Contemporary Mathematics Volume 00, 19xx

Three term theta identities

Hershel M. Farkas and Irwin Kra

Abstract. We produce three term identities among coefficients P_{N} (n) of the Ramanujan partition functions defined by

∏_{n=1}^{∞} \frac{1}{(1 − x^{n})^{N}} = ∑_{n=0}^{∞} P_{N} (n)x^{n}

by constructing degree one Γ_{o}(k)-invariant functions. The two main ingredients of our work are the use of the η-function including precise information about its behavior at the cusps of certain subgroups of the modular group and the relative Poincaré series averaging process.

1. Introduction

The Ramanujan partition functions

P_{N} : ℤ^{+} ∪ {0} → ℤ, N ∈ ℤ,

defined by convergent products and sums on the unit disc Δ

P^{N}(x) = ∏_{n=1}^{∞} \frac{1}{(1 − x^{n})^{N}} = ∑_{n=0}^{∞} P_{N}(n)x^{n}, x ∈ Δ,

have many interesting combinatorial and number theoretic properties. (We write P for P_{1} and P for P^{1}.) For positive integers N, the partition coefficients P_{N}(n) satisfy congruence conditions; for example ([8] and [9]), for all m ∈ ℤ^{+} ∪ {0},

P (5m + 4) ≡ 0 mod 5 and P (13m + 6) ≡ 11P_{−11}(m) mod 13.

Stronger results are available for negative N. For example ([7]), for all m ∈ ℤ^{+}∪{0}, we have the following two term recursion formulae^{1}

P_{−8}(5m + 3) = −5^{3}P_{−8} (\frac{m − 1}{5}) and P_{−14}(5m + 4) = 5^{6}P_{−4} (\frac{m − 2}{5}) .

1991 MSC: 11F12, 11F30, 30F30, 30F35, 14H45. Research by IK supported in part by NSF Grant DMS 9800924. Research by HMF supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas sponsored by the Minerva Foundation Germany. Research of HMF and IK supported in part by a US-Israel BSF Grant 95-348.

1We extend the definition of P_{N} to ℚ by setting P_{N} (n) = 0 for n ∈ ℚ − (ℤ^{+} ∪ {0}).

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0271-4132/00 $1.00 + $.25 per page

Our investigation of the functions P_{N} ([1] and [3]) is part of a long term study of the applications of theta functions to function and number theory; throughout this note we use the notation of the first of these two references. The case N = −24 is of particular interest because of its connection to the Ramanujan τ-function (called, in our papers, T). The main purpose of this short note is to explain the function theory behind the five three term recursion identities in

Theorem 1. For all m ∈ ℤ^{+} ∪ {0},

P_{−24}(2m + 1) = −2^{11}P_{−24} (\frac{m − 1}{2}) − 2^{3}3P_{−24}(m), (1)

P_{−12}(3m + 1) = −3^{5}P_{−12} (\frac{m − 1}{3}) − 2^{2}3P_{−12}(m), (2)

P_{−6}(5m + 1) = −5^{2}P_{−6} (\frac{m − 1}{5}) − 2 · 3P_{−6}(m), (3)

P_{−4}(7m + 1) = −7P_{−4} (\frac{m − 1}{7}) − 2^{2}P_{−4}(m) (4)

and

P_{−2}(13m + 1) = −P_{−2} (\frac{m − 1}{13}) − 2P_{−2}(m). (5)

In our study of the Ramanujan τ-function [4] we gave a new proof of the following three term recursion identity due to Mordell, [6], that for all primes k and all positive integers l,

P_{−24}(kl − 1) = P_{−24}(k − 1)P_{−24}(l − 1) − k^{11}P_{−24} (\frac{l}{k} − 1) . (6)

It is immediate that (1) is a special case for the prime 2 (k = 2 and l = m + 1) of (6). It is not at all clear whether (2), (3), (4) and (5) are also special cases of more general identities.

2. The function theoretic foundation

The connection between partitions and complex function theory is provided by the classical η-function on the upper half plane ℍ^{2}, that can, for our purposes, be defined by

η(τ) = x^{\frac{1}{24}} P^{−1}(x), x = exp(2πıτ), τ ∈ ℍ^{2}.

Let k be a prime. We are studying the field of meromorphic functions K(G) on the compactification \overline{ℍ^{2}/G} of the Riemann surface ℍ^{2}/G for finite index subgroups G of the modular group Γ = PSL(2, ℤ). Two families of groups are most important for these applications:

Γ(k, k) ⊂ Γ_{o}(k) ⊂ Γ.

A basic fact for us is that η^{24} is a cusp 6-form for Γ (η^{24} ∈ 𝔸_{6}(ℍ^{2}, Γ)). Using the more precise information provided by the multiplier system (see, for example, [5, Ch. 3]) for η and calculations of zeros and poles of functions, we conclude that^{2}

α(k) = max \{1, \frac{24}{k^{2} − 1} \} and β(k) = max \{1, \frac{k^{2} − 1}{24} \}

2For the convenience of the reader we repeat our definitions: For the prime k,

1. h_{k}(τ) = (\frac{η(kτ)}{η(\frac{τ}{k})})^{α(k)} ∈ K(Γ(k, k)) with divisor (\frac{P_{∞}}{P_{0}} )^{β(k)}; more precisely, in

terms of the local coordinate x = exp (\frac{2πıτ}{k} ) at P_{∞} on \overline{ℍ^{2}/Γ(k, k)},

h_{k}(τ) = x^{β(k)} (\frac{∏_{m=1}^{∞}(1 − x^{k^{2}m})}{∏_{m=1}^{∞}(1 − x^{m})})^{α(k)} .

2. For each N ∈ ℤ such that (k − 1)|12N, f_{k}^{N}(τ) = (\frac{η(kτ)}{η(τ)})^{\frac{24N}{k−1}} ∈ K(Γ_{o}(k)) with

divisor (\frac{P_{∞}}{P_{0}} )^{N}. In this case, in terms of the local coordinate x = exp(2πıτ) at P_{∞}

on \overline{ℍ^{2}/Γ_{o}(k)},

f_{k}^{N}(τ) = x^{N} (\frac{∏_{m=1}^{∞}(1 − x^{km})}{∏_{m=1}^{∞}(1 − x^{m})})^{\frac{24N}{k−1}} .

In particular, we can produce a univalent function (N = 1) for k = 2, 3, 5, 7 and 13 (the cases where \overline{ℍ^{2}/Γ_{o}(k)} has genus 0).

The next ingredient is an averaging ℂ-linear operator

Θ : K(Γ(k, k)) → K(Γ_{o}(k))

defined by

Θ(φ)(τ) = \frac{1}{k} ∑_{l=0}^{k−1} φ(τ + l), φ ∈ K(Γ(k, k)), τ ∈ ℍ^{2}.

For arbitrary N ∈ ℤ, we now set

F_{k,N} = Θ(h_{k}^{N}).

The function F_{k,N} ∈ K(Γ_{o}(k)) is holomorphic except perhaps at P_{∞} and P_{0}. Furthermore:

3. For N > 0, F_{k,N} has a pole at P_{0} of order β(k)N and vanishes at P_{∞} to order ⌈\frac{β(k)N}{k} ⌉.

4. F_{k,0} = 1.

5. For N < 0, F_{k,N} has a pole at P_{∞} of order

− min \{m ≥ ⌈\frac{β(k)N}{k} ⌉; P_{α(k)N}(km − β(k)N) ≠ 0\};

it is holomorphic but need not vanish at at P_{0}.

In all cases (arbitrary N), in terms of the local coordinate x = exp(2πıτ) at P_{∞} on \overline{ℍ^{2}/Γ_{o}(k)},

F_{k,N}(τ) = ∏_{m=1}^{∞}(1 − x^{km})^{α(k)N} ∑_{m=⌈\frac{β(k)N}{k} ⌉}^{∞} P_{α(k)N}(km − β(k)N)x^{m}.

Let k be such that \overline{ℍ^{2}/Γ_{o}(k)} has genus 0. The fact that in these cases we have a generator for K(Γ_{o}(k)) carries lots of information.

α(k) = β(k) \frac{24}{k^{2} − 1} .

and observe that

Theorem 2. (a) For k = 2, 3, 5, and N = k, k + 1, ..., 2k − 1,

(b) for k = 7 and N = 4, 5 and 6, and

(c) for k = 13 and N = 2 and 3, we have

F_{k,−N}(τ) = \frac{1}{k} ∑_{l=0}^{k−1} (\frac{η(\frac{τ+l}{k})}{η(k(τ + l))})^{α(k)N} = c_{0} + c_{1}f_{k}^{−1} = c_{0} + c_{1} (\frac{η(τ)}{η(kτ)})^{\frac{24}{k−1}} .

The translation of the last function theoretic identities to Laurent series and power products yields

Corollary 1. For the above values of k and N, we have

∏_{n=1}^{∞}(1−x^{kn})^{−α(k)N} ∑_{m=−1}^{∞} P_{−α(k)N}(km+β(k)N)x^{m} = c_{0}+c_{1} [x^{−1} ∏_{n=1}^{∞} (\frac{1 − x^{kn}}{1 − x^{n}} )^{\frac{24}{1−k}} ] ,

where

c_{0} = P_{−α(k)N}(β(k)N) + \frac{24}{k − 1}P_{−α(k)N}(−k + β(k)N)

and

c_{1} = P_{−α(k)N}(−k + β(k)N).

It is useful to calculate the above constants.

\begin{aligned}k & N & c_{0} & c_{1} \\ 2 & 2 & 2^{7} & 1 \\ & 3 & −2^{11} & −2^{3} · 3 \\ 3 & 3 & 0 & 1 \\ & 4 & −3^{5} & −2^{2} · 3 \\ & 5 & 3^{7} & 2 · 3 · 5 \\ 5 & 5 & 0 & 1 \\ & 6 & −5^{2} & −6 \\ & 7 & 5^{3} & 2 · 7 \\ & 8 & −5^{3} & 0 \\ & 9 & −5 · 7 · 13 & −2 · 3^{2} · 5 \\ 7 & 4 & −7 & −4 \\ & 5 & 7^{2} & 2 · 5 \\ & 6 & 7^{2} & 0 \\ 13 & 2 & −1 & −2 \\ & 3 & 13 & 0 \end{aligned} .

We can express the last corollary purely in terms of power series.

Corollary 2. For the above values of k and N, we have

∑_{m=−1}^{∞} P_{−α(k)N}(km + β(k)N)x^{m} = c_{0} ∑_{m=0}^{∞} P_{−α(k)N}(m)x^{km}

+c_{1}x^{−1} ∑_{m=0}^{∞} P_{\frac{24}{k−1} −α(k)N}(m)x^{km} ∑_{m=0}^{∞} P_{\frac{24}{1−k}} (m)x^{m}.

For the above low values of k and N = \frac{k+1}{β(k)}, we have α(k)N = \frac{24β(k)}{k−1} = \frac{24}{k−1}. Hence we obtain Corollary 3. For k = 2, 3, 5, 7 and 13,

∑_{m=−1}^{∞} P_{\frac{24}{1−k}} (km + β(k)N)x^{m} = c_{0} ∑_{m=0}^{∞} P_{\frac{24}{1−k}} (m)x^{km} + c_{1}x^{−1} ∑_{m=0}^{∞} P_{\frac{24}{1−k}} (m)x^{m},

The theorem of the introduction is obtained by equating the coefficients of like powers of x^{m} in the above formula. In general, Corollary 2 yields

Corollary 4. For the values of k and N in the last theorem and for all m ∈ ℤ^{+}{0},

P_{−α(k)N}(km + β(k)N)

= (P_{−α(k)N}(β(k)N) + \frac{24}{k − 1}P_{−α(k)N}(−k + β(k)N)) P_{−α(k)N} (\frac{m}{k})

+P_{−α(k)N}(−k + β(k)N) \begin{matrix}⌊ & \frac{m+1}{k} & ⌋ \\ & ∑ \\ & j=0 \end{matrix} P_{\frac{24}{k−1} −α(k)N}(j)P_{\frac{24}{k−1}} (m + 1 − 5j).

The cases with c_{1} = 0 produce constant functions (F_{5,−8} = −5^{3}, F_{7,−6} = 7^{2} and F_{13,3} = 13). This is only a partial list of constant functions produced by the averaging method; more complete information may be found in [7] and [2]. The two cases where c_{0} = 0 lead to the formulae F_{3,−3} = f_{3}^{−1} and F_{5,−5} = f_{5}^{−1} that also follow from the modular equations for the primes 3 and 5. (See, for example, [1]). The case k = 5, leads to

∑_{m=0}^{∞} P_{−5}(5m)x^{m−1} = \frac{1}{x} \frac{∏_{n=1}^{∞}(1 − x^{5n})^{−1}}{∏_{n=1}^{∞}(1 − x^{n})^{−6}}

which should be contrasted with a formula of Ramanujan

∑_{m=1}^{∞} P (5m − 1)x^{m} = 5x\frac{∏_{n=1}^{∞}(1 − x^{5n})^{5}}{∏_{n=1}^{∞}(1 − x^{n})^{6}} .

Together they produce the curious relation

∑_{m=0}^{∞} P_{−5}(5m)x^{m−1} ∑_{m=1}^{∞} P (5m − 1)x^{m} = 5 ∏_{n=1}^{∞}(1 − x^{5n})^{4}.

It is perhaps interesting to observe that the above formula of Ramanujan already implies the mod 5 version of the curious relation. We rewrite Ramanujan’s formula as

∏_{n=1}^{∞}(1 − x^{n})(1 − x^{n})^{5} ∑_{m=1}^{∞} \frac{P (5m − 1)}{5}x^{m−1} = ∏_{n=1}^{∞}(1 − x^{5n})^{5}.

Since

(1 − x^{n})^{5} ≡ 1 − x^{5n} mod 5,

Ramanujan’s formula can be replaced by

∏_{n=1}^{∞}(1 − x^{n}) ∑_{m=1}^{∞} \frac{P (5m − 1)}{5}x^{m−1} ≡ ∏_{n=1}^{∞}(1 − x^{5n})^{4} mod 5

or by

∑_{m=0}^{∞} P_{−1}(m)x^{m} ∑_{m=1}^{∞} \frac{P (5m − 1)}{5}x^{m−1} ≡ ∏_{n=1}^{∞}(1 − x^{5n})^{4} mod 5.

Since it is clear that

P_{−5}(5n) = P_{−1}(n) mod 5,

the above becomes

∑_{m=0}^{∞} P_{−5}(5m)x^{m} ∑_{m=1}^{∞} \frac{P (5m − 1)}{5}x^{m−1} ≡ ∏_{n=1}^{∞}(1 − x^{5n})^{4} mod 5

which is the same as

∑_{m=0}^{∞} P_{−5}(5m)x^{m−1} ∑_{m=1}^{∞} \frac{P (5m − 1)}{5}x^{m} ≡ ∏_{n=1}^{∞}(1 − x^{5n})^{4} mod 5;

the mod 5 version of the curious relation derived above.

It would appear that interesting formulae would be produced working with k = 7 = −N. However, we find that

F_{7,−7} = 2^{2} · 3^{2} · 7 + 7^{2}f_{7}^{−1} + 2^{2} · 5f_{7}^{−2}.

Remark 1. It is of interest to determine whether the list of three term recursions for partition coefficients P_{N}, N ≠ −24, in the theorem in the introduction is complete. As remarked earlier, the case k = 2 of the theorem is an application of the multiplicativity of the τ-function.

3. A related averaging process

For f(τ) ∈ K(Γ_{o}(k)), f (\frac{τ}{k} ) ∈ K(Γ(k, k)). Thus for N ∈ ℤ with (k − 1)|12N, we can set

Y_{k,N}(τ) = Θ (f_{k}^{N} (\frac{τ}{k} )) .

One can work with the function Y in a manner analogous to our development of the properties of the function F or prove a relation between them, as we do below.

Theorem 3. For every prime k and all N ∈ ℤ such that (k − 1)|12N,

F_{k, \frac{N(k+1)}{β(k)}} = f_{k}^{N}Y_{k,N}.

Proof. Both sides of the last equation are defined for all primes k. We compute, in terms of the local coordinate x,

F_{k, \frac{N(k+1)}{β(k)}} (x) = ∏_{n=1}^{∞}(1 − x^{kn})^{\frac{24N}{k−1}} ∑_{m=⌈\frac{N(k+1)}{k}⌉}^{∞} P_{\frac{24N}{k−1}} (km − N(k + 1))x^{m} =

x^{N} \frac{∏_{n=1}^{∞}(1 − x^{kn})^{\frac{24N}{k−1}}}{∏_{n=1}^{∞}(1 − x^{n})^{\frac{24N}{k−1}}} ∏_{n=1}^{∞}(1 − x^{n})^{\frac{24N}{k−1}} ∑_{m=⌈\frac{N}{k} ⌉}^{∞} P_{\frac{24N}{k−1}} (km − N)x^{m} = f_{k}^{N}Y_{k,N}(x).

The last theorem suggests that we will not learn anything new from the function Y. It, however, leads to the question for what M ∈ ℤ such that (k − 1)|12M and what N ∈ ℤ is the Γ_{o}(k)-invariant function f_{k}^{M}F_{k,N} constant?

References

[1] H.M. Farkas and I. Kra, A function theoretic approach to the Ramanujan partition identities with applications to combinatorial number theory, Proceedings of I Iberoamerican Congress on Geometry, January 1998, Chile, 75–106 and in this volume.

[2] , Ramanujan partition identities, Proceedings of I Iberoamerican Congress on Geometry, January 1998, Chile, 107–128 and in this volume.

[3] , Theta Constants, Riemann Surfaces and the Modular Group, monograph in preparation.

[4] , Theta constant identities with applications to combinatorial number theory, Contemporary Math. 211 (1997), 227–252.

[5] M.I. Knopp, Modular Functions in Analytic Number Theory, Chelsea Publishing Company, 1993.

[6] L.J. Mordell, Note on certain modular relations considered by Messers. Ramanujan, Darling, and Rogers, Proc. Lond. Math. Soc. 20 (1922), 408–416.

[7] M. Newman, An identity for the coefficients of certain modular forms, J. Lond. Math. Soc. 30 (1955), 488–493.

[8] S. Ramanujan, Some properties of p(n), the number of partitions of n, Proc. Camb. Phil. Soc. 19 (1919), 207–210.

[9] H. Zuckerman, Identities anologous to Ramanujan’s identities involving the partition function, Duke Math. J. 5 (1939), 88–110.

The Hebrew University of Jerusalem Jerusalem 91904, Israel E-mail: farkas@sunset.huji.ac.il and State University of New York at Stony Brook Stony Brook, NY 11794 E-mail: irwin@math.sunysb.edu