INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A28

THE NUMBER OF REPRESENTATIONS BY SUMS OF SQUARES AND TRIANGULAR NUMBERS

Heung Yeung Lam

Institute of Information and Mathematical Sciences, Massey University, Albany Campus, Private Bag 102

904, North Shore Mail Centre, Auckland, New Zealand

                           h.y.lam@massey.ac.nz

Received: 11/27/06, Revised: 4/12/07, Accepted: 4/22/07, Published: 6/11/07

Abstract

In this paper, we present eighteen interesting infinite products and their Lambert series expansions. From these, we deduce formulae for the number of representations of an integer n by eighteen quadratic forms in terms of divisor sums.

–Dedicated to the memory of my grandmother Yuet Kwai Mah.

1. Introduction and Statement of Results

Let τ be a fixed complex number satisfying Im (τ) > 0 and let q = e^{iπτ}, so that |q| < 1. Let

φ (q) = ∑_{j=−∞}^{∞} q^{j^{2}},

and

ψ (q) = ∑_{j=0}^{∞} q^{j(j+1)/2}.

The purpose of this paper is to study and give proofs of eighteen theorems in the area of the number of representations by sums of squares and triangular numbers. Most of these results appear to be new. The results in this paper can be found in the author’s thesis [7]. Section 2 contains preliminary results and will be used as a basis for Section 3. In Section 3, we will prove the following results.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A28 2

Theorem 1

φ (q) φ (q^{4}) = 1 + 2 ∑_{j=1}^{∞} \frac{(−1)^{j} q^{2j}}{1 + q^{4j}} − 2 ∑_{j=1}^{∞} \frac{(−1)^{j} q^{2j−1}}{1 − q^{2j−1}} ,  (1)
φ (q) ψ (q^{8}) = − ∑_{j=1}^{∞} \frac{(−1)^{j} q^{2j−2}}{1 − q^{2j−1}} − ∑_{j=1}^{∞} \frac{(−1)^{j} q^{2j−1}}{1 + q^{4j}} ,  (2)
φ (q^{4}) ψ (q^{2}) = −\frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} q^{\frac{j−1}{2}}}{1 − q^{\frac{2j−1}{2}}} − \frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} (−q)^{\frac{j−1}{2}}}{1 − (−q)^{\frac{2j−1}{2}}} .  (3)

Theorem 2

φ^{3} (q) ψ (q^{8}) = 2 ∑_{j=1}^{∞} \frac{jq^{j−1}}{1 + (−q)^{j}} − 2 ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{4j−1}}{1 + q^{4j}}
+ ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1) q^{2j−2}}{1 + q^{4j−2}} ,  (4)
φ^{2} (q) ψ^{2} (q) = \frac{1}{2} ∑_{j=1}^{∞} \frac{(2j − 1) q^{\frac{j−1}{2}}}{1 − q^{\frac{2j−1}{2}}} − \frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1) q^{\frac{j−1}{2}}}{1 + q^{\frac{2j−1}{2}}} ,  (5)
φ^{2} (q) ψ^{2} (q^{4}) = ∑_{j=1}^{∞} \frac{jq^{j−1}}{1 + (−q)^{j}} − ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{2j−1}}{1 + q^{2j}} ,  (6)
φ^{2} (q) ψ^{2} (q^{8}) = \frac{1}{2} ∑_{j=1}^{∞} \frac{jq^{j−2}}{1 + (−q)^{j}} − ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{4j−2}}{1 + q^{4j}}
+\frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1) q^{2j−3}}{1 + q^{4j−2}} + \frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{2j−2}}{1 + q^{2j}} ,  (7)
φ (q) ψ^{3} (q^{8}) = \frac{1}{8} ∑_{j=1}^{∞} \frac{jq^{j−3}}{1 + (−q)^{j}} + \frac{3}{8} ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{2j−3}}{1 + q^{2j}}
−\frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{4j−3}}{1 + q^{4j}} + \frac{1}{8} ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1) q^{2j−4}}{1 + q^{4j−2}} ,  (8)
φ (q^{4}) ψ^{3} (q^{2}) = \frac{1}{16} ∑_{j=1}^{∞} \frac{(2j − 1)q^{\frac{j−2}{2}}}{1 − q^{\frac{2j−1}{2}}} + \frac{1}{16} ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)q^{\frac{j−2}{2}}}{1 − (−q^{\frac{1}{2}} )^{2j−1}}

−\frac{1}{16} ∑_{j=1}^{∞} \frac{(−1)^{\frac{j}{2}} (2j − 1)q^{\frac{j−2}{2}}}{1 − (−q)^{\frac{2j−1}{2}}}

−\frac{1}{16} ∑_{j=1}^{∞} \frac{(−1)^{\frac{3j}{2}} (2j − 1)q^{\frac{j−2}{2}}}{1 + (−q)^{\frac{2j−1}{2}}} ,  (9)

φ (q) φ (q^{4}) ψ^{2} (q^{4}) = \frac{1}{2} ∑_{j=1}^{∞} \frac{jq^{j−1}}{1 + (−q)^{j}} − \frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{2j−1}}{1 + q^{2j}}

−\frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)q^{2j−2}}{1 + q^{4j−2}} ,  (10)

φ (q) φ^{2} (q^{2}) ψ (q^{8}) = ∑_{j=1}^{∞} \frac{jq^{j−1}}{1 + (−q)^{j}} + ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{2j−1}}{1 + q^{2j}}

−2 ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{4j−1}}{1 + q^{4j}} ,  (11)

φ (q) ψ^{2} (q^{4}) ψ (q^{8}) = \frac{1}{4} ∑_{j=1}^{∞} \frac{jq^{j−2}}{1 + (−q)^{j}} − \frac{1}{4} ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{2j−2}}{1 + q^{2j}}

+\frac{1}{4} ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)q^{2j−3}}{1 + q^{4j−2}} .  (12)

Theorem 3

φ^{4} (q) φ^{2} (q^{2}) = 1 + 2 ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)^{2} q^{2j−1}}{1 − q^{2j−1}} + 8 ∑_{j=1}^{∞} \frac{j^{2}q^{j}}{1 + q^{2j}}

−2 ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)^{2} q^{2j−1}}{1 + q^{2j−1}} ,  (13)

φ^{2} (q) φ^{4} (q^{2}) = 1 + 2 ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)^{2} q^{2j−1}}{1 − q^{2j−1}} + 4 ∑_{j=1}^{∞} \frac{j^{2}q^{j}}{1 + q^{2j}}

−2 ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)^{2} q^{2j−1}}{1 + q^{2j−1}} ,  (14)

ψ^{4} (q) ψ^{2} (q^{2}) = ∑_{j=1}^{∞} \frac{j^{2}q^{j−1}}{1 + q^{2j}} ,  (15)

φ^{2} (q) ψ^{4} (q) = ∑_{j=1}^{∞} \frac{(2j − 1)^{2}q^{j−1}}{1 + q^{2j−1}} ,  (16)

φ^{2} (q) ψ^{4} (q^{4}) = \frac{1}{4} ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)^{2} q^{2j−3}}{1 − q^{4j−2}} + \frac{1}{4} ∑_{j=1}^{∞} \frac{j^{2}q^{j−2}}{1 + q^{2j}} ,  (17)

φ^{4} (q) ψ^{2} (q^{4}) = ∑_{j=1}^{∞} \frac{(−1)^{j} (2j − 1)^{2} q^{2j−2}}{1 − q^{4j−2}} + 2 ∑_{j=1}^{∞} \frac{j^{2}q^{j−1}}{1 + q^{2j}} .  (18)

Formula (16) was given by S. Ramanujan [8, Chapter 17, Entry 17] [1, p. 139]. Proofs of (5), (6), and (16) were given by S. H. Chan [2]. In the author’s thesis [7], a total 51 identities are given; only identities (1)–(18) are stated here because either they appear to be new or involve both sums of squares and triangular numbers.

Finally, we will demonstrate an arithmetic interpretation of Theorems 1–3 in terms of divisor sums. For example, (4) implies that the number of solutions in integers x_{1}, x_{2}, x_{3}, and y_{1} of x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + (2y_{1} + 1)^{2} = m, is

k (m) ∑_{{d|m \atop d odd}} d,  (19)

where

k (m) = \{ \begin{matrix}1  : m ≡ 1 (mod 4), \\ 6  : m ≡ 2 (mod 4), \\ 3  : m ≡ 3 (mod 4), \\ 8  : m ≡ 4 (mod 8), \\ 0  : m ≡ 0 (mod 8). \end{matrix}

An arithmetic interpretation of identity (3) appeared in M. D. Hirschhorn [6].

2. Preliminary Results

Following [4, pp. 120–121], we define f_{1} (θ) , f_{2} (θ) , and f_{3} (θ) by

f_{1} (θ) = f_{1} (θ; q) = \frac{1}{2} cot \frac{θ}{2} − 2 ∑_{j=1}^{∞} \frac{q^{2j}}{1 + q^{2j}} sin jθ,  (20)

f_{2} (θ) = f_{2} (θ; q) = \frac{1}{2} csc \frac{θ}{2} + 2 ∑_{j=1}^{∞} \frac{q^{2j−1}}{1 − q^{2j−1}} sin (j − \frac{1}{2}) θ,  (21)

f_{3} (θ) = f_{3} (θ; q) = \frac{1}{2} csc \frac{θ}{2} − 2 ∑_{j=1}^{∞} \frac{q^{2j−1}}{1 + q^{2j−1}} sin (j − \frac{1}{2}) θ.  (22)

It can be shown [4, p. 121] that the series in (20)–(22) converge for −Im (2πτ) < Im (θ) < Im (2πτ). By Ramanujan’s _{1}ψ_{1} summation formula [8, Chapter 16, Entry 17], we have [4, p. 121]:

f_{1} (θ) = \frac{1}{2i} ∏_{k=1}^{∞} \begin{matrix}(1 + q^{2k−2}e^{iθ}) & (1 + q^{2k}e^{−iθ}) & (1 − q^{2k}) & 2 \\ (1 − q^{2k−2}e^{iθ}) & (1 − q^{2k}e^{−iθ}) & (1 + q^{2k}) & 2 \end{matrix} ,  (23)

f_{2} (θ) = \frac{e^{iθ/2}}{i} ∏_{k=1}^{∞} \begin{matrix}(1 − q^{2k−1}e^{iθ}) & (1 − q^{2k−1}e^{−iθ}) & (1 − q^{2k}) & 2 \\ (1 − q^{2k−2}e^{iθ}) & (1 − q^{2k}e^{−iθ}) & (1 − q^{2k−1}) & 2 \end{matrix} ,  (24)

f_{3} (θ) = \frac{e^{iθ/2}}{i} ∏_{k=1}^{∞} \begin{matrix}(1 + q^{2k−1}e^{iθ}) & (1 + q^{2k−1}e^{−iθ}) & (1 − q^{2k}) & 2 \\ (1 − q^{2k−2}e^{iθ}) & (1 − q^{2k}e^{−iθ}) & (1 + q^{2k−1}) & 2 \end{matrix} .  (25)

These are valid for all values of θ except θ = 2mπ + 2nπτ, where there are poles of order 1. Equations (23)–(25) provide an analytic continuation for the functions f_{1}, f_{2}, f_{3}. The functions f_{1}, f_{2}, f_{3} are the Jacobian elliptic functions cs, ns, and ds, respectively. See [3, p. 77] for precise identification.

From [4, p. 124] we have

f′_{1} (θ) = −f_{2} (θ) f_{3} (θ) ,  (26)
f′_{2} (θ) = −f_{1} (θ) f_{3} (θ) ,  (27)
f′_{3} (θ) = −f_{1} (θ) f_{2} (θ) .  (28)

Letting

z = z (q) = ∏_{k=1}^{∞} (1 + q^{2k−1})^{4} (1 − q^{2k})^{2} ,  (29)

x = x (q) = 16q ∏_{k=1}^{∞} \begin{matrix}(1 + q^{2k}) & 8 \\ (1 + q^{2k−1}) & 8 \end{matrix} ,  (30)

x′ = x′ (q) = ∏_{k=1}^{∞} \begin{matrix}(1 − q^{2k−1}) & 8 \\ (1 + q^{2k−1}) & 8 \end{matrix} ,  (31)

we have ([4, p. 124–134]):

x + x′ = 1,  (32)
φ (q) = ∑_{j=−∞}^{∞} q^{j^{2}} = \sqrt{z},  (33)

ψ (q) = ∑_{j=0}^{∞} q^{j(j+1)/2} = \frac{\sqrt{z}x^{\frac{1}{8}}}{\sqrt{2}q^{\frac{1}{8}}} .  (34)

Using the infinite products for f_{1}, f_{2}, f_{3}, and comparing with (29)–(31), we obtain [4, p. 129] the values in Table 1.

Table 1. Values of f_{1}, f_{2}, and f_{3}.
π  πτ π + πτ

f_{1} (θ) 0  \frac{z}{2i}  \frac{z\sqrt{x′}}{2i}

f_{2} (θ) \frac{z}{2}  0  \frac{z\sqrt{x}}{2}

f_{3} (θ) \frac{z\sqrt{x′}}{2} \frac{z\sqrt{x}}{2i}  0

We also summarize 3 transformations of the functions x′, x, and z from [1, pp. 125–126] in Table 2.

Table 2. Three transformations of the functions x′, x, and z.

\begin{matrix}q & x′ & x & z \\ q → −q & \frac{1}{x′} & −\frac{x}{x′} & z\sqrt{x′} \\ q → q^{\frac{1}{2}} & \begin{matrix}(1 − \sqrt{x}) & 2 \\ (1 + \sqrt{x}) & 2 \end{matrix} & \frac{4\sqrt{x}}{(1 + \sqrt{x})^{2}} & z (1 + \sqrt{x}) \\ q → q^{2} & \frac{4\sqrt{x′}}{(1 + \sqrt{x′})^{2}} & \begin{matrix}(1 − \sqrt{x′}) & 2 \\ (1 + \sqrt{x′}) & 2 \end{matrix} & \frac{1}{2}z (1 + \sqrt{x′}) \end{matrix}

We shall give some explanation for Table 2. For example, if we apply the transformation q → −q to the functions x′, x, and z, then the second row of Table 2 implies that x′ (−q) =

\frac{1}{x′} , x (−q) = −\frac{x}{x′} , and z (−q) = z\sqrt{x′}. The transformations q → q^{\frac{1}{2}} and q → q^{2} can be read similarly.

The results in Table 3 can be easily obtained by applying the results of Table 2.

Table 3. Four transformations of the functions x′, x, and z.

\begin{matrix}q & x′ & x & z \\ q → −q^{\frac{1}{2}} & \begin{matrix}(1 + \sqrt{x}) & 2 \\ (1 − \sqrt{x}) & 2 \end{matrix} & \frac{−4\sqrt{x}}{(1 − \sqrt{x})^{2}} & z (1 − \sqrt{x}) \\ q → −q^{2} & \frac{(1 + \sqrt{x′})^{2}}{4\sqrt{x′}} & \frac{− (1 − \sqrt{x′})^{2}}{4\sqrt{x′}} & z\sqrt[4]{x′} \\ q → iq^{\frac{1}{2}} & \begin{matrix}(\sqrt{x′} − i\sqrt{x}) & 2 \\ (\sqrt{x′} + i\sqrt{x}) & 2 \end{matrix} & \frac{4i\sqrt{xx′}}{(\sqrt{x′} + i\sqrt{x})^{2}} & z (\sqrt{x′} + i\sqrt{x}) \\ q → −iq^{\frac{1}{2}} & \begin{matrix}(\sqrt{x′} + i\sqrt{x}) & 2 \\ (\sqrt{x′} − i\sqrt{x}) & 2 \end{matrix} & \frac{−4i\sqrt{xx′}}{(\sqrt{x′} − i\sqrt{x})^{2}} & z (\sqrt{x′} − i\sqrt{x}) \end{matrix}

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A28 7

3. Proofs of Theorems 1–3 The following 3 lemmas are required to prove Theorems 1–3. Lemma 1

φ (q) φ (q^{4}) = f_{2} (π) + if_{1} (π + πτ; q^{2}) ,  (35)
φ (q) ψ (q^{8}) = \frac{1}{2q} [f_{2} (πτ) − if_{1} (π + πτ; q^{2})] ,  (36)
φ (q^{4}) ψ (q^{2}) = \frac{1}{4q^{\frac{1}{4}}} [f_{2}^{(0)} (π + πτ; q^{\frac{1}{2}} ) + \frac{1}{i^{\frac{1}{2}}} f_{2}^{(0)} (π + πτ; iq^{\frac{1}{2}} )] .  (37)

Lemma 2

φ^{3} (q) ψ (q^{8}) = −\frac{1}{q}f′_{1} (π; iq^{\frac{1}{2}} ) + \frac{1}{q}f′_{1} (π; q^{2}) + \frac{i}{q} f′_{3} (π + πτ; q^{2}) ,  (38)
φ^{2} (q) ψ^{2} (q) = \frac{1}{2q^{\frac{1}{4}}} [f′_{2} (πτ; q^{\frac{1}{2}} ) − if′_{3} (π + πτ; q^{\frac{1}{2}} )] ,  (39)
φ^{2} (q) ψ^{2} (q^{4}) = \frac{−1}{2q}f′_{1} (π; iq^{\frac{1}{2}} ) + \frac{1}{2q} f′_{1} (π) ,  (40)
φ^{2} (q) ψ^{2} (q^{8}) = \frac{−1}{4q^{2}} f′_{1} (π; iq^{\frac{1}{2}} ) + \frac{1}{2q^{2}} f′_{1} (π; q^{2}) + \frac{1}{2q^{2}} if′_{3} (π + πτ; q^{2})
−\frac{1}{4q^{2}} f′_{1} (π) ,  (41)
φ (q) ψ^{3} (q^{8}) = −\frac{1}{16q^{3}} f′_{1} (π; iq^{\frac{1}{2}} ) − \frac{3}{16q^{3}} f′_{1} (π) + \frac{1}{4q^{3}} f′_{1} (π; q^{2})
+\frac{i}{8q^{3}} f′_{3} (π + πτ; q^{2}) ,  (42)
φ (q^{4}) ψ^{3} (q^{2}) = \frac{1}{16q^{\frac{3}{4}}} f′_{2} (πτ; q^{\frac{1}{2}} ) + \frac{i}{16q^{\frac{3}{4}}} f′_{2} (πτ; −q^{\frac{1}{2}} )

−\frac{i^{\frac{1}{2}}}{16q^{\frac{3}{4}}} f′_{2} (πτ; iq^{\frac{1}{2}} ) + \frac{1}{16i^{\frac{1}{2}} q^{\frac{3}{4}}} f′_{3} (π + πτ; iq^{\frac{1}{2}} ) ,

(43)

φ (q) ψ^{2} (q^{4}) ϕ (q^{4}) = −\frac{1}{4q} f′_{1} (π; iq^{\frac{1}{2}} ) + \frac{1}{4q} f′_{1} (π) − \frac{i}{2q} f′_{3} (π + πτ; q^{2}) ,

(44)

φ (q) φ^{2} (q^{2}) ψ (q^{8}) = −\frac{1}{2q} f′_{1} (π; iq^{\frac{1}{2}} ) − \frac{1}{2q} f′_{1} (π) + \frac{1}{q}f′_{1} (π; q^{2}) ,

(45)

φ (q) ψ^{2} (q^{4}) ψ (q^{8}) = −\frac{1}{8q^{2}} f′_{1} (π; iq^{\frac{1}{2}} ) + \frac{1}{8q^{2}} f′_{1} (π) + \frac{i}{4q^{2}} f′_{3} (π + πτ; q^{2}) .

Lemma 3

φ^{4} (q) φ^{2} (q^{2}) = 4f′′_{2} (π) − 4if′′_{1} (πτ) + 4f′′_{3} (π) ,  (47)
φ^{2} (q) φ^{4} (q^{2}) = 4f′′_{2} (π) − 2if′′_{1} (πτ) + 4f′′_{3} (π) ,  (48)
ψ^{4} (q) ψ^{2} (q^{2}) = −\frac{i}{2q} f′′_{1} (πτ) ,  (49)
φ^{2} (q) ψ^{4} (q) = −\frac{2i}{q^{\frac{1}{2}}} f′′_{3} (πτ) ,  (50)
φ^{2} (q) ψ^{4} (q^{4}) = \frac{1}{4q^{2}} f′′_{2} (π) − \frac{i}{8q^{2}} f′′_{1} (πτ) − \frac{1}{4q^{2}} f′′_{3} (π) ,  (51)
φ^{4} (q) ψ^{2} (q^{4}) = \frac{1}{q}f′′_{2} (π) − \frac{i}{q} f′′_{1} (πτ) − \frac{1}{q}f′′_{3} (π) .  (52)

Proof. The proofs of Lemmas 1–3 follow by using (26)–(28), (32)–(34), and Tables 1–3. We express both sides of (35)–(52) in terms of z and x. We give complete details for φ (q^{4}) ψ^{3} (q^{2}) only; the other formulae can be proved in a similar way. By employing (33), (34), and Table 2, the left hand side of (38) can be rewritten as

φ^{3} (q) ψ (q^{8}) = \frac{1}{4q} z^{2} (1 − x^{′^{\frac{1}{4}}} ) .  (53)

Next by substituting the values of θ = π and π + πτ into (26) and (28), respectively, and then using Table 1, we find that

f′_{1} (π) = −\frac{z^{2}\sqrt{x′}}{4} ,  (54)
f′_{3} (π + πτ) = −\frac{iz^{2}\sqrt{xx′}}{4} .  (55)

Now if we employ the results of Table 2, Table 3, (54), and (55) in the right hand side of (38) we get

−\frac{1}{q}f′_{1} (π; iq^{\frac{1}{2}} ) + \frac{1}{q}f′_{1} (π; q^{2}) + \frac{i}{q} f′_{3} (π + πτ; q^{2})

= \frac{1}{4q} z^{2} − \frac{1}{8q} z^{2}x^{′^{\frac{1}{4}}} (1 + x^{′^{\frac{1}{2}}} ) − \frac{1}{8q} z^{2}x^{′^{\frac{1}{4}}} (1 − x^{′^{\frac{1}{2}}} )
= \frac{1}{4q} z^{2} (1 − x^{′^{\frac{1}{4}}} ) .  (56)

Combining (53) and (56) we obtain (38). This completes the proof of Lemmas 1–3. □

We now prove Theorems 1–3.

Proofs of Theorems 1–3. We use the series expansions of f_{1} (θ) , f_{2} (θ) , and f_{3} (θ) in (20), (21), and (22), while the right hand sides of the results in Lemmas 1–3 can be represented explicitly as Lambert series. We give complete details for (4) only; the others can be proved in a similar way.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A28 9

First, differentiating (21) and (22) with respect to θ we have

f′_{1} (θ) = −\frac{1}{4} − \frac{1}{4} cot^{2} \frac{θ}{2} − 2 ∑_{j=1}^{∞} \frac{jq^{2j}}{1 + q^{2j}} cos jθ,  (57)
f′_{3} (θ) = −\frac{1}{4} csc \frac{θ}{2} cot \frac{θ}{2} − ∑_{j=1}^{∞} \frac{(2j − 1) q^{2j−1}}{1 + q^{2j−1}} cos (j − \frac{1}{2}) θ.  (58)

Substituting θ = π into (57) we have

f′_{1} (π) = −\frac{1}{4} − 2 ∑_{j=1}^{∞} \frac{(−1)^{j} jq^{2j}}{1 + q^{2j}} .  (59)

Substituting θ = π + πτ into (58) and recalling that q = e^{iπτ}, we have

f′_{3} (π + πτ) = \frac{e^{i(π+πτ)/2} + e^{−i(π+πτ)/2}}{2 (e^{i(π+πτ)/2} − e^{−i(π+πτ)/2})^{2}}

−\frac{1}{2} ∑_{j=1}^{∞} \frac{(2j − 1) q^{2j−1}}{1 + q^{2j−1}} (e^{i(j−1/2)(π+πτ)} − e^{−i(j−1/2)(π+πτ)})

= \frac{iq^{1/2} (1 − q)}{2 (1 + q)^{2}} − \frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j−1/2} (2j − 1) q^{j−1/2}(1 − 1 + q^{2j−1})}{1 + q^{2j−1}}

+\frac{1}{2} ∑_{j=1}^{∞} \frac{(−1)^{j−1/2} (2j − 1) q^{j−1/2}}{1 + q^{2j−1}}

= \frac{iq^{1/2} (1 − q)}{2 (1 + q)^{2}} − \frac{1}{2} ∑_{j=1}^{∞} (−1)^{j−1/2} (2j − 1) q^{j−1/2}
+ ∑_{j=1}^{∞} \frac{(−1)^{j−1/2} (2j − 1) q^{j−1/2}}{1 + q^{2j−1}} .  (60)

We observe that

∑_{j=1}^{∞} (−1)^{j−1/2} (2j − 1) q^{j−1/2} = ∑_{j=1}^{∞} j(−q)^{j/2} − ∑_{j=1}^{∞} 2j(−q)^{j}
= \frac{(−q)^{1/2}}{(1 − (−q)^{1/2})^{2}} − \frac{2(−q)}{(1 − (−q))^{2}}
= \frac{iq^{1/2} (1 − q)}{(1 + q)^{2}}.

Substituting these into (60) we obtain

f′_{3} (π + πτ) = ∑_{j=1}^{∞} \frac{(−1)^{j−1/2} (2j − 1) q^{j−1/2}}{1 + q^{2j−1}} .  (61)

Using (59) and (61) in the right hand side of (38) and simplifying the results we obtain (4). This completes the proofs of Theorems 1–3. □

Next we present an arithmetic interpretation of Theorems 1–3 in terms of divisor sums.

Let k and m be positive integers. Let λ_{1}, λ_{2}, …, λ_{k} and μ_{1}, μ_{2}, …, μ_{m} be positive integers, where λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{k} and μ_{1} ≤ μ_{2} ≤ ⋯ ≤ μ_{m}. The function

r(λ_{1}□ + λ_{2}□ + ⋯ + λ_{k}□ + μ_{1}△ + μ_{2} △ + ⋯ + μ_{m}△) (n)

will denote the number of solutions in integers of

λ_{1}x_{1}^{2} + λ_{2}x_{2}^{2} + ⋯ + λ_{k}x_{k}^{2} + μ_{1} \frac{y_{1} (y_{1} + 1)}{2} + μ_{2} \frac{y_{2} (y_{2} + 1)}{2} + ⋯ + μ_{m} \frac{y_{m} (y_{m} + 1)}{2} = n, (62)

where n = 0, 1, 2, 3, …. We also define r(λ_{1}□ + ⋯ + λ_{k}□ + μ_{1}△ + ⋯ + μ_{m}△) (0) = 1.

Then the generating function for r(λ_{1}□+λ_{2}□+⋯+λ_{k}□+μ_{1}△+μ_{2} △+ ⋯+μ_{m}△) (n) is

∑_{n=0}^{∞} r(λ_{1}□ + λ_{2}□ + ⋯ + λ_{k}□ + μ_{1}△ + μ_{2} △ + ⋯ + μ_{m}△) (n) q^{n}

= φ (q^{λ_{1}}) φ (q^{λ_{2}}) ⋯ φ (q^{λ_{k}}) ψ (q^{μ_{1}}) ψ (q^{μ_{2}}) ⋯ ψ (q^{μ_{m}}) .  (63)

We remark that since (62) is equivalent to

2λ_{1}x_{1}^{2} + 2λ_{2}x_{2}^{2} + ⋯ + 2λ_{k}x_{k}^{2} + μ_{1} (y_{1} + \frac{1}{2})^{2} + μ_{2} (y_{2} + \frac{1}{2})^{2} + ⋯ + μ_{m} (y_{m} + \frac{1}{2})^{2} = 2n + \frac{m}{4}, (64)

then geometrically, 2^{m}r(λ_{1}□ + λ_{2}□ + ⋯ + λ_{k}□ + μ_{1}△ + μ_{2} △ + ⋯ + μ_{m}△) (n) counts the number of lattice points on the k + m dimensional ellipsoid centred at (0, 0, …, 0, −\frac{1}{2}, −\frac{1}{2}, …, −\frac{1}{2}), the point whose first k coordinates are 0 and remaining m coordinates are −\frac{1}{2}, with radius \sqrt{2n + \frac{m}{4}}.

Now we give complete details for an arithmetic interpretation of (4) in terms of divisor sums.

Corollary 1 For n ≥ 1,

r (□ + □ + □ + 8△) (n) = k (n) ∑_{{d|n+1 \atop d odd}} d,  (65)

where

k (n) = \{ \begin{matrix}6  : n ≡ 1 (mod 4), \\ 3  : n ≡ 2 (mod 4), \\ 8  : n ≡ 3 (mod 8), \\ 1  : n ≡ 0 (mod 4), \\ 0  : n ≡ 7 (mod 8). \end{matrix}

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A28 11

Proof. First use (63) and expand the right hand side using the geometric series in (4) to get

∑_{n=0}^{∞} r (□ + □ + □ + 8△) (n) q^{n}

= 2 ∑_{j=0}^{∞} ∑_{m=0}^{∞} (−1)^{mj} (j + 1)q^{(j+1)(m+1)−1}

+2 ∑_{j=0}^{∞} ∑_{m=0}^{∞} (−1)^{m+j} (j + 1)q^{4(j+1)(m+1)−1}

− ∑_{j=0}^{∞} ∑_{m=0}^{∞} (−1)^{m+j} (2j + 1)q^{(2j+1)(2m+1)−1}

= 2 ∑_{j=0}^{∞} ∑_{m=0}^{∞}(2j + 1)q^{(2j+1)(m+1)−1} + 2 ∑_{j=0}^{∞} ∑_{m=0}^{∞} (−1)^{m} (2j + 2)q^{(2j+2)(m+1)−1}

−2 ∑_{j=0}^{∞} ∑_{m=0}^{∞} (−1)^{(m+1)} (2j + 1)q^{4(2j+1)(m+1)−1}

+2 ∑_{j=0}^{∞} ∑_{m=0}^{∞} (−1)^{m+1} (2j + 2)q^{4(2j+2)(m+1)−1}

− ∑_{j=0}^{∞} ∑_{m=0}^{∞} (−1)^{m+j} (2j + 1)q^{(2j+1)(2m+1)−1}

= ∑_{n=0}^{∞} [2 ∑_{(2j+1)(m+1)=n+1}(2j + 1) + 2 ∑_{(2j+2)(m+1)=n+1} (−1)^{m} (2j + 1)

−2 ∑_{4(2j+1)(m+1)=n+1} (−1)^{(m+1)} (2j + 1)

+2 ∑_{4(2j+2)(m+1)=n+1} (−1)^{(m+1)} (2j + 2)

− ∑_{(2j+1)(2m+1)=n+1} (−1)^{(m+j)} (2j + 1)] q^{n}

= ∑_{n=0}^{∞} [2 ∑_{{d|n+1 \atop d odd}} d + 4 ∑_{{d|n+1 \atop n,d odd}} d − 2 ∑_{{d|n+1 \atop n≡3 (mod 4),d odd}} (−1)^{\frac{n+1}{4d}} d − 4 ∑_{{d|n+1 \atop n≡7 (mod 8),d odd}} d

− ∑_{{d|n+1 \atop n even, d odd}} (−1)^{\frac{n}{2}} d] q^{n}.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A28 12

By comparing coefficients of q^{n} on both sides we obtain

r (□ + □ + □ + 8△) (n) = 2 ∑_{{d|n+1 \atop d odd}} d + 4 ∑_{{d|n+1 \atop n,d odd}} d − 2 ∑_{{d|n+1 \atop n≡3 (mod 4),d odd}} (−1)^{\frac{n+1}{4d}} d

− 4 ∑_{{d|n+1 \atop n≡7 (mod 8),d odd}} d − ∑_{{d|n+1 \atop n even, d odd}} (−1)^{\frac{n}{2}} d.

If n ≡ 1 (mod 4), then n + 1 = 4k + 2 and so

r (□ + □ + □ + 8△) (n) = 2 ∑_{{d|n+1 \atop d odd}} d + 4 ∑_{{d|4k+2 \atop d odd}} d
= 2 ∑_{{d|n+1 \atop d odd}} d + 4 ∑_{{d|n+1 \atop d odd}} d
= 6 ∑_{{d|n+1 \atop d odd}} d.  (66)

Similarly, if n ≡ 2 (mod 4), then n + 1 = 4k + 3 and so

r (□ + □ + □ + 8△) (n) = 2 ∑_{{d|n+1 \atop d odd}} d − ∑_{{d|4k+3 \atop d odd}} (−1)^{\frac{4k+2}{2}} d
= 2 ∑_{{d|n+1 \atop d odd}} d + ∑_{{d|n+1 \atop d odd}} d
= 3 ∑_{{d|n+1 \atop d odd}} d.  (67)

If n ≡ 3 (mod 8), then n + 1 = 8k + 4 and so

r (□ + □ + □ + 8△) (n) = 2 ∑_{{d|n+1 \atop d odd}} d + 4 ∑_{{d|8k+4 \atop d odd}} d − 2 ∑_{{d|8k+4 \atop d odd}} (−1)^{\frac{8k+4}{4d}} d
= 2 ∑_{{d|n+1 \atop d odd}} d + 4 ∑_{{d|n+1 \atop d odd}} d + 2 ∑_{{d|n+1 \atop d odd}} d
= 8 ∑_{{d|n+1 \atop d odd}} d.  (68)

If n ≡ 0 (mod 4), then n + 1 = 4k + 1 and so

r (□ + □ + □ + 8△) (n) = 2 ∑_{{d|n+1 \atop d odd}} d − ∑_{{d|4k+1 \atop d odd}} (−1)^{\frac{4k}{2}} d
= 2 ∑_{{d|n+1 \atop d odd}} d − ∑_{{d|n+1 \atop d odd}} d
= ∑_{{d|n+1 \atop d odd}} d.  (69)

If n ≡ 7 (mod 8), then n + 1 = 8k + 8 and so

r (□ + □ + □ + 8△) (n) = 2 ∑_{{d|n+1 \atop d odd}} d + 4 ∑_{{d|8k+8 \atop d odd}} d − 2 ∑_{{d|8k+8 \atop d odd}} (−1)^{\frac{8k+8}{4d}} d − 4 ∑_{{d|8k+8 \atop d odd}} d
= 0.  (70)

Combining (66)–(70), we obtain (65). □

By (64), formula (65) is equivalent to (19).

4. Remarks

The results in this paper can also be proved using the theory of modular forms. I thank the referee for his/her permission to reproduce the following remark.

Consider identity (15). We have that qψ^{4} (q) ψ^{2} (q^{2}) is a modular form of weight 3 on Γ_{0} (4). On the other hand, it is easy to check that

F (q) = ∑_{j=−∞}^{∞} \frac{j^{2}q^{j}}{1 + q^{2j}}

is an Eisenstein series in that same space. To see this, let χ be the non-trivial Dirichlet character mod 4, and let σ (n) := ∑_{d|n} χ (n/d) d^{2}. Define

E (q) = ∑_{n=1}^{∞} σ (n) q^{n}.

It is known that E (q) is in the space (see [5, chapter 4] for a complete discussion). It is not hard to check that E (q) = F (q) directly. The identity qψ^{4} (q) ψ^{2} (q^{2}) = F (q) then follows by checking that the first few terms agree.

An arithmetic interpretation of other identities can be found in the author’s thesis [7]. We shall remark that Hirschhorn [6] also presented many others which give the number of representations of an integer n by various quadratic forms in terms of divisor sums.

Acknowledgements

The author would like to thank Professors Shaun Cooper and Mike Hirschhorn for their encouragement and for uncovering several misprints in an earlier version of this article. The author also would like to thank the anonymous referee for his/her valuable suggestions.

References

[1] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.

[2] S. H. Chan, Generalized Lambert series identities, Proc. London Math. Soc. (3) 91 (2005), no. 3, 598–622.

[3] S. Cooper, The development of elliptic functions according to Ramanujan and Venkatachaliengar, Proceedings of The International Conference on the Works of Srinivasa Ramanujan (C. Adiga and D. D. Somashekara, eds.), University of Mysore, Manasagangotri, Mysore - 570006, India (2001) 81–99. Available electronically at http://www.massey.ac.nz/˜ wwiims/research/letters/volume1number1/06cooper.pdf

[4] S. Cooper, On sums of an even number of squares, and an even number of triangular numbers: an elementary approach based on Ramanujan’s _{1}ψ_{1} summation formula, q-series with Applications to Combinatorics, Number theory and Physics (B. C. Berndt and K. Ono, eds.), Contemporary Mathematics, No. 291, American Mathematical Society, Providence, RI (2001) 115–137.

[5] F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, 228, Springer-Verlag, New York, 2005.

[6] M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Math. 298 (2005), no. 1-3, 205–211.

[7] H. Y. Lam, q-series in Number Theory and Combinatorics, PhD Thesis, Massey University, New Zealand, 2006.

[8] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.