\input amstex
\input amsppt.sty

\magnification1200
\hsize13cm
\vsize19cm

\NoBlackBoxes

\def\la{\lambda}
\def\suml{\sum\limits}
\def\prodl{\prod\limits}
\def\liml{\lim\limits}
\def\({\left(}
\def\){\right)}

\head {\bf Three Aspects of Partitions}\endhead
\vskip0,5cm

\centerline{by}
\vskip0,5cm

\centerline{\smc George E. Andrews\footnote {Partially supported
by National Science Foundation Grant DMS 8702695-03 and the IBM T.~J.
Watson Research Center.}}
\vskip1,5cm

\topmatter 
\address 
IBM\newline
The Thomas J. Watson Research Center \newline
Yorktown Heights, NY 10598 USA \newline
and \newline
The Pennsylvania State University \newline
University Park, PA, PA 16802 USA \newline
\endaddress
\endtopmatter

\document



{\bf 1.~Introduction}
\vskip0,5cm

In this paper we shall discuss three topics in partitions. Section 2
is devoted to partitions with difference conditions and is an
elucidation of joint work with J.~B. Olsson \cite {16}. In Section 3
we discuss certain partition problems which have their origins in
statistical mechanics. We take as the theme for this section Euler's
article: Exemplum Memorabile Inductionis Fallacis \cite {23}. The
material for this section is closely related to the work in \cite
{10}. Section 4 contains a discussion of some of Ramanujan's formulas
from both his Notebooks and lost Notebook. More extensive accounts of
this topic are found in \cite {8} and \cite {11}. 
\vskip1cm

{\bf 2.~Partitions with Difference Conditions}
\vskip0,5cm

The work in this section is based on \cite {16}, joint work with J.~B.
Olsson. In 1989, Olsson was studying Mullineux's conjecture \cite {29}
which briefly may be described as a "conjugation" map for $p$--regular
partitions (i.e\. partitions with no part repeated more than $p-1$
times). As Mullineux asserts \cite {29; p.~60}: "\dots when $p$ is
prime it is conjectured that this bijection (i.e\. conjugation) arises
in the representation theory of the symmetric group $S_n$ of degree
$n$. Farahat, M\"uller and Peel \cite {24} have shown how to form a
'good' labelling of the irreducible $p$--modular representations of
$S_n$ (for prime $p$) by $p$--regular partitions of $n$. Now the
alternating representation of $S_n$ induces in the usual way a
bijection (whose square is the identity) upon these representations
and hence induces a similar bijection upon the set of $p$--regular
partititons via the labelling. For low values of $n$ this group
theoretic bijection agrees with the one constructed here; the
verification of this has been carried out using the tables of
decomposition numbers found in Kerber and Peel (\cite {27}, $p=3,\
n\le10,\ n\ne 7$); Robinson (\cite {30}, $p=3,\ n=7$) and Wagner
(\cite {34}, $p=5,7$, $\le 8$)."

Olsson calculated the number of partitions fixed by Mullineux's map
and those fixed by the Farahat--M\"uller--Peel \cite {24} induced map.
If the two maps are the same then obviously the cardinalities of the
two sets of fixed points will be identical. This calculation led to
the following:


\proclaim{Problem} Let $\lambda =(a_1,a_2,\dots,a_k)$ be a partition
of $n$. Consider the two following sets of conditions ($p$ an odd prime).
\vskip0,3cm
 
$\cases 
\text {\rm(1A)}\hskip0,5cm p \not\kern2pt\mid a_i \ \text {and}\
2\not\kern2pt\mid 
a_i\ \text {for all}\
i,\\
\text {\rm(1B)}\hskip0,5cm a_i\ne a_{i+1}\ \text {for all}\ i. 
\endcases$
\vskip0,5cm

$\cases 
\text {\rm(2A)}\hskip0,5cm 2\mid  a_i \ \text {if and only if}\ p\mid a_i
\ \text {for all}\ i,\\
\text {\rm(2B)}\hskip0,5cm 0\le a_i-a_{i+1}\le 2p\ \text {for all $i$}\ 
(a_{k+1}=0) \\
\text {\rm(2C)}\hskip0,5cm \text {if}\ a_i=a_{i+1}\ \text {then}\ a_i \ \text {is
even}, \\
\text {\rm(2D)}\hskip0,5cm \text {if}\ a_i-a_{i+1}=2p \ \text {then}\ a_i
\ \text {is odd}.\endcases$
\endproclaim
\vskip0,5cm

Is the number of partitions of $n$ satisfying (1A)--(1B) equal to the
number of partitions satisfying (2A)--(2D)?

The simplest possible case is $p=3$. In this case conditions
(1A)--(1B) describe partitions into distinct parts $\equiv 1$ or
5 (mod 6), and this suggests the following result of Schur
\cite {32} specialized to fit this instance.  

\proclaim{Theorem 2.1 \rm\cite {32}} The number or partititons of $n$
into distinct parts congruent to 1 or 5 mod 6 equals the
number of partitions of $n$ into parts congruent to 0, 1 or 5 mod 6
with the condition that the difference between parts is at least 6 and
greater than 6 between two multiples of 6.
\endproclaim

For example there are 11 partitions of 36 into distinct parts
congruent to 1 or 5 (mod 6): 35+1, 31+5, 29+7, 25+11, 23+13, 23+7+5+1,
19+17, 19+11+5+1, 17+13+5+1, 17+11+7+1, 13+11+7+5. The second class of
eleven partitions arising from Schur's Theorem when $n=36$ is: 36,
35+1, 31+5, 30+6, 29+7, 25+11, 24+12, 24+11+1, 23+13, 23+12+1,
19+12+5.

In contrast, the eleven partitions arising from conditions (2A)--(2D)
in the Problem for $n=36$, $p=3$ are: 17+11+7+1, 13+11+7+5,
13+11+6+5+1, 12+12+7+5, 12+11+7+5+1, 12+11+6+6+1, 12+11+6+6+5,
11+7+6+6+5+1, 11+6+6+6+6+1, 7+6+6+6+6+5, 6+6+6+6+6+5+1.

Inspection shows that MacMahon's theory of modular partitions for
modulus 6 \cite {28} provides a perfect bijection between these two
latter classes of partitions. In MacMahon's representation each part
is represented by a row of 6's with the residue mod 6 tacked on at the
end. Consequently the eleven MacMahon graphs of the final set of
partitions above is: 

$$\matrix \format \l \hskip1cm & \l \hskip1cm & \l \hskip1cm  &
\l \hskip1cm & \l \\
6\quad 6\quad 5 & 6\quad 6\quad 1 & 6\quad 6 & 6\quad 6\quad 1 & 6\quad 6 \\
6\quad 5 & 6 \quad 5 & 6\quad 6 & 6\quad 5 & 6\quad 5 \\
6\quad 1 & 6\quad 1 & 6\quad 1& 6\quad 6 & 1 \\
1 & 5 & 5 & 5 & 5 \\
& & & 1 & 1 
\endmatrix$$
$$\matrix \format \l \hskip1cm & \l \hskip1cm & \l \hskip1cm  &
\l \hskip1cm & \l \hskip1cm & \l\\
6 \quad 6 & 6\quad 6 & 6\quad 5 & 6\quad 5 & 6\quad 1 & 6 \\
6\quad 5 & 6\quad 1 & 6\quad 1 & 6 & 6 & 6 \\
6 & 6 & 6 & 6 & 6 & 6 \\
6 & 6 & 6 & 6 & 6 & 6 \\
1 & 5 & 5 & 6 & 6 & 6\\
& & 1 & 1 & 5 & 5 \\
& & & & & 1  
\endmatrix$$

Now we form a new set of partitions by reading these graphs by columns
instead of rows. The result is Schur's second set of partitions, and a
little reflection shows that the above mapping always provides a
bijection between Schur's partitions and those of the second kind in
the Problem where $p=3$. 

The relationship described above suggests that the Problem may be
solved by relating it to some generalization of Schur's Theorem and
then applying MacMahon's modular theory. While there arose some
difficulties, this program eventually produced the following result. 

\proclaim{Theorem 2.2 \rm\cite {16}} Let $A=\{a_1,a_2,\dots,a_r\}$ be a
set of $r$ distinct positive integers arranged in increasing order, and
let $N$ be an integer larger than $a_r$. Let $P_1(A;N,n)$ denote the
number of partitions of $n$ into distinct parts each of which is
congruent to some $a_i$ modulo $N$. Let $P_2(A;N,n)$ denote the number
of partititons of $n$ into parts each of which is congruent to 0 or to
some $a_i$ modulo $N$, in addition only parts divisible by $N$ may be
reapeated, the smallest part is $<N$, the difference between sucessive
parts is at most $N$ and strictly less than $N$ if either part is
divisible by $N$. Then for each $n\ge 0$, 
$$P_1(A;N,n)=P_2(A;N,n).$$
\endproclaim

In the above theorem, $A=\{a_1,\dots ,a_r\}$ is an arbitrary set of
positive integers arranged in increasing order for which $a_r<N$. The
relevant generalization of Schur's theorem requires additionally: (i)
$\suml _{s=1} ^{k-1}a_s<a_k\ (k\le r)$, (ii) $\suml _{s=1} ^{r}a_s\le
N$, and (iii) all $2^r$ subsets of $A$ must have distinct sums. Let $A'$
be the set of $2^r-1$ positive sums arising from the nonempty subsets
of $A$. Let $A'_N$ denote the set of all positive integers that are
congruent to some element of $A'$ modulo $N$. Let $\rho_N(m)$ denote
the least positive residue of $m$ modulo $N$. For $m\in A'$, let
$b(m)$ be the number of terms appearing in the sum of distinct
elements of $A$ making up $m$ and let $\nu(m)$ denote the least $a_i$ in
this sum. 

In \cite {2}, the main result may be restated as follows: 
 
\proclaim{Theorem 2.3} Let $E(A'_N;n)$ denote the number of partitions
of $n$ into parts taken from $A'_N:n=c_1+c_2+\dots+c_s$, $c_i\ge
c_{i+1}$,
$$c_i-c_{i+1}\ge
Nb(\rho_N(c_{i+1}))+\nu(\rho_N(c_{i+1}))-\rho_N(c_{i+1}).$$
Then $E(A';n)=P_1(A;N,n)$. 
\endproclaim

The proof of Theorem 2.3 was successfully altered to yield Theorem
2.2. In addition when conditions (i)--(iii) listed above apply to $A$
and $N$ in Theorem 2.2, then MacMahon's modular partitions may be
utilized to show the equivalence of the two results. 

Finally it should be mentioned that C.~Bessenrodt \cite {20} has proved
a generalization of Theorem 2.2 using purely combinatorial methods.
Also K.~Alladi and B.~Gordon \cite {1} have a nice study of related
continued fractions when $A$ is the two element set $\{a_1, a_2\}$.
\vskip1cm

{\bf 3.~Euler's ``Exemplum Memorabile Inductionis Fallacis"}
\vskip0,5cm

In \cite {12}, \cite {13} and \cite {14} a model generalizing the hard
hexagon model was solved using several $q$--analogs of trinomial
coefficients. The trinomial coefficients $\binom {m} {j}_2$ may be
defined by
$$\suml _{j=-m} ^{m}{\binom {m} {j}}_2 x ^j=(1+x+x ^{-1})
^m. \tag 3.1$$
In this way for given $m$, the largest coefficient is $\binom {m}
{0}_2$. These numbers fit a modified Pascal triangle
$$\matrix \format \c\kern3pt & \c & \kern3pt \c & \kern3pt \c & 
\kern3pt \c & \kern3pt \c & \kern3pt \c & \kern3pt \c & \kern3pt \c & 
\kern3pt \c & \kern3pt \c \\
\hphantom{11}&\hphantom{11}&\hphantom{11}&\hphantom{11}&\hphantom{11}&1
&\hphantom{11}&\hphantom{11}&\hphantom{11}&\hphantom{11}&\hphantom{11}\\
&&&&1 & 1 & 1\\
&&&1 & 2 & 3 & 2 & 1\\
&&1 & 3 & 6 & 7 & 6 & 3 & 1\\
&1 & 4 & 10 & 16 & 19 & 16 & 10 & 4 & 1\\
1 & : & : & : & : & : & : & : & : & : & 1
\endmatrix$$
Euler discovered a sufficiently mysterious aspect of the central
column of this array that he wrote a short note entitled, ``Exemplum
Memorabile Inductionis Fallacis" (A Remarke Example of Misleading
Induction). 

Euler first computed $\binom {m} {0}_2$ for $0\le m \le 9$:
$$1,1,3,7,19,51,141,393,1107,3139,\dots $$ 
He then tripled each entry in a row shifted one to the right: 
$$\matrix \format\r&\r&\r&\r&\r&\r&\r&\r&\r&\r&\r\\
1,&1,&3,&7,&19,&51,&141,&393,&1107,&3139,&\dots\\
&3,&3,&9,&21,&57,&153,&423,&1179,&3321,\dots,
\endmatrix$$
and starting with the first two--entry column, he subtracted the first
row from the second:
$$2,0,2,2,6,12,30,72,182,\dots ,$$
each of which may be factored into two consecutive integers: 
$$1 \cdot 2,\ 0\cdot 1,\ 1\cdot 2,\ 1\cdot 2,\ 2\cdot
3,\ 3\cdot 4,\ 5\cdot 6, \ 8\cdot 9,\ 13 \cdot 14,\dots.$$
The first factors make up the Fibonacci sequence $F_n$ defined by
$F_{-1}=1$, $F_0=0$, $F_n=F_{n-1}+F_{n-2}$ for $n>0$.

Surprisingly, however, this marvelous rule
$$3\binom {m+1} {0}_2-\binom {m+2} {0}_2=F_m (F_m+1),\quad -1\le m\le 7,
\tag 3.2$$
is {\it false} for $m>7$. In order to understand (3.2) we define
$$E_m(a,b)=\suml _{\lambda=-\infty} ^{\infty} \( \binom {m}
{10\lambda+a}_2-\binom {m} {10\lambda+b}_2 \).\tag 3.3$$
As part of Theorem 3.1, we show that
$$2E_{m+1}(0,1)=F_m(F_m+1),\tag 3.4$$
from which (3.2) follows by inspection.

\proclaim{Theorem 3.1}
$$\gather
2E_m(1,2)=2E_{m-1}(0,3)=2E_{m+1}(0,1)=F_m(F_m+1),\tag 3.5\\
E_m(1,4)=E_{m+1}(2,3)=F_{m+1}F_m,\tag 3.6\\
2E_m(3,4)=2E_{m-1}(2,5)=2E_{m+1}(4,5)=F_m(F_m-1), \tag 3.7\\
2E_m(1,3)=F_{2m}+F_{m},\tag 3.8\\
2E_m(2,4)=F_{2m}-F_{m}, \tag 3.9\\
2E_m(1,5)=F_{2m+1}-F_{m-1}, \tag 3.10\\
2E_m(0,4)=F_{2m+1}+F_{m-1}, \tag 3.11\\
2E_m(0,2)=F_{2m-1}+F_{m+1}, \tag 3.12\\
2E_m(3,5)=F_{2m-1}-F_{m+1}, \tag 3.13\\
E_m(0,5)=F_{2m-1}+F_mF_{m-1}. \tag 3.14
\endgather$$
\endproclaim


\demo {Sketch of Proof} We note that 
$$\gather
E_m(a,b)=-E_m(b,a), \quad E_m(10r+a, 10s+b)=E_m(a,b), \tag 3.15\\
E_m(10-a,b)=E_m(a,b), \tag 3.16\\
E_m(10-a,b)=E_m(a,b)=E_m(a,10-b), \tag 3.16
\endgather$$
and that
$$E_m(a,b)=E_{m-1}(a,b)+E_{m-1}(a-1,b-1). \tag 3.17$$

Equations (3.15)--(3.17) totally define $E_m(a,b)$ together with
appropriate initial values. The rest follows by induction.\quad \quad
\qed
\enddemo

As a corollary of Theorem 3.1 it is easy to show that
$$3\binom {m+1} {0}_2-\binom {m+2} {0}_2=2\binom {m+1} {0}_2-2\binom
{m+1} {1}_2$$
$$\hskip4,3cm =2E_{m+1}(0,1)\quad \quad  (\text {for}\ m\le 7)$$
$$\hskip4,4cm =F_m(F_{m+1}+1)\quad \quad (\text {by (3.5)}).$$
The natural question that arises is: Are there $q$--analogs of at
least portions of Theorem 2.1 and if so, what are the implications for
the Rogers--Ramanujan type identities?

In \cite {10}, $q$--analogs of (3.8)--(3.11) were found. For example,
we recall Schur's polynomials $G_1(q)=G_2(q)=1,\ G_n(q)=G_{n-1}(q)+q
^{n-2}G_{n-2}(q)$ for $n>2$. Schur \cite {31} showed that
$$ G_{n+1}(q)=\suml _{\lambda=-\infty} ^{\infty}(-1) ^\lambda
q^{\lambda (5\lambda +1)/2}\bmatrix n\\ \lfloor \frac {n-5\lambda}
{2}\rfloor \endbmatrix_q.  \tag 3.18$$
where $\lfloor x \rfloor$ is the greatest integer $\le x$ and
$$\bmatrix A\\ B \endbmatrix = \bmatrix A\\ B \endbmatrix_q=\cases
\frac {(1-q ^A)(1-q ^{A-1})\cdots (1-q ^{A-B+1})} {(1-q ^B)(1-q
^{B-1})\cdots (1-q)}& 0\le B\le A ,\\
0&\text {otherwise.}\endcases
$$
The $q$--analog of trinomial coefficients appropriate for our
discussion here is
$$\binom {m;B;q} {A}_2=\suml _{j\ge 0} ^{}q ^{j(j+B)}\bmatrix m\\
j\endbmatrix \bmatrix m-j\\ j+A\endbmatrix. \tag 3.19$$
Note that
$$\binom {m;B;1} {A}_2=\binom {m} {A}_2. \tag 3.20$$
Schur [31] deduced the first Rogers-Ramanujan identity as a limiting
case of (3.18). For $q$--analogs of (3.11) and (3.10) respectively, we
discover that
$$\multline
\frac {1} {2}(G_{2m+1}(q ^{1/2}) +G_{2m+1}(-q ^{1/2}))\\
=\suml _{\la=-\infty} ^{\infty} q ^{30 \la ^2 - 2 \la} \binom
{m;10\la;q} {10\la}_2 - \suml _{\la=-\infty} ^{\infty} q ^{30\la
^2+22\la+4} \binom {m;10 \la +4;q} {10 \la +4}_2, 
\endmultline\tag 3.21$$
$$\multline
\frac {q ^{-1/2}} {2} (G_{2m+1}(q ^{1/2})-G_{2m+1}(-q ^{1/2}))\\
=\suml _{\la=-\infty} ^{\infty} q ^{30\la ^2+8\la} \binom
{m;10\la+1;q} {10\la+1}_2 - \suml _{\la=-\infty} ^{\infty} q ^{30\la
^2+32\la+8} \binom {m;10\la+5;q} {10\la+5}_2. 
\endmultline\tag 3.22$$

In contrast with Schur's identity (3.18), we find [9] that
$$G_{n+1}(q)=\suml _{\la=-\infty} ^{\infty} (-1) ^\la q
^{\la(5\la+1)/2} \binom {n;\lfloor \frac {5\la+1} {2} \rfloor;q} {\lfloor
\frac {5\la+1} {2} \rfloor}_2. \tag 3.23$$

From (3.23) one can again deduce the first Rogers-Ramanujan
identity as a limiting case; however, the simple replacement of the
Gaussian polynomial in (3.18) by a $q$-trinomial in (3.23) is at the
very least quite surprising.

Furthermore the limiting cases of (3.21) and (3.22) do not lead to the
first Rogers-Ramanujan identity, but rather to the Rogers-Ramanujan
series split into even and odd parts. Namely
$$\multline
1+\suml _{j=1} ^{\infty} \frac {q ^{j ^{2}}} {(1-q) (1-q^2) \cdots
(1-q^j)}= \frac {1} {\prodl _{n=1} ^{\infty} (1-q ^{2n})}\\
\Biggl \{\suml _{\la =-\infty} ^{\infty} (q ^{60 \la  ^2-4 \la}-q
^{60  \la^2 + 44 \la + 8})+ q \suml _{\la=-\infty} ^{\infty}(q ^{60
\la ^2 + 16 \la} -q ^{60 \la ^2 + 64 \la + 16}) \Biggr \}.
\endmultline$$

The main riddle concerning all this is precisely the combinatorics. In
[3], [4; Ch.~ 9], [15], [21], we see clearly the partition-theoretic
significance of (3.18). However the $q$-trinomial version (3.23) is
still a complete conbinatorial mystery.
\vskip0,6cm

{\bf 4.~ Ramanujan}
\vskip0,3cm

The work in the last several years stemming from Ramanujan's
discoveries has truly been amazing. Bruce Berndt at the University of
Illinois has been the chief architect of much of the work. He is
bringing out edited versions [17], [18], [19] of Ramanujan's famous
Notebooks. In addition, the book Ramanujan Revisited [7], edited by
Berndt and others, describes recent research on a number of topics
related to Ramanujan's work, and the Lost Notebook has been published
in photostatic reproduction by Springer--Narosa in 1987. The Mock Theta
Conjectures arising from the Lost Notebook were described as follows
by Ian Stewart in Nature [33]:
\vskip0,3cm

{\leftskip1cm\rightskip1cm
\noindent
One of the most unusual people in the annals of mathematical research
is Srinivasa Ramanujan, a self-taught Indian mathematician whose
premature death left a rich legacy of unproved theorems. Ramanujan was
preeminent in an unfashionable field --- the manipulation of formulas.
He tended to state his results without proofs --- indeed on many
occasions it is unclear whether he possessed proofs in the accepted
sense --- yet he had an uncanny knack of penetrating to the heart of
the matter. Over the years, many of Ramanujan's claims have been
established in full rigour, although seldom easily. The most recent
example, the `mock theta conjectures', is especially striking, because
the results in question were stated in Ramanujan's final
correspondence with his collaborator Godfrey H. Hardy. The conjectures
have recently been proved by Dean Hickerson [26]
\newline
\dots The proof involves delicate manipulations of infinite series of a kind
that would have delighted Ramanujan. The astonishing complexity of the
proof underlines, yet again, the depth of Ramanujan's genius. It is
very hard to see how anyone could have been led to such results
without getting bogged down in the fine detail. Ramanujan was the
formula man {\it par exellence}, operating in a period when formulas were
out of fashion. Today's renewed emphasis on combinatorics, inspired in
a part by the digital nature of computers, has provoked a renewed
interest in formula manipulations. The half--forgotten ideas of
Srinivasa Ramanujan are breathing new life into number theory and
combinatorics. 
\par}
\vskip0,3cm

In what follows we provide a sketch of recent work arising from
Ramanujan's Notebooks.

In [22], D. Bressoud gave a very simple proof of the Rogers--Ramanujan
identities. We may for purposes of example slightly rephrase his
proof. Namely, he noted that 
$$\alpha _n= \cases
1 & \text {if} \quad   n=0 \\
(-1) ^n \left ( zq^{ \binom {n} {2} }
+ z  ^{-n}q ^{\binom  {n+1} {2}} \right) &\text {if}\quad  n>0, 
\endcases \tag 4.1$$
and
$$\beta_n= \frac {(z)_n\,(q/z)_n} {(q)_{2n}} \tag 4.2$$
form a Bailey pair [6; p.~26], i.e\.
$$\beta_n=\suml _{j=0} ^{n} \frac {\alpha_j} {(q)_{n-j} \,(aq)_{n+j}}
\tag 4.3$$
where
$$(A)_n=(A;q)_n=(1-A)(1-Aq) \cdots (1-Aq ^{n-1}). \tag 4.4$$

A weak iterated version of Bailey's Lemma asserts that if $\alpha_n$
and $\beta_n$ form a Bailey pair, then 
$$\frac {1} {(aq;q)_\infty} \suml _{n=0} ^{\infty} q ^{kn ^2} a ^{kn}
\alpha _n
= \suml _{n_k \geq \cdots \geq n_1 \geq 0 } ^{} \frac {\alpha ^{n_1+
\cdots + n_ a} q ^{n ^2_1+ \cdots + n ^2_k} \beta_{n_1}}
{(q)_{n_k-n_{k-1}} (q)_{n_{k-1}-n_{k-2}} \cdots (q)_{n_2-n_1}}.
\tag4.5$$

Bressoud's proof [22] can be viewed as setting $z=1$ in (4.1) and
(4.2) and inserting the resulting pair in (4.5) with $k=2$.
\newline
If instead, we take (4.1) and (4.2) as they are and insert them into
(4.5) with $k=1$ and $\alpha=1$, we find

$$\align \suml _{n=0} ^{\infty} \frac {(z)_n\,(q/z)_n \,q ^{n ^2}}
{(q)_{2n}}&=
\frac {1 + \suml _{n=1} ^{\infty} (-1) ^n \left(z ^n q ^{\binom {n}
{2}} + z ^{-n} q ^{
\binom {n+1} {2}} \right)q ^{n ^2}} {(q)_\infty} \\
&=\frac {\suml _{n=-\infty} ^{\infty} (-z) ^n q ^{n(3n-1)/2}}
{(q)_\infty} \tag 4.6 \\
&=\frac {(q ^3;q ^3)_\infty (zq;q ^3)_\infty (z ^{-1}q ^2;q
^3)_\infty} {(q)_\infty }, \endalign$$
a result from the Lost Notebook.

If we differentiate each entry in (4.6) and then set $z=1$, we deduce
$$ \align \suml _{n=1} ^{\infty} \frac {(1-q) (1-q ^2) \cdots (1-q ^{n-1}) q
^{n ^2}} {(1-q ^{n+1}) (1-q ^{n+2}) \cdots (1-q ^{2n})} &= \frac {\suml
_{n=-\infty} ^{\infty} (-1) ^n nq ^{n(3n-1)/2}} {(q)_\infty} \\
&=\suml _{n=0} ^{\infty} \Biggl (\frac {q ^{3n+1}} {1-q ^{3n+1}} -
\frac {q ^{3n+2}} {1-q ^{3n+2}} \Biggr ) \\  &=\suml _{n=1} ^{\infty}
q ^n \Biggl (\suml _{d/n} ^{} \biggl (\frac {d} {3} \biggr ) \Biggr )
, \tag 4.7 \endalign $$
where $\left(\frac {d} {3} \right)$ is the Legendre symbol.

Thus just beneath the surface of (4.6) is a $q$-series (namely the
left--hand side of (4.7)) with multiplicative coefficients all
non--negative, all indeed $O(n^2)$.

This suggests that the underlying combinatorics of (4.7) is well worth
a look, and we shall not be disappointed. Following the program
outlined in [5], we consider
$$f(t,q)=\suml _{n=1} ^{\infty} \frac {t ^{2n}q ^{n ^2}(-tq)_{n-1}}
{(t)_{n+1}\,(t ^2q;q ^2)_n} . \tag 4.8$$
The function $f(t,q)$ was constructed to both satisfy a first order
nonhomogeneous $q$-difference equation
$$f(t,q)=\frac {t ^2q} {(1-t) (1-tq) (1-t ^2q)} + \frac {t ^2q(1+tq)}
{(1-t) (1-t ^2q)} f(tq,q), \tag 4.9$$
and to reduce in the case $t=-1$ to the left--hand side of (4.7):
$$f(-1,q)=\frac {1} {2} \suml _{n=1} ^{\infty} \frac {q ^{n ^2}
(q)_{n-1}} {(q ^{n+1};q)_n}.$$

If the magic of Ramanujan's mathematics is operating here, then
$f(t,q)$ should be an interesting generating function of polynomials
(in $q$), and $f(q,q ^2)$, $\liml _{t\to 1^-} f(t,q) (1-t)$ should
also exhibit interesting structure. In this regard, we find
$$f(t,q)=\suml _{n=2} ^{\infty} t ^np_n(q)$$
with
$$p_n(q)=\suml _{\la=-\infty} ^{\infty} q ^{(2\la+1)(3\la+1)} \bmatrix
n \\ \lfloor \frac {n} {2} \rfloor -3 \la -1 \endbmatrix,$$
$$\liml _{t \to 1^-} (1-t) f (t,q)= \suml _{n=1} ^{\infty} \frac {(1+q)
^2 (1+q ^2) ^2 \cdots (1+q ^{n-1}) ^2 (1+q ^n) q ^{n ^2}} {(1-q) (1-q
^2) \cdots (1-q ^{2n})} $$
$$=q \prodl _{n=1} ^{\infty} \frac {(1-q ^{12n}) (1+q ^{12n-1}) (1+q
^{12n-11})} {(1-q ^n)},$$
and finally
$$\frac {1} {1-q} + (1+q)f(q,q ^2)
 =\suml _{n=0} ^{\infty} \frac {q ^{2n ^2+2n} (-q;q ^2)_n} {(q)_{2n+1}
(-q ^2;q ^2)_n}
 =\prodl _{n=1} ^{\infty} \frac {(1-q ^{6n}) (1+q ^{6n-1}) (1+q
^{6n-5})} {(1-q ^{2n})}.$$

It should be added that the above discoveries all resulted from a
consideration of four seemingly benign identities of Ramanujan [11].
The simplest of which is 
$$\suml _{n=1} ^{\infty} \frac {(-1) ^{n-1}nq ^{\binom {n+2} {2}} (1-q
^n)} {(1+q ^n) ^2}= \biggl (\suml _{n=-\infty} ^{\infty} (-1) ^nq ^{n
^2}  \biggr ) ^2 \suml _{n=1} ^{\infty} \frac {nq ^{\binom {n+1}
{2}}}
{1-q ^n}.$$
The proof of this result relies in an essential way on Bressoud's
Bailey pair (4.1), (4.2) differentiated with respect to $z$ and with
$z$ then set equal to $1$ together with a general $q$-hypergeometric
identity of Bailey [25, p.~42, eq.~(2.10.10)].
\vskip0,5cm

{\bf 5.~ Conclusion}
\vskip0,3cm

These lectures being expository lack the details necessary for a full
understanding of the underlying proofs. In this regard, Section 2 is
an exposition of [16]; Section 3 of [10], and Section 4 of [5], [8]
and [11]. Related background material may be found in [2] and [4] for
Section 2, [9] for Section 3 and [6; Ch.9] for Section 4.
\vskip0,5cm

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\vskip0,3cm
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\enddocument

