FROBENIUS PARTITIONS AND THE COMBINATORICS OF
RAMANUJAN’S 1 ψ1 SUMMATIONSYLVIE CORTEEL AND JEREMY LOVEJOYAbstract. We examine the combinatorial significance of Ramanujan’s famous summation. In particular, we prove bijectively a partition theoretic identity which implies
the summation formula.

1. Introduction
One of the more remarkable identities in the theory of basic hypergeometric series is
Ramanujan’s product formula for the summation of the 1 ψ1 bilateral series. Namely,if |q| < 1 and | ab | < |z| < 1 then∞X(a; q)n(b; q)nn=−∞zn =(b/a; q)∞ (q; q)∞ (q/az; q)∞ (az; q)∞(b; q)∞ (b/az; q)∞ (q/a; q)∞ (z; q)∞(1)where(a; q)∞ :=∞Y(1 − aq n )n=0

and(a; q)n =(a; q)∞.
(aq n ; q)∞The 1 ψ1 summation formula is a multi-parameter generalization of Jacobi’s famous
triple product identity:∞Xq n(n+1)/2 z n = (q; q)∞ (−z −1 ; q)∞ (−zq; q)∞

(2)

n=−∞

which can be obtained from (1) by replacing z with −zq/a and letting b → 0, a → ∞.
The summation of the 1 ψ1 has been proven in several ways [1, 2, 3, 5, 7, 8, 9, 10],
typically by using some clever applications of other hypergeometric series identities.
Most notable perhaps is Ismail’s observation [8] that (1) is a corollary of the q - binomial
theorem. Here we shall demonstrate how the 1 ψ1 summation formula is equivalent to
a combinatorial statement about certain types of partitions.
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SYLVIE CORTEEL AND JEREMY LOVEJOY

2. Partitions
A partition π of n := σ(π) is a nonincreasing sequence of natural numbers whose
sum is n. We denote by µ(π) the number of parts in the partition π. Let pA,B (n)denote the number of generalized Frobenius partitions of n, that is, the number of two- rowed arraysµ¶a1 a2 ... amb1 b2 ... bm
PP
P
where ai is a partition of type A, bi is a partition of type B, and n =(ai + bi + 1).
If A denotes ”distinct nonnegative parts,” then Frobenius [6] observed thatpA,A (n) = p(n)(3)

where p(n) is the number of ordinary partitions of n. In [4], Andrews discusses how
this combinatorial identity is the essence of Jacobi’s triple product formula. It turns
out that (3) is just a specialization of a more general combinatorial identity which isessentially the 1 ψ1 summation.
Let C denote ”distinct nonnegative parts and unrestricted nonnegative overlined
parts, with n > n̄.” Let fr,s (n) be the number of generalized Frobenius partitions
counted by pC,C (n) where there are r overlined parts in the top row and s overlined
parts in the bottom row.
Now let gr,s (n) be the number of 4 - tuples of partitions (π1 , π2 , π3 , π4 ) where π1and π2 are ordinary partitions, π3 and π4 are partitions into distinct parts, σ(π1 ) +σ( π2 ) + σ(π3 ) + σ(π4 ) = n, r = µ(π2 ) + µ(π3 ), and s = µ(π2 ) + µ(π4 ). Notice thatf0,0 (n) = pA,A (n) and g0,0 (n) = p(n).
Theorem 1. For all nonnegative integers n, r, s, fr,s (n) = gr,s (n)Proof: Given a Frobenius partitionµ¶a1 a2 ... am
α=b1 b2 ... bm
counted by fr,s (n), transform α into another two-rowed array¶µc1 c2 ... cpβ=
d1 d2 ... dp
as follows:(i ) Let c1 , ..., cp be a1 , ..., am except k is inserted if 0 ≤ k ≤ a1 but k does not occur
in row 1 of α.
(ii ) Let d1 , ..., dp be the −k’s from (i) written in increasing order, followed by thenon-overlined parts of row 2 of α, incremented by 1 and written in increasingorder, followed by the overlined parts from row 2 of α, incremented by 1 andwritten in non-increasing order.

3For example, ifµα=then5̄ 3 3̄ 3̄ 1̄ 0
3 3̄ 0̄ 0̄ 0̄ 0̄¶µ¶
5̄
43 3̄ 3̄ 2 1 1̄ 0.
β=−4 −2 −1 4 4̄ 1̄ 1̄ 1̄ 1̄
Now map β to a 4 - tuple π = (π1 , π2 , π3 , π4 ) by adding, for all i, a part of size di + cito
π1 if neither ci nor di is overlinedπ if c and d are overlined2ii
π3 if ci but not di is overlined
π4 if di but not ci is overlined
In our example, we obtainπ1 : (2, 2)π : (7, 2)2π=
π
3 : (1, 7)
π4 : (3, 2, 1)
It is easy to see that π1 and π2 are ordinary partitions and that π3 and π4 are partitions into distinct parts (where π1 and π3 arePwritten in reverse order). By construction,r = µ(π2 ) + µ(π3 ), s = µ(π2 ) + µ(π4 ) and(ci + di ) = n. In other words, the imageof α is a 4 - tuple counted by gr,s (n).
This mapping is uniquely reversible. Given π, a 4-tuple of partitions counted bygr,s (n), first write π2 and π4 in reverse order. We use the notation πi,1 for the firstpart of πi and π\πi,1 for the 4-tuple of partitions π without the first part of πi . We
shall denote the empty partition by ². With the following algorithm we reconstruct
the two-rowed array β from π.
a←0β←²
While π2 6= ² or π4 6= ² doIf π4 = ² or π2,1µ≤ π4,1¶a
1. β ← β ∪.
π2,1 − a2. π ← π\π2,1else¶µa.
1. β ← β ∪π4,1 − a2. π4 ← π\π4,1
3. a ← a + 1

While π1 6= ² or π3 6= ²If π1 = ² or π1,1µ< π3,1¶a1. β ← β ∪.
π3,1 − a2. π3 ← π\π3,1elseµ¶a1. β ← β ∪.
π1,1 − a3. π1 ← π\π1,1
3. a ← a + 1

It is straightforward to recover the Frobenius partition α from β.

4SYLVIE CORTEEL AND JEREMY LOVEJOY¤3. The summation formula
We first make the substitutions z → −zqa−1 , b → −bq, and a → −a−1 to obtain theequivalent form∞(−aq; q)∞ (−bq; q)∞ X (−a−1 ; q)n (zqa)n(−zq; q)∞ (−z −1 ; q)∞=(q; q)∞ (abq; q)∞ n=−∞(−bq; q)n(bz −1 ; q)∞ (azq; q)∞(4)1
where |b| < |z| < | aq
| and |q| < 1. Notice that the coefficient of z 0 on the left handside of (4) is
Xgr,s (n)ar bs q n(5)n,r,s≥0while the coefficient of z 0 on the right hand side isX
fr,s (n)ar bs q n(6)n,r,s≥0so that the truth of (4) implies Theorem 1. In fact, it is also true that Theorem 1
implies the 1 ψ1 summation formula.
Proof of 1 ψ1 : If φ(z) denotes the right hand side of (4), then(−zq 2 ; q)∞ (−z −1 q −1 ; q)∞(bz −1 q −1 ; q)∞ (azq 2 ; q)∞(1 + z −1 q −1 )(1 − azq)(−zq; q)∞ (−z −1 ; q)∞=(1 + zq)(1 + bz −1 q −1 )(bz −1 ; q)∞ (azq; q)∞(1 − azq)=φ(z)(zq − b)φ(zq) =(7)(8)(9)

1Since φ(z) is an analytic function of z in the annulus |b| < |z| < | aq|, it has a Laurent
series∞Xφ(z) =An (a, b, q)z nn=−∞
1First we assume that |ab| < 1 so that for all z with | qb | < |z| < | aq|, applying (9) tothis series yields∞
XAn (a, b, q)zn+1 n+1qn=−∞

=∞
Xn=−∞

−∞
X

An (a, b, q)bz n q nn=−∞n

An (a, b, q)z −∞Xn=−∞An (a, b, q)aqz n+1

5so that∞
Xn∞
Xn

An−1 (a, b, q)(aq + q )z =n=−∞

An (a, b, q)(1 + bq n )z nn=−∞and hence for all integers n we have thatµ¶
1 + a−1 q n−1An (a, b, q) = aq
An−11 + bq n
If n > 0, then this implies thatAn =an q n (−a−1 ; q)nA0(−bq; q)nand if n < 0, say n = −m, then we havea−m q −m (−bq −m+1 ; q)mA0(−a−1 q −m ; q)ma−m q −m (−a−1 ; q)−m=A0(−bq; q)−mA−m =Therefore
φ(z) =∞X(−a−1 ; q)n (zqa)nn=−∞(−bq; q)nA0But it follows from Theorem 1 and equations (4)-(6) that
A0 =(−aq; q)∞ (−bq; q)∞(q; q)∞ (abq; q)∞

1By analytic continuation we can easily extend to |b| < |z| < | aq|.

¤

4. Concluding Remarks
It should be mentioned that for any integer n, it is indeed possible to bijectively
prove the equality between coefficients of z n on both sides of (4). The arguments are
just more complicated variations on Theorem 1, which is the essential identity. Wealso wish to emphasize that a more careful consideration of the bijection presented inTheorem 1 yields an elegant proof of the q−analogue of a summation of Gauss,∞X(a; q)n (b; q)n (c/ab)nn=0(q; q)n (c; q)n=(c/a; q)∞ (c/b; q)∞.
(c; q)∞ (c/ab; q)∞In fact, Frobenius partitions can be employed to give straightforward combinatorial
proofs of several identities in the theory of basic hypergeometric series. This shall bedemonstrated in a forthcoming paper.

6SYLVIE CORTEEL AND JEREMY LOVEJOY

References[1] C. Adiga, B.C. Berndt, S. Bhargava, and G.N. Watson, Chapter 16 of Ramanujan’s second
notebook: Theta-functions and q-series, Mem. Amer. Math. Soc. 315 (1985).
[2] G.E. Andrews, On Ramanujan’s summation of 1 ψ1 (a; b; z), Proc. Amer. Math. Soc. 22 (1969),552-553.
[3] G.E. Andrews, On a transformation of bilateral series with applications, Proc. Amer. Math. Soc.
25 (1970), 554-558.
[4] G.E. Andrews, Generalized Frobenius Partitions, Mem. Amer. Math. Soc. 301 (1984).
[5] G.E. Andrews and R. Askey, A simple proof of Ramanujan’s summation of the 1 ψ1 , Aequationes
Mathematicae 18 (1978), 333-337.
[6] G. Frobenius, Über die Charaktere der symmetrischen Gruppe, Sitzber. Pruess. Akad. Berlin,
1900, 516-534.
[7] W. Hahn, Beitrage zur Theorie der Heineschen Reihen, Math. Nach. 2 (1949), 340-379.
[8] M. Ismail, A simple proof of Ramanujan’s 1 ψ1 sum, Proc. Amer. Math. Soc. 63 (1977), 185-186.
[9] M. Jackson, On Lerch’s transcendant and the basic bilateral series 2 ψ2 , J.London Math. Soc. 25(1950), 189-196.
[10] K.W.J. Kadell, A probabilistic proof of Ramanujan’s 1 ψ1 sum, SIAM J. Math. Anal. 18 (1987),
1539 - 1548.
LRI, Bât 490, Université Paris-Sud, 91405 Orsay, France
E-mail address: Sylvie.Corteel@lri.fr
Dept. Of Math., University of Wisconsin, Madison, WI 53706
E-mail address: lovejoy@math.wisc.edu