q;q)_n}\;,
\tag2.3
$$
where
$$
	(a;q)_n = (1 - a)(1 - aq) \cdots (1 - aq^{n-1})\,.
\tag2.4
$$

George's proof \c{19; Ch. 7} of Gordon's generalization
follows quite directly from the elegant functional equation
found by Selberg \c{83}
$$
	C_{k,i}(x;q) - C_{k,i-1} (x;q) = x^{i-1} q^{i-1}
	(1 - xq) C_{k,k-i+1}(xq;q)\,.
\tag2.5
$$

However, what really got his attention was a review by W. Schwarz
\c{82} of the Ph.D. thesis of Heinz G\"ollnitz \c{65}.  The
first paragraph of the review reads as follows:
\block
	``Continuing work of Schur \c{S.-B. Deutsch. Akad.
	Wiss. Munich {\bf 1926}, 488--495} and Glei\ss berg
	\c{Math. Z. {\bf 28} (1928), 372--382}, the author states
	about 15 theorems on partitions; for instance, the number
	of partitions of $n$ $(n = a_1 + a_2 + \cdots, a_1 \geqq
	a_2 \geqq \cdots)$ into positive integers $a_i \equiv
	2,5$ or $11 \pmod{12}$ is equal to the number of
	partitions of $n$ into different parts $b_i \equiv 2,4$
	or $5 \pmod{6}$, and is equal to the number of partitions
	of $n$ into parts $c_i \neq 1,3$, where $c_i - c_{i+1}
	\geqq 6$ and $c_i - c_{i+1} > 6$, if $c_i \equiv 0,1$
	or $3$ and $c_{i+1} \equiv 0,1$ or $3 \pmod{6}$.''
\endblock

It was easy enough for anyone to check this theorem for say
$n \leqq 20$; however, George had no idea from his previous
work how to prove this.  So he wrote to Schwarz to see if he
could obtain a copy of the G\"ollnitz thesis, and Schwarz sent
him the review copy.  This paper was challenging to say the
least.  The proof of the ``$\mode 12$'' theorem alluded to
above was elementary but absolutely overwhelming; it is a
reasonably safe guess that no one (including George) has
ever read it in detail.  Indeed the study of the ``$\mode 12$''
identity led to one stream of thought that has continued to
this day and involved fruitful collaborations \c2, \c8, \c9, \c{15},
\c{16}, \c{46}.  However, the work in \golly's \ thesis refers to
work by \golly \ undergraduate thesis \c{64} and that had the
most profound impact on George's research.  Namely \golly \
proved the following identities: \c{64; p. 37}
$$
\displaylines{
(2.6)\quad
1 + q \fr{1 + q}{1 - q^2} + q^4 \fr{(1 + q)(1 + q^3)}
	{(1 - q^2)(1 - q^4)} \hfill\cr
\qquad\qquad{}+ q^9 \fr{(1 +q)(1 + q^3)(1 + q^5)}
	{(1 - q^2)(1 - q^4)(1 - q^6)}
+\cdots + q^{n^2} \prod_{k=1}^n
	\fr{1 + q^{2k-1}}{1 - q^{2k}}
	+ \cdots \hfill\cr
\hfill{}
= \prod_{n=0}^{\infty}
	 \fr1{(1 - q^{8n+1})(1 - q^{8n+4})(1 - q^{8n+7})};\quad\cr
(2.7)\quad
1 + q^3 \fr{1 + q}{1 - q^2} + q^8 \fr{(1 + q)(1 + q^3)}
	{(1 - q^2)(1 - q^4)}\hfill\cr
\qquad\qquad {} + q^{15} \fr{(1 +q)(1 + q^3)(1
+ q^5)}
	{(1 - q^2)(1 - q^4)(1 - q^6)}+\cdots + q^{n(n+2)}
\prod_{k=1}^n
	\fr{1 + q^{2k-1}}{1 - q^{2k}}
	+ \cdots \hfill\cr
\hfill{}= \prod_{n=0}^{\infty}
	 \fr1{(1 - q^{8n+3})(1 - q^{8n+4})(1 - q^{8n+5})}.\quad\cr}
$$

\goodbreak
\noindent
\golly \ based his entire development on proving $q$-difference
equations for the following function \c{64; p. 37}
$$\displaylines{
(2.8)\quad
	G(s) = 1 = \sum_{n=1}^{\infty} (-1)^n s^{2n}
	q^{n(4n-1)} \;(1 - sq^{4n}) \hfill\cr
\hfill{}\times
\fr{(1 + q)(1 + q^3)() \cdots
	(1 + q^{2n-1})}{(1 + sq)(1 + sq^3) \cdots (1 + sq^{2n-1})}
	\cdot \fr{(1 - sq^2)(1 - sq^4)\cdots
	(1 - sq^{2n-2})}{(1 - q^2)(1 - q^4)(1 - q^6) \cdots
	(1 - q^{2n})}\cr
	\qquad{} = 1 - s^2 q^3\fr{(1 - sq^4)(1 + x)}{(1 + sq)(1 -
q^2)}
	+ s^4 q^{14} \fr{(1 - sq^8)(1 + q)(1 + q^3)((1 - sq^2)}
	{(1 + sq)(1 + sq^3)(1 - q^2)(1 - q^4)}\hfill\cr
\hfill{} - s^6 q^{33} \fr{(1 - sq^{12})(1 + q)(1 + q^3)
	(1 + q^5)(1 - sq^2)(1 - sq^4)}{(1 + sq)(1 + sq^3)(1 + sq^5)
	(1 - q^4)(1 - q^6)} + \cdots
\cr}
$$
George saw that $G(s)$ and $C_{k,i}(x;q)$ had to lie in a
general family of such series.  By extrapolating from these
results, he was able \c7 to provide a generalization of (2.6)
and (2.7) (known as the \golly --Gordon identities \c{64},
\c{68}).  From there he was led to a study of very-well-poised
$q$-hypergeometric series \c8 and their combinatorial implications
\c9, \c{15}.  The culmination of this effort yielded the main
theorem of \c{15}.

\definition
{Definition 1}  If $\lam$ is an even integer, we denote by
$A_{\lam,k,a}(n)$ the number of partitions of $n$ into parts
such that no part $\not\equiv 0 (\mod \lam + 1)$ may be
repeated and no part is $\equiv 0, \pm (a - \fr12 \lam)
(\lam + 1)$ $(\mod(2k - \lam + 1)(\lam + 1))$.  If $\lam$
is an odd integer, we denote by $A_{\lam,k,a}(n)$ the number
of partitions of $n$ into parts such that no part $\not\equiv
0 (\mod \fr12(\lam + 1))$ may be repeated, no part is
$\equiv \lam + 1 (\mod 2 \lam + 2)$, and no part is $\equiv 0$,
$\pm (2a - \lam) \fr12(\lam + 1)$ $(\mod (2k - \lam + 1)(\lam
+ 1))$.
\enddefinition

\definition
{Definition 2}  Let $B_{\lam,k,a}(n)$ denote the number of
partitions of $n$ of the form $b_1 + b_2 + \dots + b_s$, with
$b_i \geqq b_{i+1}$, no parts $\not\equiv 0 (\mod \lam + 1)$
are repeated, $b_i - b_{i+k-1} \geqq \lam +1$ with strict
inequality if $(\lam + 1) \big| b_i$, and finally if $f_j$
denotes the number of appearances of $j$ in the partition,
then $\displaystyle{\sum_{i=j}^{\lam-j+1}} f_i \geqq a-j$
for $1 \leqq j \leqq \fr12 (\lam + 1)$, and $f_1 + f_2 +
\cdots + f_{\lam + 1} \leqq a - 1$.
\enddefinition

\proclaim
{Theorem}  If $\lam$, $k$, and $a$ are positive integers with
$\lam/2 < a \leqq k$, $k \geqq \lam$, then
$$
	A_{\lam,k,a}(n) = B_{\lam,k,a}(n)
$$
\endproclaim

The Rogers-Ramanujan identities are the cases $\lam = 0$, $k = a = 2$
and $\lam = 0$, $k = 2$, $a = 1$.  Gordon's generalization is the case
$\lam = 0$.  George's generalization of \golly -Gordon is $\lam = 1$,
and his generalization of Schur's theorem in \c{14} is $\lam = 2$.

Ironically, H. Alder \c1 proved that $B_{\lam,2,2}(n)$ is never equal
to the number of partitions of $n$ taken from a fixed subset $S$ of
the positive integers unless $\lam \leqq 2$.  The above theorem states
clearly that while the particular generalization of Schur's theorem
considered by Alder does not exist, there are indeed valid Gordon-like
generalization of Alder's non-existence theorems for
$B_{\lam,k,a}(n)$.  Indeed all that is necessary is that $k \geqq
\lam$.

This number-theoretic tour deforce made inevitable our meeting and
collaboration.  I too had been led into considering very-well-poised
series of the ordinary (i.e. $q=1$) type.  They arose naturally
in the solution of the connection coefficient problem for Jacobi
polynomials \c{55; p. 63}, \c{56}.

\head
{\bf3. \ More q-Series.}
\endhead

By now, George was thoroughly engrossed in the study of $q$-series.
His discovery that very well-poised series led to grand generalizations
of the Rogers-Ramanujan identities caused him to look at a variety of
$q$-series.  In light of his work extending Watson's proof \c{85}
of Ramanujan's fifth order mock theta function identities \c4
\c5 \c6, George eventually considered $q$-Appell series.  In a
short paper \c{11}, he showed that if
$$
	\Phi^{(1)} [a;b,b';c; x,y] = \sum_{n=0}^{\infty}\;
	\sum_{m=0}^{\infty} \;\fr{(a)_{m+n} (b)_m(b')_n x^m y^n}
	{(q)_m(q)_n (c)_{m+n}}\;,
\tag3.1
$$
and
$$
	_{r+1}\phi_r \bmatrix a_0,a_1,\dots,a_r;q,z  \\
	b_1,\dots,b_r \endbmatrix = \sum_{n=0}^{\infty}
	\fr{(a_0)_n(a_1)_n \cdots (a_r)_n z^n}{(q)_n (b_1)_n
	\cdots(b_r)_n}\;,
\tag3.2
$$
where
$$
	(a)_n = (1 - a)(1 - aq) \cdots (1 - aq^{n-1})\,,
\tag3.3
$$
then
$$
	\Phi^{(1)} [a;b,b';c;x,y] = \fr{(a)_{\infty}(bx)_{\infty}
	(b'y)_{\infty}}{(c)_{\infty}(x)_{\infty}(y)_{\infty}}\;
	_3\phi_2 \pmatrix c/a,x,y;q,a  \\ bx,b' y\endpmatrix\,.
\tag3.4
$$

I found this result quite disturbing.  At first glance, it is
unreasonable.  The Appell function $F^{(1)}$ \c{54; Ch. IX} certainly
does not reduce to an ordinary $_3 F_2$, and yet the above result
asserts that a generalization of $F^{(1)}$ reduces to an ordinary
$q$-hypergeometric function.  Indeed, as George subsequently pointed
out \c{18}, all the known theorems for $\Phi^{(1)}$ are merely
specializations of classical $_3\phi_2$ transformations.  It turns out
that there is a satisfying and benign explanation of (3.4) as a
$q$-integral which we both came to understand several years later
during his visit to Madison in 1975--76.  Namely, it is the
$q$-analog of the integral representation of $F^{(1)}$.

In addition to this work, George was also considering $q$-series
from the point of view of Rota's theory of functions of binomial
type.  Again he produced a study that disturbed me in quite a
different way.  In his paper on Eulerian differential operators
\c8, George suggests that there is probably no $q$-analog of the
Rodrigues formula.  His discussion was clearly
inadequate to say the least; indeed, we found numerous $q$-Rodrigues
formulas during the year he spent in Madison.

\head
{\bf 4. \ The Evanston Meeting.}
\endhead

Given the variety of ways that our interests converged (although
starting from vastly different viewpoints), I wanted to get George and
$q$-series involved in the world of special functions.  The most
immediate opportunity for this was the AMS Special Session on Special
Functions that I was organizing for the regional meeting of the
A.M.S. at Northwestern on April 27--28, 1973.

Inviting George to this meeting had several beneficial effects.
First it did introduce him to many workers in special functions.
Second, it induced him to prepare a SIAM Review survey article
\c{16} out of the talk he gave; this was the first survey in a
long time (if ever) that attempted to provide a variety of
applications of $q$-series.

Finally, it provided one of those rare moments when one discovers that
someone else shares one of your own pet peeves.  For years I had been
trying to point out that the rather confused world of binomial
coefficient summations is best understood in the language of
hypergeometric series identities.  Time and again I would find
first-rate mathematicians who had never heard of this insight and who
would waste considerable time proving some apparently new binomial
coefficient summation which almost always turned out to be a special
case of one of a handful of classical hypergeometric identities.

To my great delight, George devoted a substantial portion of his talk
to this exact topic.  He even made the point more emphatically by
illustrating that the hypergeometric understanding led naturally to
the almost automatic construction of $q$-analogs.  He finished his
comments on this topic by asserting that 80\% of the formulas in Table
3 of Henry Gould's Tables \c{70} yielded to this approach.  This was
the first portion of his talk with which I disagreed.  I pointed out
during the question period that the correct percentage was at least
90\% if not 95\%.  Independently, I had worked through the same
chapter.

\head
{\bf5. \ The Madison Special Functions Meeting.}
\endhead

After the Evanston meeting, George and I corresponded extensively,
and I invited him to the CBMS conference in Blacksburg during
June, 1974.

The following year I invited George to give one of the addresses
at the MRC Advanced Seminar on Special Functions (March 31--April 2,
1975).

I had only heard George speak twice before; so I had reasonable
confidence that he would give an interesting and comprehensible
talk.  He spoke in Session IV, at 3:15 pm on Tuesday, April 1,
1975.  My confidence in his mathematical taste started to
disintegrate immediately.  His first slide read

\vfill\eject
\head
{\bf Q-ANALOG OF EXTENDED MEIJER'S G-FUNCTION}
\endhead

$$
	G_{p,t,s,r}^{n,\nu_1,\nu_2,m_1,m_2} \left[
	\matrix x \\ {} \\  y  \endmatrix \left| \matrix (\epsilon_p) \\
	(\gamma_t);(\gamma_t')   \\ (\delta_s)   \\ (\beta_r);
	(\beta_r')  \endmatrix \right| q\right] =
$$
$$
	\sum_{h=1}^{m_1}\;\sum_{k=1}^{m_2} \; x^{\beta_h}
	y_{\beta_k'} \; \fr{\displaystyle{\prod_{j=\nu_1 + 1}^t}
	(q/\gamma_j \beta_h)_{\infty}\displaystyle{\prod_{j=\nu_2 + 1}^t}
	(q/\gamma_j'\beta_k')_{\infty}}{\displaystyle{\prod_{j=1}^{\nu_1}}
	(\gamma_j \beta_h)_{\infty} \displaystyle{\prod_{j=1}^{\nu_2}}
	(\gamma_j' \beta_k')_{\infty}}
$$
$$
	\cdot \quad  \fr{\displaystyle{\prod_{j=m_1 + 1}^r}
	(q\,\beta_h/\beta_j)_{\infty}\displaystyle{\prod_{j=m_2 + 1}^r}
	(q\,\beta_k'/\beta_j')_{\infty}}{\displaystyle{\prod_{\Sb j=1  \\
	j \neq k \endSb}^{m_2}}
	(\beta_j'/ \beta_k)_{\infty} \displaystyle{\prod_{j=1}^{n}}
	(q\beta_h' \beta_k'/\varepsilon_j)_{\infty}}
$$
$$
	\cdot \quad \fr{\displaystyle{\prod_{j=n+1}^{p}} (\varepsilon_j/
	\beta_h \beta_k')_{\infty} \displaystyle{\prod_{j=1}^{s}}
	(\delta_j \beta_h \beta_k')_{\infty}}
	{\displaystyle{\prod_{\Sb j=1 \\ j \neq h \endSb}^{m_1}}
	(\beta_j/\beta_h)_{\infty}}
$$
To my horror a second slide was required just to complete this
definition.  Here is the content of the second slide:
$$
	\cdot\quad \Phi  \left[ \matrix p \\ {} \\ t \\ {} \\ s \\ {}
	\\ r-1\endmatrix
	\left| \aligned & (q\,\beta_h \beta_k'/\varepsilon_p)  \\
	& (\gamma_t \beta_h) ; \;(\gamma_t'\beta_k')   \\
	& (\delta_s\beta_h \beta_k')    \\
	& (q\,\beta_h/\beta_r)_{h\neq r} ;\; (q\,\beta_k'/\beta_r')_{k\neq r}
	\endaligned\right| \matrix (-1)^{m_1 + p-n+t-\nu_1} x  \\
	{} \\ {}  \\ (-1)^{m_2 + p-n+t-\nu_2} y  \endmatrix \right]
$$
where
$$
	\Phi  \left[ \;\matrix p \\ {} \\ t \\ {} \\ s \\ {} \\ r \endmatrix
	\;\left| \;\aligned  & \varepsilon_1,\varepsilon_2, \dots,
	\varepsilon_p
	\\ & \gamma_1,\gamma_1',\dots,\gamma_t, \gamma_t'  \\
	& \delta_1,\delta_2,\dots, \delta_s  \\
	& \beta_1,\beta_1',\dots,\beta_r,\beta_r'  \endaligned  \;\;
	\right| \;
	\matrix  x \\ {} \\ {}  \\ y \endmatrix \;\right]_q
$$
$$
	= \sum_{m=0}^{\infty}\;\sum_{n=0}^{\infty} \fr{(\varepsilon_1)_{m+n}
	\cdots(\varepsilon_p)_{m+n} (\gamma_1)_m (\gamma_1')_n \cdots
	(\gamma_t)_m (\gamma_t')_n x^m y^n}{(\delta_1)_{m+n} \cdots
	(\delta_s)_{m+n}(\beta_1)_m(\beta_1')_n \cdots (\beta_r)_m
	(\beta_r')_n (q)_m(q)_n}
$$
where
$$
\aligned
	(a)_n & = (a;q)_n = (1 - a)(1 - aq) \cdots(1 - aq^{n-1})\,,
	\\
	(a)_{\infty} & = \lim_{n\rightarrow\infty} (a)_n\,.
\endaligned
$$
Had he lost his mind?  This was the worst possible beginning
of a talk on $q$-series that I could imagine.  I started to
hiss at him.  Apparently he was waiting for this reaction,
because the third slide followed saying:

\bigskip\medskip
\centerline{\bf APRIL}

\centerline{\bf FOOL}

\bigskip\medskip
What a relief!  His talk (without reference to these first 3
slides) appears in the proceedings of the conference \c{18}.
It contained the seeds of several subsequent important research
topics.

This is where the $q$-analog of the Dyson conjecture first
appeared.  Eventually Doron Zeilberger and David Bressoud \c{86}
proved the conjecture.  More importantly Ian Macdonald recognized
the relationship of this conjecture with his own work \c{76} on
identities, and he subsequently made much more general conjectures
which have led to an explosion of research \c{77}.

As bad as those first 3 slides were, George managed to put up
a few more complicated formulas, such as Watson's $q$-analog
of Whipple's theorem \c{57, p. 69}:
$$
	_8\phi_7  \bmatrix
	a,g\sqrt{a}, - q\sqrt{a}, b, c, d, e, q^{-N}; q,X  \\
	\sqrt{a}, - \sqrt{a}, \fr{aq}{b}, \fr{aq}{c}, \fr{aq}{d},
	\fr{aq}{e}, aq^{N+1}
	\endbmatrix
\tag5.1
$$
$$
	= \fr{(a)_N(aq/ef)_N}{(aq/e)_N(aq/f)_N}  _4\phi_3
	\bmatrix \fr{aq}{cd}, e,f, q^{-N}; q,q  \\
	efq^{-N}/a,  \fr{aq}{c},\fr{aq}{d}  \endbmatrix\;,
$$
where
$$
	X = \fr{a^2 q^{N+2}}{bcde}\;.
$$
George presented the natural full generalization of this in
\c{18}, and this contains the basic mechanism of the Bailey
chains that he first described fully in \c{26}.

\vfill\eject
Namely, for $k \geqq 1$, $N$ a nonnegative integer,
$$
\displaylines{
	_{2k + 4}{\phi}_{2k + 3} \bmatrix
	a, q\sqrt{a}, -q\sqrt{a},b_1,c_1,b_2,c_2, \dots,b_k,c_k,
	q^{-N}; q, \fr{a^k q^{k+N}}{b_1\dots b_k c_1 \dots c_k}
	\\
	\sqrt{a},-\sqrt{a},aq/b_1,aq/c_1,aq/b_2,aq/c_2,\dots,
	aq/b_k,aq/c_k,aq^{N+1}
	\endbmatrix
	\hfill\cr
	= \fr{(aq)_n(aq/b_k c_k)_N}{(aq/b_k)_N (aq/c_k)_N}
\!\!	\sum_{m_1,\dots,m_{k-1}}\kern-15pt
\fr{(aq/b_1c_1)_{m_1}(aq/b_2 c_2)_{m_2}
	\dots (aq/b_{k-1}c_{k-1})_{m_{k-1}}}{(q)_{m_1}(q)_{m_2} \dots
	(q)_{m_{k-1}}}
	\cr
\kern3cm{}	\times
\fr{(b_2)_{m_1}(c_2)_{m_1}(b_3)_{m_1+m_2}(c_3)_{m_1
	+ m_2} \dots }{(aq/b_1)_{m_1}(aq
	/c_1)_{m_1} (aq/b_2)_{m_1 + m+2}(aq/c_2)_{m_1 + m_2} \dots
	}\hfill\cr
\kern3cm{}\times\fr{(b_k)_{m_1 + \dots + m_{k-1}}}
{(aq/b_{k-1})_{m_1 + \dots + m_{k-1}}}
\cdot
	\fr{(c_k)_{m_1 + \dots + m_{k-1}}}{(aq/c_{k-1})_{m_1 + \dots
	+ m_{k-1}}} \hfill\cr
\kern3cm{}\times \fr{(q^{-N})_{m_1 + m_2 + \dots +
	m_{k-1}}}{(b_k c_k q^{-N}/a)_{m_1 + m_2 \dots + m_{k-1}}}
	\hfill\cr
\kern3cm{}\times	\fr{(aq)^{m_{k-2} + 2m_{k_3} + \dots +
(k-2)m_{1}}q^{m_1
	+ m_2 + \dots + m_{k-1}}}{(b_2 c_2)^{m_l}(b_3 c_3)^{m_1 + m_2}
	\dots (b_{k-1}c_{k-1})^{m_1 + m_2 + \dots + m_{k-2}}}\;.
\hfill\cr}
$$
Now let's be honest.  This proves George is an analyst.  No one
but a hard core analyst would have the nerve to extol a formula
like this especially after his outrageous April Fool joke.

His comments \c{18; p. 205} about the work of Holman,
Biedenham and Louck \c{75} caught the attention of Steve
Milne (see for example \c{78}) who has subsequently in a
work spaning two decades revealed what a rich theory was
being hinted at in \c{75}.

As is frequently the case, there was a sequel to his marvelous April
Fool joke.  A bit over a year ago, Doron Zeilberger sent out a three
page paper titled ``Mathematical Genitalysis: \ A Powerful New
Combinatorial Theory that Obviates Mathematical Analysis.''  The
abstract claimed that this new combinatorial theory would supersede
and sometimes trivialize mathematical analysis, and illustrated this
by an exact determination of Bloch's constant, and two other results.
The last paragraph of the paper commented favorably on the NCTM
Standards and Calculus Reform, and said that the new theory propounded
in the note sent was consistent with these important movements.

George wrote me a note which said in part:  ``It is 100\% certain
that Doron sent this to you\dots\ Whatever the merits of the
sketchy body of the text, the last paragraph is vintage
stuff.''

My reply was to suggest that George look at the date this was posted
on, April 1.  He replied: ``Why do I always fall for these things?''
To which, my reply was: ``Since you gave one of the best April Fool's
jokes I have seen or heard, it is fitting that you get fooled at
times.''

\head
{\bf 6. \ The Year in Madison.}
\endhead

By 1975, we were both aware of many areas of common interest.
In particular each of us had looked seriously at the papers
that Wolfgang Hahn had published in the late 1940's and early
1950's (see especially \c{71}).  Each of us recognized that
Hahn was pioneering a topic that was quite important.  So I
obtained money from the Mathematics Research Center and the Graduate
School of the University of Wisconsin for George to spend the
1975-1976 academic year in Madison.  Our plan which we followed
fairly closely was to work through Hahn's paper \c{71}.

This seminar led to our extended collaboration culminating
many, many years later in the publication of Special Functions
\c{42} (a project whose appearance would have never occurred
without the resolute efforts of our co-author, Ranjan Roy).

In addition to our own collaboration, my two students, Jim Wilson
and Dennis Stanton were introduced to $q$-series, and each has
made substantial contributions to this topic.

The year (so George tells me) was one of the most fruitful of
his career.  He wrote and submitted The Theory of Partitions
during that year.  While he was writing Chapter 11 on plane partitions,
I happened to mention to him a recent Russian book I had just obtained
on the evaluation of determinants.  This discussion reminded
George of the tale he describes in Section 5 of \c{38}, and
simultaneously, unknown to me at the time, he was studying
the Bender-Knuth paper \c{59} which reduced MacMahon's then
75 year old conjecture on symmetric plane partitions to the
following identity:

If
$$\leqalignno{
	g_j(q)& = \prod_{i=1}^s \left\{ \fr{(1 - q^{j+2i-1})}
	{(1 - q^{2i-1})} \prod_{h=i+1}^s \fr{(1 - q^{2(j+i+h-1})}
	{(1 - q^{2(i+h-1})}\right\}\;,&(6.1)\cr
\noalign{\hbox{then}}
	g_{2n}(q) &= \det(C_{i-j} + C_{i+j-1})_{n\times n}\,,\cr
\noalign{\hbox{and}}
	g_{2n+1}(q) &= \left[ \prod_{i=1}^m (1 - q^{2i-1}) \right]
	\det (C_{i-j} - C_{i+j})_{n\times n}\,,
&(6.2)\cr
\noalign{\hbox{with}}
	C_k& = q^{k^2} \binom{2m}{m+k}_2\,.
&(6.3)\cr}
$$
$\binom{N}{M}_r$ is the Gaussian polynomial (or $q$-binomial
coefficient) defined by
$$
	\binom{N}{M}_r = \left\{
	\alignedat3
	& \fr{(1 - q^{Nr})
	(1 - q^{(n-1)r}) \cdots (1 - q^{(N-M+1)r})}{(1 - q^{Mr})
	(1 - q^{(M-1)r})\cdots (1 - q^r)}\;, &\    &\  0 < M
\leqslant N ;
	\\
	& 1\,,   & \   	& M = 0\, ;
	\\
  	& 0\,,   & \    	& M < 0\,, \  M > N\,.
	\endalignedat
	\right.
\tag6.4
$$

This problem so consumed him that, apart from his appearances
at our seminar, I saw almost nothing of him throughout October,
1975.  Fortunately he was able to prove the MacMahon conjecture
during that month \c{20}, \c{22} and his interest in Hahn's
paper resumed.

The seminar yielded a number of joint papers, for example \c{39},
\c{40}, and \c{41}.  Probably \c{39} and \c{41} are most
representative of our work.  In \c{39} we developed the full
solution of the connection coefficient problem for little
$q$-Jacobi polynomials.  Namely, if
$$\displaylines{
	p_n (x;\alpha,\beta|q) = \sum_{j=0}^n\;
	\fr{(q^{-n};q)_j(\alpha\beta q^{n+1};q)_j q^j x^j}
	{(q;q)_j (\alpha q;q)_j},\cr
\noalign{\vskip-3pt}
\noalign{\hbox{then}}
\noalign{\vskip-3pt}
	p_n(x;\gamma,\delta|q) = \sum_{k=0}^n a_{k,n} p_k
	(x;\alpha,\beta|q),\cr
\noalign{\hbox{where}}
\aligned
	a_{k,n} & = \fr{(-1)^k q^{k(k+1)/2}(\gamma\delta q^{n+1};q)_k
	(q^{-n};q)_k(\alpha q;q)_k}{(q;q)_k (\gamma q;q)_k (\alpha
	\beta q^{k+1}; q)_k}
	\\
	& \qquad \times \sum_{j=0}^{n-k}  \fr{(q^{-n+k};q)_j
	(\gamma\delta q^{n+k+1};q)_j (\alpha q^{k+1};q)_j q^j}
	{(q;q)_j(\gamma q^{k+1};q)_j (\alpha \beta q^{2h+w};
	q)_j}\;.
\endaligned\cr
\noalign{\vskip-3pt}}
$$

{}From here it is a simple matter to deduce Watson's $q$-analog
of Whipple's theorem (namely (5.1)).  This was all based on the
analogous results for the classical Jacobi polynomials \c{55; p. 63}.

This inexorably led George to a mild generalization in \c{23} and
eventually to the Bailey chains \c{26}, a powerful method that
has its genesis in a theorem of Bailey \c{58}.  It was George's
good fortune that Bailey was so diffident about his result
that he only gave the recipe for it rather than displaying
it and realizing its power.

In the next section, I shall describe the most bizarre event
connected with George's year in Madison.

\head
{\bf 7. \ Ramanujan's Lost Notebook.}
\endhead

In the spring of 1976, Dominique Foata invited George to speak at the
Table Ronde, Combinatoire et Repr\'esentation du Groupe
Sym\'etrique to be held in Strasbourg on April 26--30, 1976.
George felt that his proof of MacMahon's conjecture was old
news by now and that our joint work on $q$-analogs of the
classical orthogonal polynomials was not ready for
presentation.

He decided to prove the Bender-Knuth conjecture, a conjecture related
to MacMahon's conjecture.  Basil Gordon had announced a proof of the
Bender-Knuth conjecture but did not publish anything on it until 1983
\c{69} and George assumed that his own methods would at least be
novel.  This was a dismaying prospect; I could envision him
effectively disappearing for another month, if not more.  Fortunately
he took the approach of proving that Bender-Knuth and MacMahon were
equivalent, a task that only took a few days \c{21}.

As the time of the conference came closer, fate, in the guise of
international airline fare irrationality, took a hand.  In the spring
of 1976, if you stayed in Europe at least 3 weeks, you could purchase
a ticket for only a fraction of the amount required for any shorter
stay.  George asked for and received permission to stay in Europe for
two weeks after the conference.  He layed out an itinerary that
included talks in Paris and Southampton, and a visit to Lucy Slater in
Cambridge.  Slater had told him about boxes deposited in the Trinity
College Library which contained papers from the estate of the late
G. N. Watson, the English analyst.  This seemed at the time to be a
rather minor aspect of his trip.  Watson was a good analyst and had
done good work (after all, he was the Watson of Whittaker and Watson);
however, it would have been overly optimistic to expect to find much in
these boxes.  To his surprise, in one of the boxes was a 100+ page
manuscript in Ramanujan's inimitable handwriting \c{22}.

In his own contribution to this volume, George has told the story
of his discovery, so I won't repeat it here.  I first found out
about it the day he arrived back in Madison.  ``How did the trip go?''
I asked.  ``Pretty well,'' he said.  ``I have in my briefcase,
a Xerox of a 100+ page manuscript in Ramanujan's handwriting.
I'm charging 25$\not\!{c}$ a peek!''

It would fill most of this volume if I were to recount in any
detail the cornucopia of results that flowed from the Lost
Notebook.  George provided a survey in his introduction to the
published version of the Lost Notebook \c{29} in 1988.
Currently he and Bruce Berndt are collaborating on a fully
edited version of the Lost Notebook.

A couple of summers later, George stopped in Madison on his
way to the summer math meetings where he was to talk on the
Lost Notebook.  He gave a general talk and a specialized
talk in Madison.  At the second of these, there were six to
eight people in the room and all of them knew a reasonable
amount about q-series.  I told George that this would
probably be the largest audience he would have of people
who knew a lot about basic hypergeometric series, so he
could use the standard notation without fear.  Little did
either of us know how this field would develop, so that
now an audience of 50 experts is not uncommon.

\head
{\bf 8. \ The Mock Theta Function and Bailey Chain.}
\endhead

I have already touched on the themes of Bailey chains and
Ramanujan's Lost Notebook.  In the early 80's, George started
substantial use of computer algebra packages.  This combined
with the Bailey chains led to real breakthroughs in the study
of mock theta functions.

The study of mock theta functions began with Ramanujan's last
letter to Hardy in January, 1920, four months before he died.
Here are a few excepts from that \c{80; pp. 354--355}.

\goodbreak
``If we consider a $\vartheta$-function in the transformed Eulerian
form, e.g.,
$$
\displaylines{
	(A) \quad 1 + \fr{q}{(1 - q)^2} + \fr{q^4}{(1 - q)^2(1 -
q^2)^2}
	+ \fr{q^9}{(1 - q)^2(1 - q^2)^2(1 - q^3)^2} + \cdots\,,
	\hfill\cr
	(B) \quad 1 + \fr{q}{1 - q} + \fr{q^4}{(1 - q)(1 - q^2)}
	+ \fr{q^9}{(1 - q)(1 - q^2)^2(1 - q^3)} + \cdots\,,
\hfill\cr}
$$
and determine the nature of the singularities at the points
$$
	q = 1,q^2 = 1,q^3 = 1,q^4 = 1,q^5 = 1,\dots,
$$
we know how beautifully the asymptotic form of the function can
be expressed in a very neat and closed exponential form.  For
instance, when $q = e^{-t}$ and $t \to 0$,
$$
\gathered
	(A) = \sqrt{\left(\fr{t}{2\pi}\right)} \exp \left(
	\fr{\pi}{6t} - \fr{t}{24}\right) + o(1)\,,
	\\
	(B) = \sqrt{\left(\fr{2}{5 - \sqrt{5}}\right)} \exp \left(
	\fr{\pi}{15t} - \fr{t}{60}\right) + o(1)\,,
\endgathered
$$
and similar results at other singularities.

If we take a number of functions like (A) and (B), it is only in
a limited number of cases the terms close as above; but in the
majority of cases they never close as above.  For instance, when
$q = e^{-t}$ and $t \to 0$,
$$
\displaylines{
	(C) \quad 1 + \fr{q}{(1 - q^2)} + \fr{q^3}{(1 - q^2)(1 -
q^2)^2}
	+ \fr{q^6}{(1 - q^2)(1 - q^2)^2(1 - q^3)^2} + \cdots
	\hfill\cr
\hfill{}
	= \sqrt{\left(\fr{t}{2\pi\sqrt{5}}\right)} \exp \left[
	\fr{\pi^{2}}{5t} + a_1 t + a_2 t^2 +
	\cdots + O(a_k t^k)\right]\,,
\cr}
$$
where $a_1 = 1/8\sqrt{5}$, and so on.  The function (C) is a simple
example of a function behaving in an unclosed form at the
singularities.

Now a very interesting question arises.  Is the converse of the
statements concerning the forms (A) and (B) true?  That is to say: \
Suppose there is a function in the Eulerian form and suppose that all
or an infinity of points are exponential singularities, and also
suppose that at these points the asymptotic form of the function
closes as neatly as in the cases of (A) and (B).  The questions is: \
Is the function taken the sum of two functions one of which is an
ordinary $\vartheta$-function and the other a (trivial) function which
is $O(1)$ at all the ponts $e^{2m\pi i/n}$?  The answer is {\it it is
not necessarily so}.  When it is not so, I call the function a Mock
$\vartheta$-function.  I have not proved rigorously that {\it it is no
necessarily so}.  But I have constructed a number of examples in which
it is inconceivable to construct a $\vartheta$-function to cut out the
singularities of the original function.  Also I have shown that if
{\it it is necessarily so} then it leads to the following
assertion---viz. it is possible to construct two power series in
$x$, namely $\sum a_n x^n$ and $\sum b_n x^n$, both of which have {\it
essential singularities} on the unit circle, are convergent when $|x|
< 1$, and tend to {\it finite limits at every point} $x = e^{2 r\pi
i/s}$, and that at the same time the limit of $\sum a_n x^n$ at the
point $x = e^{2 r\pi i/s}$ is equal to the limit of $\sum b_n x^n$ at
the point $x = e^{-2 r\pi i/s}$.

This assertion seems to be untrue.  Anyhow, we shall go to the
examples and see how far our assertions are true.''

Ramanujan concludes the letter with a list of mock theta functions
together with identities satisfied by them.  In G. N. Watson's
LMS Presidential Address \c{84}, he provides a method (relying on
(5.1)) for a deep analysis of one collection of mock theta
function (those that Ramanujan named ``third order'').  However
Watson failed to produce an analysis of any comparable depth for the
fifth order and seventh order functions.

In the Lost Notebook, George found a number of identities which would
provide the missing analysis.  For example, if $\phi_0(q)$ defined by
$$
	\phi_0(q) = 1 + \sum_{n=1}^{\infty} q^{n^2}(1 + q)(1 + q^3)
	\cdots(1 + q^{2n-1})
$$
is one of the fifth-order functions, then in the ``lost'' notebook
we find a result equivalent to
$$
\displaylines{
	\phi_0(-q) =  \prod_{n=0}^{\infty} \fr{(1 - q^{5n+5})
	(1 + q^{5n+2})(1 + q^{5n+3})}{(1 - q^{10n+2})(1 - q^{10n+8})}
	\hfill\cr
\hfill{} + 1 - \sum_{n=0}^{\infty} \fr{q^{5n^2}}{ (1 - q)(1 -
q^6)
	\cdots (1 - q^{5n+1})(1 - q^4)(1 - q^9)\cdots
(1 - q^{5n-1})}\;.
\cr}
$$

It follows immediately by an application of (5.1), that
$$
\displaylines{
	\phi_0(-q) = \prod_{n=0}^{\infty} \fr{(1 - q^{5n+5})
	(1 + q^{5n+2})(1 + q^{5n+3})}{(1 - q^{10n+2})(1 - q^{10n+8})}
	\hfill\cr
\hfill{} + 1 - \prod_{n=0}^{\infty}  (1 - q^{5n+5})^{-1}
\left\{
	\fr1{1 - q} + (1 - q^{-1}) \sum_{n=1}^{\infty}
	\fr{(-1)^n q^{n(15n+5)/2}(1 + q^{5n})}{(1 - q^{5n+1})
	(1 - q^{5n-1})}\right\}.
\cr}
$$

This and similar identities for the other fifth order mock theta
functions were central to their study as George noted \c{22}.

The key to unlocking such formulas lay in a subtle application of the
Bailey chain.  George has given an account of the basic properties of
the chain in Some Debts I Owe; so I will restrict myself to one
portion of the study.  Namely the objects he calls Bailey pairs.
Sequences of functions $\{ \alpha_n\}$ and $\{\beta_n\}$
satisfying
$$
	\beta_n = \sum_{r=0}^n
	\fr{\alpha_r}{(q;q)_{n-r}(q;q)_{n+r}}\;.
$$
The pair key to the understanding of $\phi_0(q)$ is
$$
	\beta_n = \fr{(-1)^n q^{-n(n - 1)/2}}{(q;q)_n}
$$
and
$$
	\alpha_n = q^{n^2 + n} \sum_{j=-n}^n (-1)^j
	q^{-j(3j+1)/2} - q^{n^2 - n} \sum_{j=1 - n}^{n-1}
	(-1)^j q^{-j(3j+1)/2}\;.
$$

The form of $\alpha_n$ is sufficiently surprising not to mention
complicated that without the help of SCRATCHPAD (a.k.a. AXIOM) to
compute many $\alpha_n$'s, these discoveries never would have
occurred.

{}From this point on, Dean Hickerson played a vital role eventually
proving all the Mock Theta conjectures \c{73} and proving comparable
theorems for the seventh order functions \c{74}.

\head
{\bf 9. \ Physics.}
\endhead

George's collaborations with physicists began with Rodney Baxter
and Peter Forrester in \c{46}; the resulting model, generalizing
Baxter's Hard Hexagon Model, is succinctly called the ABF Model.

The mathematics background that George brought to bear on this
is best laid out in his only single author physics publication
\c{25}.  Here he observes that a number of the functions produced
by the method of Corner Transfer Matrices are limits of nice
polynomials.  The prototype example is Schur's theorem \c{81}.

If $D_0 = D_1 = 1$, and $D_n = D_{n-1} + q^{n-1} D_{n-2}$ for
$n \geqq 2$, then
$$
	D_n = \sum_{\lam = -\infty}^{\infty}  (-1)^{\lam}
	q^{\lam(5\lam + 1)/2} \bmatrix n \\ \left\lfloor
	\fr{n - 5\lam}{2}\right\rfloor\endbmatrix
$$
where $\lfloor x\rfloor$ is the greatest integer in $x$ and
$\bmatrix A \\ B\endbmatrix$ is the familiary Gaussian polynomial
defined in (6.4).

George had already studied generalizations of Schur's theorem
in \c{12} and \c{13}.  It turned out that the ABF model could be
treated by an analysis of polynomials similar in nature to
Schur's polynomial version of the Rogers-Ramanujan identities.

In a subsequent collaboration with Baxter \c{45}, they discover
$q$-analogs of the trinomial numbers.  The latter are the entries
in the following table where each entry is the sum of the 3
entries directly above it
$$
\gathered
	1  \\
	1 \quad 1 \quad 1   \\
	1 \quad 2 \quad 3 \quad 2 \quad 1   \\
	1 \quad 3 \quad 6 \quad 7 \quad 6 \quad 3 \quad 1   \\
	1 \quad 4 \quad 10 \quad 16 \quad 19 \quad 16 \quad 10
	\quad 4 \quad 1   \\
	- \quad - \quad - \quad - \quad - \quad - \quad -
	\quad - \quad -   \\
	- \quad - \quad - \quad - \quad - \quad - \quad -
	\quad - \quad -	\quad - \quad -   \\
\endgathered
$$

An example of the polynomials in question is \c{43; p. 299}
$$
	\pmatrix n;B;q   \\ A \endpmatrix_2 =
	\sum_{j\geqq 0} \fr{q^{j(j+B)}(q;q)_n}{(q;q)_j (q;q)_{j+A}
	(q;q)_{n-2j-A}}\;.
$$

Besides solving the model with Baxter in \c{43}, \c{44} and
\c{45}, George found a variety of applications including a full
explanation of a mathematical mystery that had been called by
Euler:  \ ``A remarkable example of misleading induction.''
\c{31}.  More recently he has worked with physicists A. Berkovich
\c{47} and A Schilling and O. Warnaar \c{54} on further
extensions of the Bailey chain.

\head
{\bf 10. \ The Crank.}
\endhead

There are many sides to George Andrews.  The one which will probably
have the most long lasting impact is the problem solver.  His ability
in this regard can be illustrated in many ways.  Here is one instance
of what happened when he felt challenged in an area that most of us
think of as his.

Ramanujan discovered a number of very surprising congruences for the
number of partitions of some infinite families of numbers.  There were
three infinite families, with the first of each family being the
following:
$$
\align
	& p(5n + 4) \equiv 0 \;\pmod{5}   \tag10.1  \\
	& p(7n + 5) \equiv 0 \;\pmod{7}   \tag10.2  \\
	& p(11n + 6) \equiv 0 \;\pmod{11}  \tag10.3
\endalign
$$

For the first two of these congruences, Freeman Dyson \c{61}
discovered a combinatorial reason for the existence of these facts,
and his conjecture was proven by Atkin and Swinnerton-Dyer \c{3}.
The statistic found by Dyson did not work in the third case, so he
expressed a hope that another statistic could be found.  These names
were appropriate, ``rank'' for the one he found, and ``crank'' for the
one still undiscovered one.  Over 40 years later, Frank Garvan, one of
George's Ph.D. students, found a pseudo crank, and talked about this
at the Ramanujan meeting in Urbana in 1987.  Garvan felt that there
was a real version of this unknown statistic, and told a number of
people about an identity which he felt was the key.

George was struck by Frank's observation that the pseudo crank
generating function
$$
	\prod_{n=1}^{\infty} \fr{(1 - q^n)}{(1 - zq^n)(1 - z^{-1}q^n)}
\tag10.4
$$
has only one negative coefficient (that of $z^0 q^1$).  Previously
they had assumed that the appearance of one negative
coefficient suggested that negative coefficients would abound.
Since the specialization $z = 1$ in (10.1) produced
$$
	\prod_{n=1}^{\infty}  \fr1{1 - q^n} = \sum_{n\geqq 0}
	p(n) q^n\,,
\tag10.5
$$
Frank's observation suggests strongly that the pseudo crank
generating function might indeed generate the crank at least
for $n > 1$.

On Saturday, June, 1987, the day after the Ramanujan meeting,
George undertook a melancholy journey to Onarga, Illinois to
visit the graves of his parents (who were born in the mid-west
and chose to be buried there even though they spent all their
adult lives in Oregon).  He returned to Urbana in the early
afternoon in a very somber mood.  He hoped to discuss mathematics
with a few people who had remained after the conference, but no
one was around in the dormitory where he was staying.  To pull
himself out of the doldrums he sat down to study (10.4) with
the hope of tracking down the crank.

By the early evening, he had found the crank \c{49; p. 168}.

Now he faced a nonmathematical problem.  He was low on cash, had
no phone card and only had access to pay phones.  How could he
let Frank know?  He proceeded to call his wife collect.  ``I
want you to call Richard Askey who will know Frank Garvan's
phone number.  I want you to say that I would like to write
a joint paper with Frank in which we show that the crank of a
partition is given by:  [he then read the definition of the crank
symbol by symbol].''

This provided the first crank.  It should be noted that further
cranks (i.e. partition statistics that provide combinatorial
interpretations of (10.3) were found by Garvan, Kim and Stanton,
\c{63} and that Dyson \c{62} provided many further insights about the
Garvan-Andrews crank.

\head
{\bf 11. \ Conclusion.}
\endhead

It would be incorrect to conclude that the bulk of George's
research concluded in the 1980's.  Indeed a full account of
his work in the 1990's would perhaps require another paper
comparable to this one.  I shall only mention a few of his
themes refer you to the literature.  It's not clear which of
his current projects will have staying power similar to those
I have already described.  Also the breadth of projects widens
partially due to the fact that he is undertaking many more
collaborations now than in the past; indeed 27 of his 44 lifetime
collaborators have written papers with him in the 1990's.

His interest in partition identities sparked a continuing
collaboration with K. Alladi that began with \c2.

The Liouville mystery \c{37} (referred to in Some Debts I Owe)
is essentially work on generalized Lambert series and relates
to his work with Crippa and Simon \c{30}.

His Pfaff Trilogy as he calls it (\c{34}, \c{35} and \c{36}) is
both part of his continuing interest in computer algebra methods
and applications to plane partitions.

The papers with the Knopfmachers (beginning with \c{50}) on generalizations
of the Engel expansion suggest a try new line of research.  The idea
here is to produce an algorithm to expand an infinite product or
modular form into an Eulerian or $q$-series.  If this method
can be expanded it may well take its place as perhaps the
converse of Euler's ancient algorithm \c{19; p. 98} for obtain
infinite product representations of generating functions.

Also in this volume we find one of his papers (joint with P. Paule)
\c{52} on the computer algebra implementation
of the Partition Analysis of P. A. MacMahon.  This project clearly
points beyond MacMahon's horizon in that the authors have observed and
implemented the fact that Partition Analysis is a purely algorithmic
process.  The success of Geroge's work on the Omega Package (joint
with Paule and Riese \c{51}, \c{52}, \c{53}) suggests a number of
further refinements and applications.

There is much more to the story and I have probably left out a number
of George's favorite items.  This is to be expected.  The topics that
interested me most are the ones I can most easily discuss.

I want to express my thanks to George.  First, he provided some of
the details in this paper.  I remember the start of his April 1
lecture in Madison, but not the specific forumlas which were given
above.  He provided them and some other facts.  More importantly,
he taught me about basic hypergeometric series.  My mathematical
life would have been significantly different without his
teaching.   There are quite a few people who could say the
same, their mathematical life would have been poorer without
the aid which George Andrews provided them.

\vfill\eject


\vglue2cm
\Refs

\ref
  \no 1
  \by H. L. Alder
  \paper The nonexistence of certain identities in the theory
	of partitions and compositions
  \jour Bull. Amer. Math. Soc.
  \vol 54
  \yr 1948
  \pages 712-722
\endref

\ref
  \no 2
  \by K. Alladi, G. E. Andrews and B. Gordon
  \paper Generalizations and refinements of a partition theorem of
	G\"ollnitz
  \jour J. reine und angew. Math.
  \vol 460
  \yr 1995
  \pages 165--188
\endref

\ref
  \no 3
  \by A. O. L. Atkin and H. P. F. Swinnerton-Dyer
  \paper Some properties of partitions
  \jour Proc. London Math. Soc. (3)
  \vol 4
  \yr 1953
  \pages 84--106
\endref

\ref
  \no 4
  \manyby G. E. Andrews
  \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 5
  \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 6
  \bysame
  \paper $q$-identities of Auluck, Carlitz, and Rogers
  \jour Duke Math. J.
  \vol 33
  \yr 1966
  \pages 575--582
\endref

\ref
  \no 7
  \bysame
  \paper A generalization of the Gollnitz-Gordon partition
	theorems
  \jour Proc. Amer. Math. Soc.
  \vol 8
  \yr 1967
  \pages 945--952
\endref

\ref
  \no 8
  \bysame
  \paper On $q$-difference equations for certain well-poised basic
	hypergeometric series
  \jour Quart. J. Math.
  \vol 19
  \yr 1968
  \pages 433--447
\endref

\ref
  \no 9
  \bysame
  \paper A generalization of the classical partition theorems
  \jour Trans. Amer. Math. Soc.
  \vol 145
  \yr 1968
  \pages 205--221
\endref

\ref
  \no 10
  \bysame
  \paper On the foundations of combinatorial theory, V.:
	Eulerian differential operators
  \jour Studies in Applied Math
  \vol 50
  \yr 1971
  \pages 345--375
\endref

\ref
  \no 11
  \bysame
  \paper Summations and transformation for basic Appell series
  \jour J. London Math. Soc.
  \vol 4(2)
  \yr 1972
  \pages 618--622
\endref

\ref
  \no 12
  \bysame
  \paper Sieves for theorems of Euler, Rogers and Ramanujan
  \jour Lecture Notes in Mathematics \#251
  \finalinfo Springer-Verlag, NY, (1972) pp. 1--20
\endref

\ref
  \no 13
  \bysame
  \paper Sieves in the theory of partitions
  \jour Amer. J. Math.
  \vol 94
  \yr 1972
  \pages 1214--1230
\endref

\ref
  \no 14
  \bysame
  \paper Partition identities
  \jour Advances in Math.
  \vol 9
  \yr 1972
  \pages 10--51
\endref

\ref
  \no 15
  \bysame
  \paper On the general Rogers-Ramanujan theorem
  \jour Memoir. Amer. Math. Soc.
  \vol 152
  \yr 1974
  \pages 86
\endref

\ref
  \no 16
  \bysame
  \paper Applications of basic hypergeometric functions
  \jour SIAM Rev.
  \vol 16 (4)
  \yr 1974
  \pages 441--484
\endref

\ref
  \no 17
  \bysame
  \paper An analytic generalization of the Rogers-Ramanujan
	identities for odd moduli
  \jour Proc. Nat. Acad. Sci.
  \vol 71 (10)
  \yr 1974
  \pages 4082--4085
\endref

\ref
  \no 18
  \bysame
  \paper Problems and prospects for basic hypergeometric functions
  \jour Theory and Application of Special Functions
  \finalinfo R. Askey, ed., Academic Press, NY, (1975) pp. 191--214
\endref

\ref
  \no 19
  \bysame
  \paper The Theory of Partitions
  \paperinfo The Encyclopedia of Mathematics and Its Applications
	Series, Addison-Wesley Pub. Co., NY, (1976) pp. 300.
	Reissued, Cambridge University Press, New York, 1998
\endref

\ref
  \no 20
  \bysame
  \paper MacMahon's conjecture on symmetric plane partitions
  \jour Proc. Nat. Acad. Sci.
  \vol 74 (2)
  \yr 1977
  \pages 426--429
\endref

\ref
  \no 21
  \bysame
  \paper Plane partitions, II:  The equivalence of the Bender-Knuth
	and MacMahon conjectures
  \jour Pacific J. Math.
  \vol 72 (2)
  \yr 1977
  \pages 283--291
\endref

\ref
  \no 22
  \bysame
  \paper An introduction to Ramanujan's ``lost'' notebook
  \jour Amer. Math. Monthly
  \vol 86 (2)
  \yr 1979
  \pages 89--108
\endref

\ref
  \no 23
  \bysame
  \paper Connection coefficient problems and partitions
  \jour Proc. Symp. in Pure Math.
  \vol 34
  \yr 1979
  \pages 1--24
\endref

\ref
  \no 24
  \bysame
  \paper Plane partitions, III:  The weak Macdonald conjecture
  \jour Invent. Math.
  \vol 53
  \yr 1979
  \pages 193--225
\endref

\ref
  \no 25
  \bysame
  \paper The hard-hexagon model and the Rogers-Ramanujan type
	identities
  \jour Proc. Natl. Acad. Sci.
  \vol 78
  \yr 1981
  \pages 5290--5292
\endref

\ref
  \no 26
  \bysame
  \paper Multiple series Rogers-Ramanujan type identities
  \jour Pac. J. Math.
  \vol 114 (2)
  \yr 1984
  \pages 267--283
\endref

\ref
  \no 27
  \bysame
  \paper The fifth and seventh order mock theta functions
  \jour Trans. Amer. Math. Soc.
  \vol 293 (1)
  \yr 1986
  \pages 113--134
\endref

\ref
  \no 28
  \bysame
  \paper Plane partitions, IV: A conjecture of Mills-Robbins-Rumsey
  \jour Aequationes Mathematicae
  \vol 33
  \yr 1987
  \pages 230--250
\endref

\ref
  \no 29
  \bysame
  \paper Introduction to {\bf Srinivasa Ramanujan: The Lost
	Notebook and Other Unpublished Papers}
  \paperinfo Narosa, New Delhi (1987)
\endref

\ref
  \no 30
  \bysame
  \paper Mock theta functions
  \jour Proc. Symposia in Pure Math.
  \vol 49
  \yr 1989
  \pages 283--298
\endref

\ref
  \no 31
  \bysame
  \paper Euler's ``Exemplum Memorabile Inductionis Fallacis'' and
	$q$-trinomial coefficients
  \jour J. Amer. Math. Soc.
  \vol 3
  \yr 1990
  \pages 653--669
\endref

\ref
  \no 32
  \bysame
  \paper Bailey chains and generalized Lambert series, I:
	Four identities of Ramanujan
  \jour Illinois J. Math.
  \vol 36
  \yr 1992
  \pages 251--274
\endref

\ref
  \no 33
  \bysame
  \paper Plane partitions, V: The T.S.S.C.P.P. conjecture
  \jour J. Combinatorial Th. (A)
  \vol 66 (1)
  \yr 1994
  \pages 28--39
\endref

\ref
  \no 34
  \bysame
  \paper Pfaff's Method (II): Diverse applications
  \jour J. Computational \& Appl. Math.
  \vol 68
  \yr 1996
  \pages 15--23
\endref

\ref
  \no 35
  \bysame
  \paper Pfaff's Method (III): Comparison with the WZ method
  \jour Electronic J. Combin. (Foata Festschrift)
  \vol 3 (2): R21
  \yr 1996
  \pages 18
\endref

\ref
  \no 36
  \bysame
  \paper Pfaff's Method (I):  The Mills-Robbins-Rumsey determinant
  \jour Discrete Math.
  \vol 193
  \yr 1998
  \pages 43--60
\endref

\ref
  \no 37
  \bysame
  \paper Three-quadrant Ferrers graphs
  \paperinfo (submitted to volumes for the B. N. Prasad Centenary
	Commemoration, Allahabad Math. Society)
\endref

\ref
  \no 38
  \bysame
  \paper Some debts I owe
  \paperinfo (submitted to Proceedings of 42nd S\'eminaire
	Lotharingien de Combinatoire, held in my honor at
	Maratea, Italy, September, 1998)
\endref

\ref
  \no 39
  \bysame (with R. Askey)
  \paper Enumeration of partitions:  The role of Eulerian
	series and $q$-orthogonal polynomials
  \jour Higher Combinatorics
  \finalinfo M. Aigner, ed., Reidell Publ. Co., Dordrecht,
	Holland, (1977) pp. 3--26
\endref

\ref
  \no 40
  \bysame (with R. Askey)
  \paper A simple proof of Ramanujan's $_1\psi_1$
	summation
  \jour Aequa. Math.
  \vol 18
  \yr 1978
  \pages 333--337
\endref

\ref
  \no 41
  \bysame (with R. Askey)
  \paper Another $q$-extension of the $\beta$-function
  \jour Proc. Amer. Math. Soc.
  \vol 81 (1)
  \yr 1981
  \pages 97--100
\endref

\ref
  \no 42
  \bysame (with R. Askey)
  \paper Classical orthogonal polynomials
  \jour Proc. Conf. on Orthog. Polys. at Bar-le duc
  \finalinfo Lecture Notes in Math., Springer, Berlin,
	{\bf 1171} (1985) pp. 36--62
\endref

\ref
  \no 43
  \bysame (with R. J. Baxter)
  \paper Lattice gas generalization of the hard hexagon
	model, I: Star triangle relation and local densities
  \jour J. Stat. Phys.
  \vol 44 (1/2)
  \yr 1986
  \pages 259--271
\endref

\ref
  \no 44
  \bysame (with R. J. Baxter)
  \paper Lattice gas generalization of the hard hexagon
	model.  II. The local densities as elliptic functions
  \jour J. Stat. Phys.
  \vol 44
  \yr 1986
  \pages 713--728
\endref

\ref
  \no 45
  \bysame (with R. J. Baxter)
  \paper Lattice gas generalization of the hard hexagon
	model.  III. $q$-trinomial coefficients
  \jour J. Stat. Phys.
  \vol 47 (3/4)
  \yr 1987
  \pages 297--330
\endref

\ref
  \no 46
  \bysame (with R. J. Baxter and P. J. Forrester)
  \paper Eight-vertex SOS model and generalized
	Rogers-Ramanujan-type identities
  \jour J. Stat. Phys.
  \vol 35
  \yr 1984
  \pages 193--266
\endref

\ref
  \no 47
  \bysame (with A. Berkovich)
  \paper A trinomial analogue of Bailey's lemma and
	$N = 2$ superconformal invariance
  \jour Comm. in Math. Physics
  \vol 192
  \yr 1998
  \pages 245--260
\endref

\ref
  \no 48
  \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 49
  \bysame and F. G. Garvan
  \paper Dyson's crank of a partition
  \jour Bull. Amer. Math. Soc.
  \vol 18
  \yr 1988
  \pages  167--171
\endref

\ref
  \no 50
  \bysame (with A. and J. Knopfmacher)
  \paper Engel expansions and the Rogers-Ramanujan
	identities
\endref

\ref
  \no 51
  \bysame (with P. Paule and A. Riese)
  \paper MacMahon's partition analysis III:  The omega
	package
\endref

\ref
  \no 52
  \bysame (with P. Paule)
  \paper MacMahon's partition analysis IV:  Summation of
	series
\endref

\ref
  \no 53
  \bysame (with P. Paule)
  \paper Magic squares and partition analysis
  \paperinfo (in preparation)
\endref

\ref
  \no 54
  \bysame (with O. Warnaar and A. Schilling)
  \paper An $A_2$ Bailey lema and Rogers-Ramanu\-jan-type
	identities
  \paperinfo Accepted for publication J. Amer. Math. Soc.
\endref

\ref
  \no 55
  \by R. Askey
  \paper Orthogonal Polynomials and Special Functions
  \paperinfo Regional Conf. Series in Applied Math.,
	No. 22, S.I.A.M., Philadelphia, 1975
\endref

\ref
  \no 56
  \by R. Askey and S. Wainger
  \paper A transplantation theorem for ultraspherical
	coefficients
  \jour Pac. J. Math.
  \vol 16
  \yr 1966
  \pages 393--405
\endref

\ref
  \no 57
  \manyby W. N. Bailey
  \paper Generalized Hypergeometric Series
  \paperinfo 1935, Cambridge, London and New York
	(reprinted:  Hafner, New York, 1964)
\endref

\ref
  \no 58
  \bysame
  \paper Identities of the Rogers-Ramanujan type
  \jour Proc. London Math. Soc. (2)
  \vol 50
  \yr 1949
  \pages 1--10
\endref

\ref
  \no 59
  \by E. A. Bender and D. E. Knuth
  \paper Enumeration of plane partitions
  \jour J. Comb. Th.,
  \vol 13
  \yr 1972
  \pages 40--54
\endref

\ref
  \no 60
  \by F. Dyson
  \paper Three identities in combinatory analysis
  \jour J. London Math. Soc.
  \vol 18
  \yr 1943
  \pages 35--39
\endref

\ref
  \no 61
  \by F. J. Dyson
  \paper Some guesses in the theory of partitions
  \jour Eureka (Cambridge)
  \vol 8
  \yr 1944
  \pages 10--15
\endref

\ref
  \no 62
  \by F. J. Dyson
  \paper Mappings and symmetries of partitions
  \jour J. Comb. Th. (A)
\endref

\ref
  \no 63
  \by F. Garvan, D. Kim and D. Stanton
  \paper  Cranks and $t$-cores
  \jour Inv. Math.
  \vol 101
  \yr 1990
  \pages 1--17
\endref

\ref
  \no 64
  \manyby H. G\"ollnitz
  \paper Einfache Partitionen
  \paperinfo (unpublished), Diplomarbeit, W. S. 1960,
	G\"ott\-ingen, 65 pp
\endref

\ref
  \no 65
  \bysame
  \paper Partitionen mit Differenzenbedingungen
  \paperinfo Dissertation,  G\"ottingen, 1963, ii + 62 pp.
\endref

\ref
  \no 66
  \bysame
  \paper Partitionen mit Differenzenbedingungen
  \jour J. Reine. Angew. Math.
  \vol 225\break
  \yr 1967
  \pages 154--190
\endref

\ref
  \no 67
  \manyby B. Gordon
  \paper A combinatorial generalization of the
	Rogers-Ramanujan identities
  \jour
  \vol 83
  \yr 1961
  \pages 393--399
\endref

\ref
  \no 68
  \bysame
  \paper Some continued fractions of the Rogers-Ramanujan
	type
  \jour Duke Math. J.
  \vol 31
  \yr 1965
  \pages 741--748
\endref

\ref
  \no 69
  \bysame
  \paper A proof of the Bender-Knuth conjecture
  \jour Pac. J. Math.
  \vol 108
  \yr 1983
  \pages 99--113
\endref

\ref
  \no 70
  \by H. W. Gould
  \paper Combinatorial Identities
  \paperinfo West Virginia University, Morgan
\endref

\ref
  \no 71
  \by W. Hahn
  \paper \"Uber Orthogonalpolynome, die
		$q$-Differenzengleichungen gen\"ugen
  \jour Math. Nachr. Berlin
  \vol 2
  \yr 1949
  \pages 4--34
\endref

\ref
  \no 72
  \by G. H. Hardy and E. M. Wright
  \paper Introduction to the Theory of Numbers
  \paperinfo 4th ed., Oxford University Press, London
	and New York, 1960
\endref

\ref
  \no 73
  \manyby D. Hickerson
  \paper A proof of the mock theta conjectures
  \jour Inv. Math
  \vol 94
  \yr 1988
  \pages 639--660
\endref

\ref
  \no 74
  \bysame
  \paper On the seventh order mock theta functions
  \jour Inv. Math.
  \vol 94
  \yr 1988
  \pages 661--677
\endref

\ref
  \no 75
  \by W. J. Holman III, L. C. Biedenharn and J. D.
	Louck
  \paper On hypergeometric series well-poised in $SU(n)$
  \jour S.I.A.M. J. Math. Anal.
  \vol 7
  \yr 1976
  \pages 529--541
\endref

\ref
  \no 76
  \manyby I. G. Macdonald
  \paper Affine root systems and Dedekind's $\eta$-function
  \jour Inv. Math.
  \vol 15
  \yr 1972
  \pages 91--143
\endref

\ref
  \no 77
  \bysame
  \paper Some conjectures for root systems
  \jour S.I.A.M. J. Math. Anal.
  \vol 13
  \yr 1982
  \pages 988--1007
\endref

\ref
  \no 78
  \by S. Milne
  \paper A $q$-analog of hypergeometric series well-poised
	in $SU(n)$ and invariant $G$-functions
  \jour Adv. Math.
\endref

\ref
  \no 79
  \by H. Rademacher
  \paper Lectures on Analytic Number Theory
  \paperinfo Tata Institute, Bombay, 1954--1955
\endref

\ref
  \no 80
  \by S. Ramanujan
  \paper Collected Papers
  \paperinfo Cambridge University Press, London and
	New York, 1927 (Reprinted:  Chelsea, New York
	1962)
\endref

\ref
  \no 81
  \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 82
  \by W. Schwarz
  \paper Review 1179
  \jour Math Reviews
  \vol 32
  \yr 1966
  \pages 202--203
\endref

\ref
  \no 83
  \by A. Selberg
  \paper \"Uber einige arithmetische Identit\"aten
  \jour Avhl. Norske Vid.
  \vol 8
  \yr 1936
  \pages 23
\endref

\ref
  \no 84
  \manyby G. N. Watson
  \paper The final problem
  \jour J. London Math. Soc.
  \vol 11
  \yr 1936
  \pages 55--80
\endref

\ref
  \no 85
  \bysame
  \paper The mock theta functions (2)
  \jour Proc. London Math. Soc. (2)
  \vol 42
  \yr 1937
  \pages 274--304
\endref

\ref
  \no 86
  \by D. Zeilberger and D. M. Bressoud
  \paper A proof of Andrews' $q$-Dyson conjecture
  \jour Discrete Math.
  \vol 54
  \yr 1985
  \pages 201--224
\endref
\endRefs
\enddocument
