A GENERALIZATION OF THE HIRSCHHORN-FARKAS-KRA SEPTAGONAL NUMBERS IDENTITY

F. G. GARVAN

Abstract. We provide elementary proofs of the Farkas-Kra septagonal numbers identity and some k-th order generalizations.

1. Introduction

Two important combinatorial identities are the Jacobi triple product identity

∏_{n=1}^{∞}(1 − zq^{n−1})(1 − z^{−1}q^{n})(1 − q^{n}) = ∑_{n∈ℤ}(−1)^{n}z^{n}q^{n(n−1)/2} =: θ(z, q),

and the quintuple product identity

∏_{n=1}^{∞}(1 − zq^{n−1})(1 − z^{−1}q^{n})(1 − z^{2}q^{2n−1})(1 − z^{−2}q^{2n−1})(1 − q^{n})

= ∑_{n∈ℤ}(z^{−3n} − z^{3n+1})q^{n(3n+1)/2}.

The first combinatorial proof of the triple product identity is due to Sylvester [14]. Other more recent proofs have been found by Andrews [2] and Lewis [10]. The quintuple product identity is usually attributed to Watson [15]. However it can be found in Ramanujan’s lost notebook [12, p. 207]. Also see [3, p.83], [4],

[8] and [6] for more history and some proofs. Recent proofs have been found by Alladi [1], by Hirschhorn [9], and by Farkas and Kra [5]. The triple and quintuple products are special cases of the Macdonald [11] identities for affine root systems. See Stanton [13] for an elementary treatment. The triple product corresponds to the root system A_{1}, and the quintuple product corresponds to the root system BC_{1}.

Farkas and Kra [5] considered a generalization of the quintuple product identity different from the BC_{n} case of the Macdonald identities. The exponent of q in the right side of the quintuple product identity is the pentagonal number n(3n + 1)/2. They obtained the following identity involving the septagonal number n(5n + 1)/2:

Date: July 8, 1999.

1991 Mathematics Subject Classification. Primary: 05A19; Secondary: 11P81, 33D10.

Key words and phrases. Product identities, triple product, quintuple product, septagonal.

Research supported in part by the NSF under grant number DMS-9870052.

∏_{n=1}^{∞}(1 − zq^{n−1})(1 − z^{−1}q^{n})(1 − z^{2}q^{n−1})(1 − z^{−2}q^{n})(1 − q^{n})^{2} (1.1)

= J_{0}(q) ∑_{n∈ℤ}(−1)^{n}(z^{5n+3} + z^{−5n})q^{n(5n+3)/2}

−J_{1}(q) ∑_{n∈ℤ}(−1)^{n}(z^{5n+2} + z^{1−5n})q^{n(5n+1)/2},

where

J_{0}(q) = ∑_{n∈ℤ}(−1)^{n}q^{n(5n+1)/2}, J_{1}(q) = ∑_{n∈ℤ}(−1)^{n}q^{n(5n+3)/2}.

Earlier, this identity had been discovered independently by Hirschhorn [7, (3.1)]. Observe that the left side of the identity above is the product of two theta functions. Farkas and Kra’s proof of this identity involves the function theory of theta functions with rational characteristics. Foata and Han [6] have found an elementary proof. Their proof utilizes both the triple product and quintuple product identities. In the theorem below we provide a generalization of the septagonal identity and our proof depends only on the triple product identity.

Farkas and Kra found that certain k fold products of theta functions have similar expressions. In particular, for any odd k ≥ 3 we consider the function

(1.2)  f(z, q) := θ(z^{2}, q)θ(z, q)^{k−4}.

Farkas and Kra [5, p.776] showed that there exist functions c_{j}(q) such that

(1.3)  f(z, q) = ∑_{j=0}^{(k−3)/2} c_{j}(q)φ_{j}(z, q),

where each φ_{j}(z, q) is the sum of two certain theta functions. Our main result below gives explicit expressions for the c_{j}(q) in this case, provided k ≥ 5. We note that the case k = 3 corresponds to the quintuple product identity. Theorem 1.1. Let k ≥ 5 be odd, and ℓ = (k − 5). Then

∏_{n=1}^{∞}(1 − q^{n})^{k−3}(1 − z^{2}q^{n−1})(1 − z^{−2}q^{n}) [(1 − zq^{n−1})(1 − z^{−1}q^{n})]^{k−4} (1.4)

= ∑_{r=0}^{(k−3)/2}(−1)^{r}J_{r}(q) ( ∑_{j=−∞}^{∞}(−1)^{j}z^{kj+r}q^{j(kj+2r+2−k)/2}

+ ∑_{j=−∞}^{∞}(−1)^{j}z^{kj+k−2−r}q^{j(kj−2r−2+k)/2}) ,

where for r = 0, 1, 2, …, k − 1,

(1.5)  J_{r}(q) := ∑_{n⃗∈ℤ^{k−4}}(−1)^{n_{0}}q^{Q_{r}(n⃗)},

n⃗ = (n_{0}, n_{1}, n_{2}, …, n_{ℓ}), 1 = (1, 1, 1, …, 1),
Q_{r}(n⃗) := Δ(r − n_{0} − n⃗ · 1) + Δ(n_{0}) + Δ(n_{1}) + Δ(n_{2}) + ⋯ + Δ(n_{ℓ}),

A GENERALIZATION OF THEHIRSCHHORN-FARKAS-KRA SEPTAGONAL NUMBERS IDENTITY 3

and

Δ(n) := \frac{1}{2}n(n − 1).

The case k = 5 corresponds to the Farkas-Kra septagonal identity (1.1).

2. Proof of the main result

It will be useful to record some elementary properties of Δ(n):

(2.1)  Δ(n + m) = Δ(n) + Δ(m) + nm
(2.2)  Δ(−n + a) = Δ(n) + Δ(a) + n(1 − a)

where m, n and a are any integers.

We proceed as follows:

Step 1. First we expand the left side of (1.4) using Jacobi’s triple product identity

and obtain a multiple q-series. Step 2. We split the resulting series according the residue of the exponent of z

modulo k, thus obtaining k functions of z. Then we change variables in the summation so that we can factor a single theta-series out of each of the k series.

Step 3. We use an affine linear transformation to show that the coefficient of one

of the k series is zero. Step 4. We use another affine linear transformation to simplify the final identity.

Step 1. Using Jacobi’s triple product identity we obtain

∏_{n=1}^{∞}(1 − q^{n})^{k−3}(1 − z^{2}q^{n−1})(1 − z^{−2}q^{n}) [(1 − zq^{n−1})(1 − z^{−1}q^{n})]^{k−4}

= ∑_{n∈ℤ}(−1)^{n}z^{2n}q^{Δ(n)} ∑_{m∈ℤ}(−1)^{m}z^{m}q^{Δ(m)}

· ∑_{n⃗=(n_{1},n_{2},...,n_{ℓ})∈ℤ^{ℓ}}(−1)^{n_{1}+n_{2}+···+n_{ℓ}}z^{n_{1}+n_{2}+···+n_{ℓ}}q^{Δ(n_{1})+Δ(n_{2})+···+Δ(n_{ℓ})}

Step 2. In the summation we let m = kj + r − 2n − n_{1} − n_{2} − ⋯ − n_{ℓ} where j and r are integers with 0 ≤ r ≤ (k − 1). Then we change variables in the summation:

n ↦ n + 2j, n_{1} ↦ n_{1} + j, n_{2} ↦ n_{2} + j, ⋯, n_{ℓ} ↦ n_{ℓ} + j

4 F. G. GARVAN

and obtain

∏_{n=1}^{∞}(1 − q^{n})^{k−3}(1 − z^{2}q^{n−1})(1 − z^{−2}q^{n}) [(1 − zq^{n−1})(1 − z^{−1}q^{n})]^{k−4}

= ∑_{r=0}^{k−1} ∑_{j∈ℤ}(−1)^{n+j+r}z^{kj+r}

· ∑_{n,n_{1},n_{2},...,n_{ℓ}∈ℤ} q^{Δ(kj+r−2n−n_{1}−n_{2}−···−n_{ℓ})+Δ(n)+Δ(n_{1})+Δ(n_{2})+···+Δ(n_{ℓ})}

= ∑_{r=0}^{k−1} ∑_{j∈ℤ}(−1)^{n+j+r}z^{kj+r}

· ∑_{n,n_{1},n_{2},...,n_{ℓ}∈ℤ} q^{Δ(j+r−2n−n_{1}−n_{2}−···−n_{ℓ})+Δ(n+2j)+Δ(n_{1}+j)+Δ(n_{2}+j)+···+Δ(n_{ℓ}+j)}

Now using (2.1) the exponent of q in the summation above may be written Δ(j + r − 2n − n_{1} − n_{2} − ⋯ − n_{ℓ}) + Δ(n + 2j)

+Δ(n_{1} + j) + Δ(n_{2} + j) + ⋯ + Δ(n_{ℓ} + j)

= (ℓ + 1)Δ(j) + Δ(2j) + jr + Δ(r − 2n − n_{1} − n_{2} − ⋯ − n_{ℓ}) + Δ(n)

+Δ(n_{1}) + Δ(n_{2}) + ⋯ + Δ(n_{ℓ}),
= j(kj + 2r + 2 − k)/2 + Q(n, n_{1}, n_{2}, …, n_{ℓ}).

In this way we obtain

∏_{n=1}^{∞}(1 − q^{n})^{k−3}(1 − z^{2}q^{n−1})(1 − z^{−2}q^{n}) [(1 − zq^{n−1})(1 − z^{−1}q^{n})]^{k−4}

= ∑_{r=0}^{k−1} ∑_{j∈ℤ}(−1)^{j+r}z^{kj+r}q^{j(kj+2r+2−k)/2}J_{r}(q),

where J_{r}(q) is defined in (1.5). Step 3. We show

(2.3)  J_{k−1} = 0.

Now

J_{k−1}(q) = ∑_{n⃗=(n_{0},n_{1},n_{2},...,n_{ℓ})∈ℤ^{ℓ+1}}(−1)^{n_{0}}q^{Q_{k−1}(n⃗)}.

In the summation we change variables:

n_{0} ↦ −n_{0} + 3, n_{1} ↦ −n_{1} + 2, n_{2} ↦ −n_{2} + 2, ⋯, n_{ℓ} ↦ −n_{ℓ} + 2 Under this transformation

(2.4)  (−1)^{n_{0}} ⟼ − (−1)^{n_{0}}
(2.5)  Q_{k−1}(n⃗) ⟼ Q_{k−1}(n⃗)

(2.4) is immediate. (2.5) follows from a calculation using (2.2). From (2.4) and (2.5) it follows that

J_{k−1}(q) = −J_{k−1}(q),

and hence J_{k−1}(q) = 0.

Step 4. We show that

(2.6)  J_{r}(q) = −J_{k−2−r}(q),

for r = 0, 1, …, (k − 3)/2. This time we consider the transformation

n⃗ ⟼ −n⃗ + 1.

The result follows by showing under this transformation

(2.7)  (−1)^{n_{0}} ⟼ − (−1)^{n_{0}}
(2.8)  Q_{r}(n⃗) ⟼ Q_{k−2−r}(n⃗)

(2.7) is immediate. (2.8) follows from an easy calculation utilizing (2.2) with a = 1. This completes the proof of (1.4).

3. Remarks

What makes our result interesting are the two properties of J_{r}(q), namely (2.3) and (2.6) and the fact that we were able to explicitly identify the coefficient functions. We only considered products of the form given in (1.2). Farkas and Kra [5] considered more general products. It would be interesting to investigate which other theta products had nice evaluations and if there are any connections to the Macdonald identities [11].

References

[1] Krishnaswami Alladi, The quintuple product identity and shifted partition functions, J. Comput. Appl. Math., 68 (1996), 3–13.

[2] George E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc., 49 (1984), no. 301.

[3] B. C. Berndt, Ramanujan’s notebooks. Part III (Springer, New York, 1991).

[4] Bruce C. Berndt. Ramanujan’s theory of theta-functions, in: Theta functions: from the classical to the modern, volume 1 of CRM Proc. Lecture Notes (Amer. Math. Soc., Providence, RI, 1993) 1–63.

[5] Hershel M. Farkas and Irwin Kra, On the quintuple product identity, Proc. Amer. Math. Soc., 127 (1999), 771–778.

[6] D. Foata and G.-N. Han, The triple, quintuple and septuple product identities revisited, Sém. Lothar. Combin., 42 (1999), Art. B42o (electronic).

[7] M. D. Hirschhorn, A simple proof of an identity of Ramanujan, J. Austral. Math. Soc. Ser. A, 34 (1983), 31–35.

[8] M. D. Hirschhorn, A generalisation of the quintuple product identity, J. Austral. Math. Soc. Ser. A, 44 (1988), 42–45.

[9] M. D. Hirschhorn, On the expansion of Ramanujan’s continued fraction, Ramanujan J., 2 (1998), 521–527.

[10] R. P. Lewis, A combinatorial proof of the triple product identity, Amer. Math. Monthly, 91 (1984), 420–423.

[11] I. G. Macdonald, Affine root systems and Dedekind’s η-function, Invent. Math., 15 (1972), 91–143.

[12] Srinivasa Ramanujan, The lost notebook and other unpublished papers (Springer-Verlag, Berlin, 1988).

[13] Dennis Stanton, An elementary approach to the Macdonald identities, in: q-series and partitions (Minneapolis, MN, 1988), volume 18 of IMA Vol. Math. Appl. (Springer, New York, 1989) 139–149.

[14] J. J. Sylvester, A constructive theory of partitions, arranged in three acts, an interact and an exodion, Amer. J. Math., 5 (1882), 251–331.

[15] G. N. Watson, Theorems stated by Ramanujan (vii): Theorems on continued fractions, J. London Math. Soc., 4 (1929), 39–48.

6 F. G. GARVAN

Department of Mathematics, University of Florida, Gainesville, Florida 32611