MORE LACUNARY PARTITION FUNCTIONSJEREMY LOVEJOY

1. introduction
A partition of n is a nonincreasing sequence of natural numbers whose sum
is n. The rank of a partition is the largest part minus the number of parts.
Inspired by Ramanujan’s lost notebook, Andrews, Dyson, and Hickerson were
led to study certain partitions weighted according to the rank [5]. They
showed that if R± (n) is equal to the number of partitions of n into distinct
parts with even (odd) rank, then(i) R+ (n) − R− (n) = k has infinitely many solutions for any integer k.
(ii) R+ (n)−R− (n) is lacunary. That is, #{k ≤ n : R+ (k) 6= R− (k)}/n →0 as n → ∞.
This was rather unlike anything seen before, and there are still only a few
examples of partition functions exhibiting such behavior.
Recently we focused on the second property and discussed how to construct infinite families of lacunary q-series which are not theta functions. We
interpreted two such families partition theoretically in terms of the Durfee
rectangle dissection of a Ferrers diagram [7]. Although this is natural in thecontext of generalizations of q-series identities arising from the Bailey chain[2], such geometric decompositions require some effort to compute. Therefore, we would like to present another family of lacunary partition functionsof a much simpler nature - an arithmetic example in terms of parts in certain
congruence classes.
Theorem 1.1. Let Qm,k (n) denote the number of partitions into distinct
non-negative parts which are 0 or m modulo k with the condition that thesmallest part λ1 is a multiple of k and the part 2λ1 + m does not occur. Let
Q±m,k (n) denote the number of such partitions where the number of parts iseven (odd). Then(1.1)

N−#{n ≤ N : Q+.
m,k (n) 6= Qm,k (n)} ¿ √log N

Date: January 21, 2004.
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JEREMY LOVEJOY

The proof of Theorem 1.1 depends on a lemma that converts a case of theHeine transformation [6, p. 241, (III.1)]
(1.2)

∞
X
(a, b; q)n z nn=0(c, q; q)n=∞(az, b; q)∞ X (c/b, z; q)n bn(z, c; q)∞ n=0 (az, q; q)ninto a statement about Bailey pairs. Here we have employed the standard
q-series notation(1.3)(a1 , ..., aj ; q)n :=n−1Y(1 − a1 q k ) · · · (1 − aj q k ).

k=0

We note that statements such as (1.2) are subject to covergence conditions
which will not be stated explicitly.
2. Proof of the main theorem
Before proving the key lemma, we recall [3] that to say a pair of sequences(αn , βn ) forms a Bailey pair with respect to a means (among other things)
thatn(1 − aq 2n ) X (a; q)n+j (−1)n−j q (n−j)(n−j−1)/2 βj(2.1)
αn =.
(1 − a) j=0(q; q)n−j
Lemma 2.2. If (αn , βn ) is a Bailey pair with respect to b, then
(2.2)

∞
X
(1 − b)z n αnn=0

Proof.
∞
X(1 − b)z n αnn=0(1 − bq 2n )(1 − bq 2n )=====∞
X
n=0∞
Xzn= (b, z; q)∞

n,j≥0

j=0∞
X
j=0∞
X
j=0

(z, q; q)n.

nX(b; q)n+j (−1)n−j q (n−j)(n−j−1)/2 βj(q; q)n−jj=0(−1)j q j(j+1)/2 βj

j=0∞
XX bn z j q 2nj βj∞X(−z)n q n(n−1)/2−nj (b; q)n+j(q; q)n−jn=jz j βj.

∞X(−z)n q n(n−1)/2 (b; q)n+2j(q; q)nn=0z j βj (b; q)2j∞
X(−z)n q n(n−1)/2 (bq 2j ; q)nn=0z j βj (z, b; q)∞(q; q)n∞X
bn q 2jn,(z, q; q)nn=0

where in the last step we have employed (1.2) with z = z/a, c = 0, b = bq 2j ,and a → ∞.
¤MORE LACUNARY PARTITION FUNCTIONS

3

Proof of Theorem 1.1. It is known [7] that (αn , βn ) form a Bailey pairwith respect to q, where2n(1 − q 2n+1 )q 2n +n X
(2.3)
αn =
(−1)j q −j(3j+1)/2 andβn = 1.
(1 − q)j=−nInserting this pair into (2.2), we obtain the transformation∞XX
2qn(2.4) (z, q; q)∞=z n (−1)j q 2n +n−j(3j−1)/2 ,2n )(z,q;q)(1−zqnn≥0n=0|j|≤nkmwhich with q = q and z = zq becomes∞XX
2q kn (zq kn+m , q kn+k ; q k )∞(2.5)
=z n (−1)j q 2kn +(k+m)n−kj(3j−1)/2 .
2kn+m )
(1−zqn≥0n=0|j|≤n

When z = 1 the elementary theory of partitions [1] tells us that the left handside generates the partitions described in the theorem, while Theorem 3 from
[7] provides the asymptotic estimate.
¤3. Further results
A number of comments are in order regarding the equations in the previoussection. First, we may set z = −1 in (2.5) to obtain another family like theone in Theorem 1.1.
±
Theorem 3.3. Let Tm,k(n) equal the number of partitions counted by Qm,k (n)where the number of parts divisible by k is even (odd). ThenN+−.
(3.1)#{n ≤ N : Tm,k(n) 6= Tm,k(n)} ¿ √log N

Moreover, we could consider the coefficient of z ` , which is a polynomial on
the right side and again a simple generating function on the left. Specifically,
we haveTheorem 3.4. Let Q±m,k,` (n) denote the number of partitions counted byQ±(n)inwhich
thenumberof parts not divisible by k is equal to `. Thenm,k(3.2)((−1)j , if n = 2k`2 + (k + m)` − kj(3j − 1)/2, |j| ≤ `+−Qm,k,` (n)−Qm,k,` (n) =0,otherwise.
Next it is worth noting that the condition that 2λ1 + m does not occur inthe partitions counted by Qm,k (n) is not superfluous. For if we set q = q k , z =zq m /a, b = q k , and let a → ∞ in (1.2), we have simply a theta-type series:∞∞XX(3.3)(zq kn+m , q kn+k ; q k )∞ q kn =(−z)n q kn(n−1)/2+mn .
n=0

n=0

4JEREMY LOVEJOYThis may also be regarded as arising from inserting into (2.2) the unit Bailey
pair with respect to q (which follows from (2.1)),(1, n = 0
(1 − q 2n+1 )(−1)n q n(n−1)/2and
βn =.
(3.4)
αn =
(1 − q)0, n > 0
In fact, there are a number of different pairs that can be inserted into (2.2)
which allow the sum over j on the right to be simplified by an application theb = c case of (1.2),∞
X(a; q)n z n(az; q)∞
(3.5)
=.
(q;q)
(z; q)∞
nn=0
For example, (αn , βn ) form a Bailey pair with respect to q [8] if
2(1 − q 2n+1 )(−1)n q n1andβn = 2 2 .
(1 − q)(q ; q )nThis results in the transformation∞∞
XX
2q n (z; q 2 )n(3.7)(−z)n q n = (q; q)∞ (zq; q 2 )∞,(q, z; q)nn=0
n=0

(3.6)

αn =

which is (5.3) in [9].
Finally we emphasize that although it was sufficient for our purposes,
Lemma 2.2 does not represent the full extent of the interaction between
Heine’s transformation and Bailey pairs. One may similarly prove a companion to (2.2) in which the roles of αn and βn are reversed, and these transformations can be extended to include the well-poised Bailey pairs [4].
References[1] G.E. Andrews, The Theory of Partitions, Cambrige Univ. Press, Cambridge, 1998.
[2] G.E. Andrews, Partitions and Durfee dissection, Amer. J. Math. 101 (1979), no.3,735-742.
[3] G.E. Andrews, q-series: Their Development and Application in Analysis, Number
Theory, Combinatorics, and Computer Algebra, CBMS 66, American MathematicalSociety, Providence, 1986.
[4] G.E. Andrews and A. Berkovich, WP Bailey pairs, preprint.
[5] G.E. Andrews, F.J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms,Invent. Math. 91 (1988), 391-407.
[6] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press,Cambridge, 1990.
[7] J. Lovejoy, Lacunary partitions functions, Math. Res. Lett. 9 (2002), 191-198.
[8] L.J. Slater, A new proof of Rogers’s transformations of infinite series, Proc. London
Math. Soc. (2) 53 (1951), 460-475.
[9] S.O. Warnaar, Partial theta functions. I. Beyond the lost notebook, preprint.
CNRS, LaBRI, 351 cours de la Libération, 33405 Talence Cedex, FRANCE
E-mail address: lovejoymath.wisc.edu