\magnification=\magstep1
\input amstex
\documentstyle{amsppt}
\NoRunningHeads
\NoBlackBoxes


\def\c{\cite}
\def\Z{\Bbb Z}
\def\cdc{,\ldots,}
\def\fr{\frac}
\def\os{\overset{\infty}\to{\underset{n= - \infty}\to \sum}}
\def\oj{\overset{\infty}\to{\underset{j= 0}\to \sum}}
\def\qq{q;q}
\def\an{\alpha_n} 
\def\bn{\binom{n}{2}}


\topmatter
\title
Ramanujan's Method in $q$-Series Congruences
\endtitle

\author
by \\
\\
George E. Andrews\footnote"{$^{(1)}$}"{Partially supported by National Science 
Foundation Grant DMS-8702695-04 \ \ \ \ \ \ \ \ \ \ \ \ \ } and Ranjan Roy
\endauthor

\dedicatory
Written in honor of Herb Wilf's $65^{th}$ birthday
\enddedicatory

\abstract{We show that the method developed by Ramanujan to prove 
$5|p (5n + 4)$ and $7|p (7n + 5)$ may, in fact, be extended to a wide 
variety of $q$-series and products including some with free parameters.}
\endabstract

\endtopmatter


\document

\baselineskip20pt

\subhead{1. \ Introduction} \endsubhead

Ramanujan \c{11} is the discoverer of the surprising fact that the partition 
function, $p(n)$, satisfies numerous congruences.  Among the infinite family 
of such congruences, the two simplest examples are
$$
	p (5n+4) \equiv 0 \quad \pmod {5}
\tag1.1
$$
and 
$$
	p (7n+5) \equiv 0 \quad \pmod {7}.
\tag1.2
$$


Ramanujan used an ingenious and elementary argument to prove these 
congruences which relied on Jacobi's famous formula \c{10; last eqn. p.5}:
$$
	(q; q)^3_{\infty} = \overset{\infty}\to{\underset{n=1}\to \prod} 
	( 1-q^n)^3 = \sum^{\infty}_{j=0} (-1)^j (2j+1) q^{j(j+1)/2}, 
\tag1.3
$$
where 
$$
	(A)_N = (A;q)_N = \overset{N-1}\to{\underset{j=0}\to \prod} 
	(1-Aq^j).
\tag1.4
$$
A rather more general result of this nature was proved in \c{3; p. 27,
Th. 10.1} to account for certain congruences connected with generalized 
Frobenius partitions.  

Indeed J. M. Gandhi \c{7}, \c{8}, \c{9}, J. Ewell \c{5}, L. Winquist \c{12} 
and many others (cf., Gupta \c{10; Sec. 6.3}) have proved partition 
function congruences based on this idea.  In all these theorems, the 
underlying generating functions were either modular forms or simple linear 
combinations thereof.

The point of this paper is to show that Ramanujan's original method is 
applicable to an infinite number of congruence theorems including many 
non-modular functions defined by $q$-series.

Our main result is:

\proclaim
{Theorem 1} \ Suppose $p$ is a prime $> 3$, and $0 < a < p$ and $b$ are 
integers.  Also, $- a$ must be a quadratic nonresidue mod $p$.  Suppose 
$\{\alpha_n\}^{\infty}_{n =  - \infty} = \{\an (z_1, z_2\cdc z_j)\}$ 
is a doubly infinite 
sequence of Laurent polynomials over $\Z$ with variables $z_1 \cdc
z_j$ independent of $q$.  Then there is an integer $c$ such that the
coefficient of $z^{m_1}_1 z^{m_2}_2 \cdots z^{m_j}_j \mathbreak
q^{pN}$ in
$$
	\fr{q^c \overset{\infty}\to{\underset{n= -\infty} \to\sum} 
	\alpha_n q^{a \binom{n}{2} + bn}}
	{(q; q)^{p-3}_{\infty}}
\tag1.5
$$
is divisible by $p$.  For each integer $m$, we shall denote by 
$\overline m$ the multiplicative inverse of $m$ mod $p$.  The integer 
$c = c_p (a, b)$ may be chosen as the least nonnegative integer congruent to 
$\bar 8 (a(2b \bar a - 1)^2 +1)$ mod $p$.
\endproclaim

In Section 2, we shall prove this result.  In Section 3, we examine the 
implications of Theorem 1 for a variety of modular forms.  In Section 4, we 
collect a number of congruences for the coefficients in several $q$-series.  


\subhead
{2. \ The Proof of Theorem 1} 
\endsubhead

With the various hypotheses of the theorem, 
we note that

\newpage

$$
	\fr{q^c \overset{\infty}\to{\underset{n= - \infty}\to \sum} \an 
	q^{a\bn + bn}}{(\qq)^{p-3}_{\infty}}  \tag2.1
$$
\vskip -.2in
$$
\aligned
	&= \fr{q^c \overset{\infty}\to{\underset{n= - \infty}\to \sum} \oj 
	(-1)^j (2j+1) \an q^{a\bn + bn + j (j+1)/2}}
	{(\qq)^p_{\infty}} \\
%
	&\equiv \fr{q^c \overset{\infty}\to{\underset{n= - \infty}\to \sum} 
	\oj (-1)^j (2j+1) \an q^{a\bn + bn + j (j+1)/2}}
	{(q^p; q^p)_{\infty}} \pmod {p}.
\endaligned
$$

We see that in this last expression the denominator is a function of
$q^p$.  Let us now examine the exponent of $q$ in the numerator; for
ease of computation we multiply by 8:
$$
	8 \left(c + a \bn + bn + j (j+1)/2 \right) \tag2.2
$$
\vskip -.2in
$$
\aligned &= 8 c + a (4n^2 - 4n) + 8bn + 4j^2 + 4j \\
%
	&\equiv a (2n + 2b\bar a-1)^2 + (2j+1)^2 \quad \pmod {p} 
\endaligned
$$

Now we observe (by the definition of $c$) that if $j \equiv (p-1)/2$ mod $p$ (i.e. $(2j+1) \equiv 0 
\pmod {p}$), then the last expression above is congruent to $0$ mod $p$ 
precisely when 
$$
	n \equiv (1- 2b\overline a) \overline 2 \equiv \fr{p+1}2 - b 
	\overline a \pmod {p}.
$$

If $j \not\equiv \fr{p-1}2 \pmod {p}$, then the last expression in
(2.2) can never be congruent to zero mod $p$ because by the conditions
on $a$
$$
	-a (2n + 2b \overline a - 1)^2
$$
is either $0$ or a quadratic nonresidue mod $p$ and so cannot be congruent to 
a quadratic residue (i.e. $(2j+1)^2$) mod $p$.

Hence the coefficients of $q^{pN}$ in (2.1) will all be linear 
combinations over $p\Z$ of various $\an$ (which are Laurent polynomials in 
several variables over $\Z$).   $\square$

\subhead{3. \ Modular Forms} \endsubhead

Ramanujan, Ewell, Gandhi (and probably many others) have proved instances of 
Theorem 1 (as mentioned in Section 1).

Congruence (1.1) follows from Theorem 1 with $p=5, a=3, b=1, c_5 (3,
1) = 1$ and $\alpha_m = (-1)^m$.  Congruence (1.2) follows from
Theorem 1 with $p=7, a=b=1, c_7 (1,1) = 2$ and $\alpha_m = (-1)^m (2m
+ 1)$ if $m\geqq 0, \alpha_m = 0$ if $m < 0$.

Gandhi's Theorem IV in \c{7} corresponds to $\alpha_m = \delta_{m,0}$,
while Theorem 2 in \c{8} corresponds to $a=b=1$ and $\alpha_m = (-1)^m
(2m+1)$ if $m \geqq 0$, $\alpha_m = 0$ if $m < 0$.  Finally, Theorem 4
in \c{8} corresponds to $a=3, b=1$ and $\alpha_m = (-1)^m$.

Theorem 10.1 of \c{3} is the case $p=5, a=2, b=1, c_5 (2,1) = 2$; in that 
result the $\alpha_m$ were assumed to be $0$ if $m < 0$ and to be integers 
otherwise.

The generality of Theorem 1 allows for a variety of other modular forms.  To 
illustrate, we consider
$$
	\overset{\infty}\to{\underset{n=0}\to\sum} V_n q^n = 
	\fr{\overset{\infty}\to{\underset{n=0}\to\sum} p(n) q^n}
	{\overset{\infty}\to{\underset{n=0}\to\sum} (-1)^n r_2 (n)q^n},
$$
where $r_2 (n)$ is the number of representations of $n$ as a sum of
two squares.  We note that
$$
\aligned
	\overset{\infty}\to{\underset{n=0}\to\sum} V_n q^n &= 
	\fr1{(q)_{\infty} \left(\os (-1)^n q^{n^2}\right)^2} \\
%
	&= \fr{(-q)^2_{\infty}}{(q)^3_{\infty}} \\
%
	&= \fr{(q^2; q^2)_{\infty}}{(q)^4_{\infty} (q;q^2)_{\infty}} \\
%
	&= \fr{\overset{\infty}\to{\underset{m=0}\to\sum}  q^{m(m+1)/2}}
	   {(q)^4_{\infty}}.
\endaligned
$$

Now by Theorem 1 with $p=7, a=b=1, c_7 (1,1) =2, \alpha_m = 1$ if $m\ge 0$, 
and $0$ if $m < 0$, we see that 
$$
	V_{7m+5} \equiv 0 \;\; \pmod {7}.
$$

\subhead{4. \ $q$-Series} \endsubhead

Of course, our point here is not to extend slightly Ramanujan's basic idea to 
a few more modular forms.  Rather we hope to illustrate its applicability to 
$q$-series.

\proclaim
{Theorem 2} \ For any prime $p\equiv 3 \pmod {4}$ with $4c \equiv 1
\pmod {p}$, the coefficient of $z^m q^{pn-c}$ in 
$$
	\fr{(zq)_{\infty}}{(q)^{p-4}_{\infty}}
	\overset{\infty}\to{\underset{n=0}\to\sum} 
	\fr{q^n}{(q)_n (zq)_n}
$$
is divisible by $p$.
\endproclaim

\demo
{Proof} \ By Heine's transformation \c{1; Cor. 2.3, p. 19} 
$$
	\overset{\infty}\to{\underset{n=0}\to\sum} \fr{q^n}{(q)_n (zq)_n}
	= \fr1{(q)_{\infty} (zq)_{\infty}} 
	\overset{\infty}\to{\underset{n=0}\to\sum} (-1)^n q^{n (n+1)/2} z^n.
$$
Now apply Theorem 1 with $a=b=1$ and $c \equiv \overline 4 \pmod {p}$.  
$\square$
\enddemo


\proclaim
{Theorem 3} \ For any prime $p \equiv 5$ or $7 \pmod {8}$ with $8 c \equiv 
1 \pmod {p}$, the coefficient of $z^m q^{pn-c}$ in 
$$
	\fr1{(q)^{p-3}_{\infty}} \overset{\infty}\to{\underset{n=0}\to\sum} 
	\fr{(z)_{n+1} z^n}{(-zq)_n}
$$
is divisible by $p$.
\endproclaim

\demo
{Proof} \ By the Rogers-Fine identity \c{6; p. 15, eqn. (14.31)} 
$$
	\overset{\infty}\to{\underset{n=0}\to\sum} \fr{(z)_{n+1}z^n}
	{(- zq)_n} = 1+2 \underset{n\ge 1}\to\sum  (-z^2)^n q^{n^2}.
$$
Now apply Theorem 1 with $a=2, b=1$ noting that for $p\equiv 5 \pmod {8}$ $2$ 
is a non-quadratic residue, and for $p\equiv 7 \pmod {8}$, $2$ is a quadratic 
residue.  Also we must have $c\equiv \overline 8 \pmod {p}$.  $\square$
\enddemo


\proclaim
{Theorem 4} \ For any prime $p\equiv 5$ or $11 \pmod {12}$, the coefficient of 
$q^{pn-(p+1)/2}$ in
$$
	\fr1{(q)^{p-4}_{\infty}} \overset{\infty}\to{\underset{n=0}\to\sum} 
	\left[\matrix 2n \\ n \endmatrix \right] q^n
$$
(where $\left[\matrix A \\ B \endmatrix \right] = (q)_A/ ((q)_B (q)_{A-B})$ 
is the $q$-binomial coefficient) is divisible by $p$.
\endproclaim
 
\demo
{Proof} \ By Lemma 3 of \c{2; p. 159}, 
$$
	\fr1{(q)_{\infty}} \overset{\infty}\to{\underset{n=0}\to\sum} (-1)^n 
	q^{3n(n+1)/2} = \overset{\infty}\to{\underset{n=0}\to\sum} 
	\left[\matrix 2n \\ n \endmatrix \right] q^n.
$$
Now apply Theorem 1 with $a=b=3$ noting that for $p\equiv 5 \pmod {12}$, $3$ is 
a non-quadric residue and for $p\equiv 11 \pmod {12}$, $3$ is a quadratic 
residue.
\enddemo

\proclaim
{Theorem 5} \ For any prime $p\equiv 5$ or $11 \pmod {12}$ and
$6c\equiv 1 \pmod {p}$, the coefficient of $q^{pn-c}$ in 
$$
	\fr1{(q)^{p-4}_{\infty}} \, 
	\overset{\infty}\to{\underset{n=1}\to\sum} \,
	\fr{q^{n^2}}{(1-q^n)\left[\matrix 2n \\ n \endmatrix \right]}
$$
is divisible by $p$.
\endproclaim

\demo
{Proof} \ By (5.1) of \c{4; p. 272}
$$
\gathered
	\overset{\infty}\to{\underset{n=1}\to\sum} \, 
	\fr{q^{n^2}}{(1-q^n)\left[\matrix 2n \\ n \endmatrix \right]} \\
%
	= \overset{\infty}\to{\underset{n=1}\to\sum} 
	\fr{q^{n^2}(q)_{n-1}}{(q^{n+1})_n} = \fr1{(q)_{\infty}} 
	\overset{\infty}\to{\underset{n= - \infty}\to\sum} (-1)^{n-1} 
	n \, q^{n(3n-1)/2}.
\endgathered
$$
Now apply Theorem 1 with $a=3, b=1$.  As in Theorem 4, $p$ must be $\equiv 5$ 
or $11$ mod $12$, and now $6c =1 \pmod {p}$.   $\square$ 
\enddemo

\subhead{5. \ Conclusion} \endsubhead

There are undoubtedly significant extensions of the ideas we have presented 
here.  We note that Winquist's proof that 
$$
	p (11 n + 6) \equiv 0 \pmod {11}
$$
does not fit into Theorem 1.  However now that the application of
(1.3) to Theorem 1 has been made, it should be possible to find a
variety of congruences for the coefficients of other $q$-series.  In
addition, the function given in Theorem 1 may be multiplied by any
function $f_1(q) \in \Z [[q]]$ for which
$$
	f_1(q) \equiv f_2 (q^p) \; \pmod {p}.
$$

Finally, it may well be asked what happens when $a$ is a quadratic residue mod
$p$ and $p\equiv 1 \pmod {4}$ or $a$ is a quadratic non-residue and $p\equiv 3
\pmod {4}$.  It is not difficult to show that our analysis produces
indices $j \not\equiv (p-1)/2 \pmod {p}$ which make the exponent on
$q$ congruent to zero mod $p$ irrespective of $c_p (a, b)$.
Consequently one would have to invoke special conditions on the
$\alpha_m$ to produce coefficients that are multiples of $p$.


\Refs

\ref
\no  1
\by  G.E. Andrews
\paperinfo  The Theory of Partitions, 
Encyclopedia of Math. and Its Applications, Vol. 2, Addison-Wesley, Reading, 
1976 (Reissued:  Cambridge University Press, Cambridge, 1985)
\endref

\ref
\no  2
\by  G.E. Andrews
\paper Ramanujan's ``lost'' notebook: I.  partial $\theta$-functions 
\jour  Adv. in Math.
\vol  41
\yr  1981
\pages  137--172
\endref

\ref
\no  3
\by  G.E. Andrews
\paperinfo  Generalized Frobenius partitions, Memoirs of the Amer. Math. Soc., 
49(1984), No. 301, iv+, 44 pp
\endref

\ref
\no  4
\by  G.E. Andrews
\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  5
\by  J.A. Ewell
\paper  Completion of a Gaussian derivation
\jour  Proc. Amer. Math. Soc.
\vol  84
\yr  1982
\pages  311--314
\endref

\ref
\no  6
\by  N.J. Fine
\paperinfo  Basic Hypergeometric Series and Applications, 
Math. Surveys No. 27, Amer. Math. Soc., Providence, 1988
\endref

\ref
\no  7
\by  J.M. Gandhi
\paper  Congruences for $p_r (n)$ and Ramanujan's $\tau$ function
\jour  Amer. Math. Monthly
\vol  70
\yr  1963
\pages  265--274
\endref

\ref
\no  8
\by  J.M. Gandhi
\paper  Generalization of Ramanujan's congruences $p(5m+4) \equiv 0 \pmod {5}$ 
and $p (7m+5) \equiv 0 \pmod {7}$
\jour  Monatsch. Math.
\vol  69
\yr  1965
\pages  389--392
\endref

\ref
\no  9
\by  J.M. Gandhi
\paper  Some congruences for $k$-line partitions of a number
\jour  Amer. Math. Monthly
\vol  74
\yr  1967
\pages  179--181
\endref

\ref
\no  10
\by  H. Gupta
\paper  Partitions -- a survey
\jour  Journal of Res. of Nat. Bur. Standards - B Math. Sciences
\vol  74B
\yr  1970
\pages  1--29
\endref

\ref
\no  11
\by  S. Ramanujan
\paperinfo  Some properties of $p(n)$, the number of partitions of $n$, Proc. 
Cambridge Phil. Soc., 19(1919), 207--210  (Reprinted: Coll. Papers., Cambridge Univ. Press, Cambridge, 1927, Reissued:  Chelsea, New York, 1962)
\endref

\ref
\no  12
\by  L. Winquist
\paper  An elementary proof of $p(11 m + 6) \equiv 0 \pmod {11}$
\jour  J. Combinatorial Theory
\vol  6
\yr  1969
\pages  56--59
\endref
\endRefs

\vskip .2in

\baselineskip 13pt

\line{The Pennsylvania State University\hfil}
\line{University Park, PA  16802\hfil}
\line{email:  andrews\@math.psu.edu\hfil}
\vskip .02in
\line{and\hfil}
\vskip .02in
\line{Beloit College\hfil}
\line{Beloit, Wisconsin\hfil}
\line{email:  royr\@beloit.edu \hfil}
\enddocument