OVERPARTITIONS AND THE q-BAILEY IDENTITYSYLVIE CORTEEL AND JEREMY LOVEJOY
Abstract. Using the framework of overpartitions, we give a combinatorial interpretation andproof of the q-Bailey identity. We then deduce from this identity a couple of facts about
overpartitions. We show that the method of proof of the q-Bailey identity also applies to the(first) q-Gauss identity.

1. Statement of Results
In 1973 Andrews [1] established the q-series identity
X (−a, −q/a)n bn q n(n+1)/2n≥0(bq)n (q 2 ; q 2 )n=(−abq, −bq 2 /a; q 2 )∞
.
(bq)∞

(1.1)

Here we employ the usual basic hypergeometric series notation [9],(x, y)n := (x, y; q)n :=

n−1
Y
k=0

(1 − xq k )(1 − yq k ).

(1.2)

When q = 1, equation (1.1) reduces to a result of Bailey [13, p.243, Eq. (III.7)] on ordinary
hypergeometric series, and hence we call it the q-Bailey identity. While this identity gets mentioned from time to time (e.g. [9, 10, 11]), not much has been written about it in the three
decades since Andrews’ paper appeared. As we shall see, however, there is indeed something
more to be said about the q-Bailey identity, particularly in the context of overpartitions.
We begin with a combinatorial interpretation and proof of (1.1). Recalling that an overpartition is simply a partition wherein we may overline the first occurrence of a number, it isclear enough that the right hand side is a generating function for overpartitions, the numerator
generating the overlined parts and the denominator generating the non-overlined parts. But
what about the left-hand side? It turns out that the summation variable n can be viewed as thesize of the generalized Durfee square of an overpartition, a generalization of the usual Durfee
square which has recently arisen in combinatorial studies of Rogers-Ramanujan type identities[8]. For our purposes, we need only know that the size of the generalized Durfee square of an
overpartition λ, denoted D(λ), is defined to be the largest number n such that the number ofoverlined parts plus the number of non-overlined parts greater than or equal to n is at least n.
For example, the overpartition λ = (10, 9, 8, 8, 7, 4, 3, 2, 2) has D(λ) = 5.
Date: January 26, 2009.
2000 Mathematics Subject Classification. 11P81, 05A17, 33D15.
Key words and phrases. partitions, overpartitions, q-Bailey identity, generalized Durfee square.
The authors were partially supported by an ACI “Jeunes Chercheurs et Jeunes Chercheuses”.
1

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SYLVIE CORTEEL AND JEREMY LOVEJOY

By arguing combinatorially that the right hand side of (1.3) below satisfies a recurrence andinitial condition also satisfied by the product on the left hand side, we will establish Theorem
1.1, from which the q-Bailey identity will then easily follow:
Theorem 1.1. Let fn (r, m) be the number of overpartitions of m into n parts such that (i)all non-overlined parts are at least n and (ii) r is the number of odd overlined parts minus thenumber of even overlined parts. Then
X(−a, −q/a)n q n(n+1)/2
fn (r, m)ar q m .
(1.3)=(q 2 ; q 2 )nr∈Z
m≥0

Next we examine two Rogers-Ramanujan type identities contained in the q-Bailey identity,√
the case a = i q and b = 1,(−q 3 ; q 4 )∞
,
(q)∞

(1.4)

(q 9 ; q 12 )∞.
(q 3 ; q 4 )∞ (q 2 ; q 2 )∞(1.5)

X (−q; q 2 )n q n(n+1)/2n≥0(q)n (q 2 ; q 2 )nand the case q = q 2 , a = ζ6 q, and b = 1,
X (q 3 ; q 6 )n q n2 +nn≥0(q)2n (q 4 ; q 4 )n==Here ζ6 is a primitive sixth root of unity.
We interpret these as overpartition identities by viewing the summation variable n in a second
way - as the number of columns in the Frobenius representation of an overpartition. We recall[6] that the Frobenius representation of an overpartition of m is a two-rowed array,
a1 a2 · · · an,
(1.6)
b1 b2 · · · bn
where the top row is a partition into distinct parts, the bottom row is an overpartition into nonnegative parts, and the sum of all the entries is m. Using a bijection between an overpartition and
its Frobenius representation [7], we also interpret (1.4) in terms of the standard representationof an overpartition. (This can also be done for (1.5), but the result does not have the desired
elegance.)
Here and throughout we identify an overpartition λ with a pair of partitions (ρ, δ), the firstelement of the pair containing the non-overlined parts and the second containing the overlined
parts. We also employ the notations ℓ(·) for the largest part and ν(·) for the number of parts.
Theorem 1.2. Let A1 (m) denote the number of overpartitions λ = (ρ, δ) of m such that (i)ℓ(δ) ≡ ν(δ) (mod 2), (ii) the overlined parts alternate in parity, (iii) the largest D(λ) − ν(δ)
non-overlined parts alternate in parity, and (iv) if D(λ) − ν(δ) > 0 then ℓ(ρ) ≡ ν(δ) (mod 2).
Let B1 (m) denote the number of overpartitions of m whose Frobenius representations have abottom row which is a partition without repeated odd parts. Let C1 (m) denote the number ofoverpartitions whose overlined parts are congruent to 3 modulo 4. Then A1 (m) = B1 (m) =C1 (m).
Theorem 1.3. Let B2 (m) denote the number of overpartitions whose Frobenius representations
have a top row in which the smallest part as well as the differences between successive parts are

OVERPARTITIONS AND THE q-BAILEY IDENTITY

3

congruent to 2 modulo 4, and a bottom row in which (i) odd parts are overlined, and (ii) if 2k
occurs, then k is positive and 2k + 2, 2k + 1, 2k, 2k − 1, and 2k − 2 do not occur. Let C2 (m)denote the number of ordinary partitions of m where odd parts are congruent to 3 modulo 4 andoccur at most twice.
We take a moment to illustrate Theorem 1.2 by recording the overpartitions counted by A1 (5),(5), (4, 1), (4, 1), (2, 2, 1), (3, 1, 1),(3, 1, 1), (2, 1, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1),
the Frobenius representations of the overpartitions counted by B1 (5),         
5
4
3
2
1
,
,
,
,
,
0
1
2
3
4

 
 
 

2 1
3 1
3 2
4 1
,
,
,
,
2 0
1 0
0 0
0 0
and the overpartitions counted by C1 (5),(5), (4, 1), (3, 2), (3, 2), (3, 1, 1),(3, 1, 1), (2, 2, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1).
In each case there are 9 overpartitions.
Finally, since there have recently [4, 5, 14] been some simple and straightforward overpartitiontheoretic proofs of the (first) q-Gauss identity,
X (−1/a, −1/b)n (abcq)n(−acq, −bcq)∞
=
,
(1.7)
(q, cq)n(cq, abcq)∞n≥0

we will show that the method of proof used in establishing the q-Bailey identity may also be
applied to prove this identity.
2. Proof of the q-Bailey identity
2.1. Proof of Theorem 1.1. Let Fn (a) denote the right hand side of (1.3). Clearly F0 (a) = 1.
For n ≥ 1, we shall establish thatFn (a) =qn((a + q n−1 )Fn−1 (1/a) + q n (1/a + q n−1 )Fn−1 (a)).
1 − q 2n

(2.1)

Start with an overpartition λ counted by Fn (a). If neither 1 nor n (non-overlined) occurs in λthen remove 1 from each part of λ. If there is still no 1 or n, then remove 1 from each partagain, continuing this process until either there is a 1 or n (or both).
Call the resulting overpartition λ′ . If 1 occurs in λ′ , then remove it and then subtract 1 fromeach remaining part. The result λ′′ is an overpartition into n − 1 parts whose non-overlined
parts are all at least n − 1. If we subtracted 1 from each part an even number of times in passing
from λ to λ′ , then λ′′ is an overpartition counted by
aq n
Fn−1 (1/a),
1 − q 2n4

SYLVIE CORTEEL AND JEREMY LOVEJOY

otherwise it is counted by
q 2n
Fn−1 (a).
a(1 − q 2n )Now if 1 does not occur in λ′ , then it has at least one occurrence of n. We remove one part ofsize n and then subtract 1 from each remaining part to obtain λ′′ , which is again an overpartition
into n − 1 parts whose non-overlined parts are at least n − 1. If we subtracted 1 from each partan even number of times in passing from λ to λ′ , then λ′′ is an overpartition counted by
q 2n−1
Fn−1 (1/a),
1 − q 2notherwise it is counted by

q 3n−1
Fn−1 (a).
1 − q 2nPutting the four cases together gives (2.1). A simple computation shows that the product on
the left hand side of (1.3) also satisfies the recurrence in (2.1):!
(a + q n−1 )(−1/a, −aq)n−1 q n(n−1)/2 q n (a + q n−1 )(−a, −q/a)n−1 q n(n−1)/2
qn+
1 − q 2n(q 2 ; q 2 )n−1
(q 2 ; q 2 )n−1
q n(n+1)/2
qn=a(−1/a)
(−aq)
+
(−a)
(−q/a)nn−1
nn−1(q 2 ; q 2 )naq n(n+1)/2 a(−a)n (−q/a)n (1 + 1/a) q n (−a)n (−q/a)n+=(q 2 ; q 2 )n (1 + a)
(1 + q n /a)a(1 + q n /a)=q n(n+1)/2 (−a, −q/a)n(1 + q n /a)(q 2 ; q 2 )n (1 + q n /a)=(−a, −q/a)n q n(n+1)/2.
(q 2 ; q 2 )nTogether with the initial condition
this implies Theorem 1.1.

q 0(0+1)/2 (−a, −q/a)0 /(q 2 ; q 2 )0 = 1,2.2. The q-Bailey identity from Theorem 1.1. Clearly the right hand side of the q-Bailey
identity is the generating function for the number of overpartitions, where the exponent of b
counts the number of parts and the exponent of a counts the number of odd overlined parts
minus the number of even overlined parts. For the left hand side, suppose that the generalized
Durfee square of an overpartition λ has size n. Then λ may be decomposed into an overpartitionµ1 into exactly n parts whose non-overlined parts are at least n and an ordinary partition µ2
into (non-overlined) parts at most n. Letting the exponent of a keep track of the difference
between the number of odd overlined parts and the number of even overlined parts and letting
the exponent of b count the number of parts, Theorem 1.1 tells us that the generating function
for the overpartitions µ1 is
bn q n(n+1)/2 (−a, −q/a)n.
(q 2 ; q 2 )n

OVERPARTITIONS AND THE q-BAILEY IDENTITY

5

Of course, the generating function for the partitions µ2 is 1/(bq)n . Putting these together andsumming over all n gives (1.1).

3. Overpartition identities from the q-Bailey identity
3.1. Proof of Theorem 1.2. The right hand side of (1.4) is clearly the generating function
for C1 (m), those overpartitions of m whose overlined parts are congruent to 3 modulo 4. To
discover the function B1 (m), we decompose the summand into two pieces, corresponding toq n(n+1)/2 /(q)n and (−q; q 2 )n /(q 2 ; q 2 )n . The first piece is the generating function for partitions
into n distinct positive parts. The second piece is the generating function for partitions without
repeated odd parts and whose parts are at most 2n, or equivalently, partitions into exactly n
non-negative parts without repeated odd parts. This equivalence may be deduced by reading thecolumns of the 2-modular diagram of a partition without repeated odd parts and whose parts
are at most 2n. For example, take n = 6 and the partition (9, 7, 6, 6, 4, 3), whose 2-modular
diagram is displayed in Fig. 1. Reading the columns, we obtain a partition into 6 non-negative
parts, (12, 11, 8, 3, 1, 0).
2
2
2
2
2
2

2
2
2
2
2
1

2 2 1
2 1
2
2

Figure 1. The 2-modular diagram of (9,7,6,6,4,3)
Evidently, the two pieces generate Frobenius representations with n columns counted by
B1 (m).
Now to see that the coefficient of q m on the left hand side is A1 (m), we use a simple bijection(generalizations of which were presented in [7]). Let us call µ1 the partition into distinct parts
contributed by the top row of the Frobenius symbol and µ2 (resp. µ3 ) the partition into evenparts (resp. odd parts) coming from the bottom row. Notice that the number of parts in µ2
plus the number of parts in µ3 is necessarily equal to the number of parts in µ1 .
We now make a diagram with µ1 , µ2 , and µ3 . This is illustrated in Fig. 2, wherein n = 5,
µ1 = (9, 8, 5, 3, 1), µ2 = (6, 6), and µ3 = (7, 5, 3). First, draw the Ferrers diagram for µ1 in thenormal way, except that each part is shifted one unit to the right of the preceding part. This
creates a diagonal (d1 , d2 , . . . , dn ) with n boxes, running from the Northwest to the Southeast.
Second, add the ith largest part of µ2 as a row to the left of the diagonal entry di . Third, add
the jth smallest part of µ3 as a column under the diagonal entry dn−j+1 . Finally, draw a verticalline just to the left of the column containing the largest part of µ3 (If µ3 is empty, this line goes
just to the right of the diagonal entry dn ). The rows to the left of the line form a partition δ into
distinct parts and the columns to the right of the line form an ordinary partition ρ. Together
these give an overpartition λ = (ρ, δ). In our example, we obtain (10, 9, 8, 8, 7, 4, 3, 2, 2).
It is clear that n is now the size of the generalized Durfee square. Verifying the conditions(i) − (iv) in the statement of the theorem is routine. For example, the fact that the parts added

6

SYLVIE CORTEEL AND JEREMY LOVEJOY

d1
d2
d3
d4
d5

Figure 2. A bijection between an overpartition and its Frobenius representationin the rows to the left of the diagonal entries are even together with the fact that the verticalline cuts out a staircase between itself and these even rows gives conditions (i) and (ii).

3.2. Proof of Theorem 1.3. We proceed as in the proof of Theorem 1.2, this time examining
the identity (1.5). First, we write the product side as
12(q ; q 2 )∞k≡3Y

(1 + q k + q 2k ).

(3.1)(mod 4)This is clearly the generating function for C2 (m).
2
For B2 (m), we break the summand on the left hand side into two pieces, q n +n /(q 4 ; q 4 )n and(q 3 ; q 6 )n /(q)2n . The first piece is certainly the generating function for the top row of a Frobenius
representation counted by B2 (m). Now rewrite the second piece asnY
1(1 + q 2k−1 + q 4k−2 ).
22(q ; q )nk=1

This is the generating function for partitions into parts at most 2n where odd parts occur atmost twice. By reading the columns of the 2-modular diagram of such a partition and overlining
any part whose column contains a 1, we see that this piece is the generating function for those
overpartitions into n non-negative parts such that odd parts are overlined and if 2k occurs, thenk is positive and 2k + 2, 2k + 1, 2k, 2k − 1, and 2k − 2 do not occur. For example, take n = 6
and the partition (9, 7, 5, 5, 4, 3). Its 2-modular diagram is displayed in Figure 3. Reading thecolumns, we obtain the overpartition (12, 11, 6, 3, 1). This is the bottom row.

OVERPARTITIONS AND THE q-BAILEY IDENTITY

7

2 2 2 2 1
2
2
2
2
2

2 2 1
2 1
2 1
2
1

Figure 3. The 2-modular diagram of (9,7,5,5,4,3)
4. The q-Gauss identity
We now prove (1.7) using the same kind of argument as in Section 2. To begin, the righthand side of (1.7) is clearly the generating function for overpartition pairs (λ, µ) such that theexponent of c is ν(λ) + ν(µ), the exponent of a is the number of overlined parts of λ plus thenumber of non-overlined parts of µ, and the exponent of b is the number of parts of µ.
We shall also interpret the summand of the left hand side as this same generating function
for overpartitions pairs (λ, µ), with a restriction introduced by the summation variable. Thissummation variable n will be the largest n such that ν(µ) plus the number of overlined parts in
λ plus the number of non-overlined parts in λ which are greater than or equal to n is at least n.
Let us write the summand as
Gn (a, b)cn
,
(cq)n
with
(−1/a)n (−1/b)n(abq)n .
Gn (a, b) =(q)nNow define Fn (a, b) to be the generating function for overpartition pairs (λ, µ) such that ν(λ) +ν(µ) = n, the exponent of a is the number of overlined parts of λ plus the number of nonoverlined parts of µ, the exponent of b is the number of parts of µ, and the non-overlined partsof λ are greater than or equal to n.
Since 1/(cq)n is the generating function for partitions into parts at most n, with the exponentof c tracking the number of parts, we will be done if we can show that Gn (a, b) = Fn (a, b).
Clearly G0 (a, b) = F0 (a, b) = 1. For n ≥ 1, we shall establish thatFn (a, b) =abq
(1 + q n−1 /a)(1 + q n−1 /b)Fn−1 (a, b).
1 − qn(4.1)

This is obviously true with F replaced by G.
We require an auxiliary function. Let F̃n (a, b) be the generating function for overpartition
pairs (λ, µ) counted by Fn (a, b) with the extra condition that the non-overlined parts of λ are
greater than n. This definition implies that(Fn (a, b) − q n F̃n−1 (a, b), if n ≥ 0,F̃n (a, b) =(4.2)
0,otherwise.
Now take an overpartition pair (λ, µ) counted by Fn (a, b). If neither 1 nor n (non-overlined)
occurs in λ and neither 1 nor 1 (non-overlined) occurs in µ, then remove 1 from each part of λ

8

SYLVIE CORTEEL AND JEREMY LOVEJOY

and 1 from each part of µ. Continue this process until one of the above conditions is filled. Callthe resulting overpartition pair (λ′ , µ′ ).
If 1 (non-overlined) occurs in µ′ , then remove it from µ′ . The result (λ′′ , µ′′ ) is an overpartition
pair counted byabq
F̃n−1 (a, b).
A1 =1 − qnIf 1 occurs in λ′ and neither 1 nor 1 occur in µ′ , then remove 1 from λ′ and then subtract 1
from each remaining part of λ′ and µ′ . The result (λ′′ , µ′′ ) is an overpartition pair counted by
A2 =

aq n
Fn−1 (a, b).
1 − qnIf 1 occurs in µ′ while 1 (non-overlined) does not occur in µ′ and 1 does not occur in λ, thenremove 1 from µ′ and then subtract 1 from each remaining part of λ′ and µ′ . The result (λ′′ , µ′′ )is an overpartition pair counted by
A3 =

bq n
Fn−1 (a, b).
1 − qnIf 1 occurs in µ′ and λ′ and 1 (non-overlined) does not occur in µ′ , then remove the 1 from
Then subtract 1 from each remaining part of λ′ and µ′ . The result (λ′′ , µ′′ ) is an
overpartition pair counted by
abq nF̃n−2 (a, b).
A4 =1 − qn(Notice that n must be at least 2 for the above to happen.)
Finally, if n occurs in λ′ , 1 does not occurs in λ′ and 1 or 1 do not occur in µ, then removen from λ′ and then subtract 1 from each remaining part of λ′ and µ′ . The result (λ′′ , µ′′ ) is an
overpartition pair counted by
q 2n−1
Fn−1 (a, b).
A5 =1 − qnNow all the cases have been covered, and putting them all together givesµ′ and λ′ .

Fn (a, b) = A1 + A2 + A3 + A4 + A5
1
(aq n + bq n + q 2n−1 )Fn−1 (a, b) + abq(F̃n−1 + q n−1 F̃n−2 )=1 − qnabq=(1 + q n−1 /a)(1 + q n−1 /b)Fn−1 (a, b),
1 − qnby an application of equation (4.2). This is the recurrence (4.1), completing the proof.



5. Concluding Remarks
We close with a few questions. First, can the combinatorial method employed in the proofs of(1.1) and (1.7) be applied to other basic hypergeometric series identities? Second, is it possible
to prove families of overpartition identities generalizing Theorem 1.2 or 1.3 by using the Bailey
machinery [2] to embed (1.4) or (1.5) in an infinite family of q-series identities? Finally, what
about other identities coming from the q-Bailey identity? Drew Sills has kindly pointed out tous that some instances of (1.1) occur in Ramanujan’s lost notebook (e.g. [3, (5.35),(5.38),5.39)])OVERPARTITIONS AND THE q-BAILEY IDENTITY

9and in Slater’s list of identities of the Rogers-Ramanujan type (e.g. [12, Eq. (110), corrected]),Undoubtedly some of these have nice consequences for overpartitions as well.
References[1] G.E. Andrews, On the q-analogue of Kummer’s theorem and applications, Duke Math. J. 40 (1973), 525-528.
[2] G.E. Andrews, Bailey’s transform, lemma, chain and tree, Special Functions 2000, eds. J. Bustoz et al., pp.
1-22. Kluwer, (2001).
[3] G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook II, Springer, New York, 2009.
[4] S. Corteel, Particle seas and basic hypergeometric series, Adv. Appl. Math. 31 (2003), no 1, 199-214.
[5] S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ramanujan’s 1 ψ1 summation, J.
Combin. Theory Ser. A 97 (2002), 177-183.
[6] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), 1623-1635.
[7] S. Corteel, J. Lovejoy, and A.J. Yee, Overpartitions and generating functions for generalized Frobenius
partitions, in: Mathematics and Computer Science III: Algorithms, Trees, Combinatorics, and Probabilities(2004) 15-24.
[8] S. Corteel and O. Mallet, Overpartitions, lattice paths, and the Rogers-Ramanujan identities, J. Combin.
Theory Ser. A 114 (2007), 1407-1437.
[9] G. Gasper and M. Rahman, Basic Hypergeomtric Series, Cambridge Univ. Press, Cambridge, 1990.
[10] J. Joichi and D. Stanton, Bijective proofs of basic hypergeometric series identities, Pacific J. Math. 127(1987), 103-120.
[11] X. Ma, Generalized Ismail’s argument and (f, g)-expansion Formula, Adv. Appl. Math. 38 (2007), 227-257.
[12] L.J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. 54 (1952), 147-167.
[13] L.J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966.
[14] A.J. Yee, Combinatorial proofs of Ramanujan’s 1 ψ1 summation and the q-Gauss summation, J. Combin.
Theory Ser. A 105 (2004), 63-77.
LRI, CNRS and Université Paris-Sud, Bâtiment 490, 91405 Orsay, FRANCE
E-mail address: corteel@lri.fr
CNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCE
E-mail address: lovejoy@liafa.jussieu.fr