An elliptic analogue of Fukuhara's trigonometeric identities

TL;DR

This paper introduces new elliptic function identities that extend Fukuhara's trigonometric identities, revealing reciprocity laws for elliptic Dedekind sums through Laurent expansion coefficients.

Contribution

It presents novel elliptic identities as an analogue to known trigonometric identities and links them to reciprocity laws for elliptic Dedekind sums.

Findings

New elliptic identities derived as analogues of Fukuhara's trigonometric identities

Laurent expansion coefficients lead to reciprocity laws for elliptic Dedekind sums

Establishes connections between elliptic functions and number theory

Abstract

We obtain new elliptic function identities, which are an elliptic analogue of Fukuhara's trigonometric identities. We show that the coefficients of Laurent expansions at $z=0$ of our elliptic identities give rise to some reciprocity laws for elliptic Dedekind sums.

Equations

$$\frac{1}{a}\sum _{\nu =1}^{a-1}\cot \left(\frac{\pi b\nu }{a}\right)\cot \left(\pi \left(z+\frac{\nu }{a}\right)\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}\cot \left(\frac{\pi a\nu }{b}\right)\cot \pi \left(\left(z+\frac{\nu }{b}\right)\right)$$

$$=-\cot (\pi az)\cot (\pi bz)+\frac{1}{ab}\csc (\pi z{)}^2-1.$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\cot \left(\frac{\pi b\nu }{a}\right)\cot \left(\pi \left(z+\frac{\nu }{a}\right)\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}\csc \left(\frac{\pi a\nu }{b}\right)\cot \left(\pi \left(z+\frac{\nu }{b}\right)\right)$$

$$=-\csc (\pi az)\cot (\pi bz)+\frac{1}{ab}\csc (\pi z{)}^2.$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\cot \left(\frac{\pi b\nu }{a}\right)\csc \left(\pi \left(z+\frac{\nu }{a}\right)\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}\csc \left(\frac{\pi a\nu }{b}\right)\csc \left(\pi \left(z+\frac{\nu }{b}\right)\right)$$

$$=-\csc (\pi az)\cot (\pi bz)+\frac{1}{ab}\csc (\pi z)\cot (\pi z).$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\csc \left(\frac{\pi b\nu }{a}\right)\cot \left(\pi \left(z+\frac{\nu }{a}\right)\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}(-1{)}^{\nu }\csc \left(\frac{\pi a\nu }{b}\right)\cot \left(\pi \left(z+\frac{\nu }{b}\right)\right)$$

$$=-\csc (\pi az)\csc (\pi bz)+\frac{1}{ab}\csc (\pi z{)}^2.$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\csc \left(\frac{\pi b\nu }{a}\right)\csc \left(\pi \left(z+\frac{\nu }{a}\right)\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}(-1{)}^{\nu }\csc \left(\frac{\pi a\nu }{b}\right)\csc \left(\pi \left(z+\frac{\nu }{b}\right)\right)$$

$$=-\csc (\pi az)\csc (\pi bz)+\frac{1}{ab}\csc (\pi z)\cot (\pi z).$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}\cot \left(\frac{\pi b\nu }{a}\right)\cot \left(\frac{\pi \nu }{a}\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}\cot \left(\frac{\pi a\nu }{b}\right)\cot \left(\frac{\pi \nu }{b}\right)$$

$$=\frac{a^2+b^2+1-3ab}{3ab}.$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\cot \left(\frac{\pi b\nu }{a}\right)\cot \left(\frac{\pi \nu }{a}\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}\csc \left(\frac{\pi a\nu }{b}\right)\cot \left(\frac{\pi \nu }{b}\right)$$

$$=\frac{-a^2+2b^2+2}{6ab}.$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\cot \left(\frac{\pi b\nu }{a}\right)\csc \left(\frac{\pi \nu }{a}\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}\csc \left(\frac{\pi a\nu }{b}\right)\csc \left(\frac{\pi \nu }{b}\right)$$

$$=\frac{-a^2+2b^2-1}{6ab}.$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\csc \left(\frac{\pi b\nu }{a}\right)\cot \left(\frac{\pi \nu }{a}\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}(-1{)}^{\nu }\csc \left(\frac{\pi a\nu }{b}\right)\cot \left(\frac{\pi \nu }{b}\right)$$

$$=\frac{-a^2-b^2+2}{6ab}.$$

$$\frac{1}{a}\sum _{\nu =1}^{a-1}(-1{)}^{\nu }\csc \left(\frac{\pi b\nu }{a}\right)\csc \left(\frac{\pi \nu }{a}\right)+\frac{1}{b}\sum _{\nu =1}^{b-1}(-1{)}^{\nu }\csc \left(\frac{\pi a\nu }{b}\right)\csc \left(\frac{\pi \nu }{b}\right)$$

$$=\frac{-a^2-b^2-1}{6ab}.$$

$$\frac{1}{a}\sum _{\begin{array}{c}\mu ,\nu =0\\ (\mu ,\nu )\ne (0,0)\end{array}}^{a-1}(-1{)}^{\mu }\mathrm{cs}\left(2Kb\frac{\mu \tau +\nu }{a},k\right)\mathrm{cs}\left(2K\left(z+\frac{\mu \tau +\nu }{a}\right),k\right)$$

$$+\frac{1}{b}\sum _{\begin{array}{c}\mu ,\nu =0\\ (\mu ,\nu )\ne (0,0)\end{array}}^{b-1}(-1{)}^{\mu }\mathrm{cs}\left(2Ka\frac{\mu \tau +\nu }{b},k\right)\mathrm{cs}\left(2K\left(z+\frac{\mu \tau +\nu }{b}\right),k\right)$$

$$=-\mathrm{cs}(2Kaz,k)\mathrm{cs}(2Kbz,k)+\frac{1}{ab}\mathrm{ds}(2Kz,k)\mathrm{ns}(2Kz,k),$$

$$\frac{1}{b}\sum _{\begin{array}{c}\mu ,\nu =0\\ (\mu ,\nu )\ne (0,0)\end{array}}^{b-1}(-1{)}^{\mu }\mathrm{cs}\left(2Ka\frac{\mu \tau +\nu }{b},k\right)\mathrm{cs}{}^{(N)}\left(2K\frac{\mu \tau +\nu }{b},k\right),$$

$$\frac{1}{a}\sum _{\begin{array}{c}\mu ,\nu =0\\ (\mu ,\nu )\ne (0,0)\end{array}}^{a-1}(-1{)}^{\mu }\mathrm{cs}\left(2Kb\frac{\mu \tau +\nu }{a},k\right)\mathrm{cs}\left(2K\left(\frac{\mu \tau +\nu }{a}\right),k\right)$$

$$+\frac{1}{b}\sum _{\begin{array}{c}\mu ,\nu =0\\ (\mu ,\nu )\ne (0,0)\end{array}}^{b-1}(-1{)}^{\mu }\mathrm{cs}\left(2Ka\frac{\mu \tau +\nu }{b},k\right)\mathrm{cs}\left(2K\left(\frac{\mu \tau +\nu }{b}\right),k\right)$$

$$=\frac{a^2+b^2+1}{6ab}\left(2-\lambda (\tau )\right),$$

$$s(a;b):=\frac{1}{4b}\sum _{\nu =1}^{b-1}\cot \left(\frac{\pi a\nu }{b}\right)\cot \left(\frac{\pi \nu }{b}\right).$$

$$e(x):=e^{2\pi \sqrt{-1}x},q:=e(\tau ).$$

$${\theta }_1(z,\tau ):=$$

$$=$$

$${\theta }_2(z,\tau ):=$$

$$=$$

$${\theta }_3(z,\tau ):=$$

$$=$$

$${\theta }_4(z,\tau ):=$$

$$=$$

$$k=k(\tau ):=\frac{{\theta }_2(0,\tau {)}^2}{{\theta }_3(0,\tau {)}^2},\lambda =\lambda (\tau ):=k(\tau {)}^2,K=K(\tau ):=\frac{\pi }{2}{\theta }_3(0,\tau {)}^2$$

$$\mathrm{sn}(2Kz,k):=$$

$$\mathrm{cn}(2Kz,k):=$$

$$\mathrm{dn}(2Kz,k):=$$

$$\Gamma(2):=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z})\,\Bigg{|}\,a\equiv d\equiv 1\,(\,\mathrm{mod}\,2),\,\,b\equiv c\equiv 0\,(\,\mathrm{mod}\,2)\right\}.$$

Full text

1 Introduction

Our starting point is the following identities. Let $a$ and $b$ be relatively prime positive integers and $z\in\mathbb{C}$.

(0) ((1.1) in [2]) For any complex number $z$,

[TABLE]

(1) ((1.2) in [2]) If $a$ is even, then

[TABLE]

(2) ((1.4) in [2]) If $a$ is odd, then

[TABLE]

(3) ((1.3) in [2]) If $a+b$ is even, then

[TABLE]

(4) ((1.5) in [2]) If $a+b$ is odd, then

[TABLE]

The formula (1.1) was given by Fukuhara [2]. Precisely, in [2], Fukuhara pointed out that (1.1) is derived from specialization of Dieter’s formula (Theorem 2.4 of [1]) and proved (1.2) - (1.5). Further, he compared the coefficients of Laurent expansions at $z=0$ of identities (1.1) - (1.5) and obtained reciprocity laws for Dedekind-Apostol sums. The most simplest case of his reciprocity laws are the following.

(0) ((1.11) in [2]) If $a$ and $b$ are relatively prime positive integers, then

[TABLE]

(1) ((1.12) in [2]) If $a$ is even, then

[TABLE]

(2) ((1.14) in [2]) If $a$ is odd, then

[TABLE]

(3) ((1.13) in [2]) If $a+b$ is even, then

[TABLE]

(4) ((1.15) in [2]) If $a+b$ is odd, then

[TABLE]

On the other hand, Fukuhara and Yui obtained the following elliptic function identity (Theorem 2.1 in [3]) which is regarded as an elliptic analogue of the trigonometric identity (1.1). If $a+b$ is odd, then

[TABLE]

where $\mathrm{cs}\,(z,k)$ is the Jacobi elliptic function (see Section 2). From this elliptic function identity, Fukuhara and Yui also gave reciprocity laws (Theorem 2.2 in [3]) for the elliptic Dedekind-Apostol sums

[TABLE]

where $\mathrm{cs}\,^{(N)}(z)$ is the $N$-th derivative of the $\mathrm{cs}\,(z)$. The simplest case of Fukuhara-Yui’s reciprocity is the following (Lemma 3.1 in [3])

[TABLE]

which is an elliptic analogue of the reciprocity law (1.6) for the classical Dedekind sum

[TABLE]

In this article, we give an elliptic analogue of (1.2) - (1.5) and (1.7) - (1.10). The content of this paper is as follows. In Section 2, we introduce the elliptic functions $\mathrm{cs}\,(z,k)$, $\mathrm{ds}\,(z,k)$, $\mathrm{ns}\,(z,k)$, and list their fundamental properties. Section 3 is the main part of this article and we prove our main results (Theorem 3, Theorem 4 and Corollary 5). In Section 4, we give all the examples of Theorem 3 and Corollary 5.

2 Preliminaries

Throughout the paper, we denote the ring of rational integers by $\mathbb{Z}$, the field of real numbers by $\mathbb{R}$, the field of complex numbers by $\mathbb{C}$ and the upper half plane $\mathfrak{H}:=\{z\in\mathbb{C}\mid\mathrm{Im}\,{z}>0\}$. For $\tau\in\mathfrak{H}$, we put

[TABLE]

First, we recall the Jacobi theta functions

[TABLE]

Further we put

[TABLE]

and introduce the Jacobi elliptic functions

[TABLE]

As is well known, the Jacobi elliptic functions $\mathrm{sn}\,(2Kz,k)$, $\mathrm{cn}\,(2Kz,k)$ and $\mathrm{dn}\,(2Kz,k)$ only depend on $\lambda(\tau)=k(\tau)^{2}$ (elliptic lambda function) that is a modular function of the modular subgroup

[TABLE]

Therefore under the following we restrict $\tau$ to the fundamental domain of $\Gamma(2)$

[TABLE]

The elliptic functions $\mathrm{cs}\,(2Kz,k)$, $\mathrm{ds}\,(2Kz,k)$ and $\mathrm{ns}\,(2Kz,k)$ are defined by

[TABLE]

The elliptic function $\mathrm{cs}\,(2Kz,k)$ is regarded as an elliptic analogue of $\cot{(\pi z)}=\frac{\cos{(\pi z)}}{\sin{(\pi z)}}$. Similarly, $\mathrm{ds}\,(2Kz,k)$ and $\mathrm{ns}\,(2Kz,k)$ are regarded as an elliptic analogue of $\csc{(\pi z)}=\frac{1}{\sin{(\pi z)}}$. According to the wolfram functions site [6] and [5], we list fundamental properties of $\mathrm{cs}\,(2Kz,k)$, $\mathrm{ds}\,(2Kz,k)$ and $\mathrm{ns}\,(2Kz,k)$.

Lemma 1**.**

(1)* (parity)*

[TABLE]

http://functions.wolfram.com/EllipticFunctions/JacobiCS/04/02/01/

http://functions.wolfram.com/EllipticFunctions/JacobiDS/04/02/01/

http://functions.wolfram.com/EllipticFunctions/JacobiNS/04/02/01/

(2)* (periodicity) For any $\mu,\nu\in\mathbb{Z}$, *

[TABLE]

http://functions.wolfram.com/EllipticFunctions/JacobiCS/04/02/03/

http://functions.wolfram.com/EllipticFunctions/JacobiDS/04/02/03/

http://functions.wolfram.com/EllipticFunctions/JacobiNS/04/02/03/

(3)* (Laurent expansions at $z=0$) *

[TABLE]

http://functions.wolfram.com/EllipticFunctions/JacobiCS/06/01/01/

http://functions.wolfram.com/EllipticFunctions/JacobiDS/06/01/01/

http://functions.wolfram.com/EllipticFunctions/JacobiNS/06/01/01/

(4)* (Partial fraction expansions) (5.1) in [5]*

[TABLE]

where $\sum_{n\in\mathbb{Z}}{}^{\!e}$ is the Eisenstein convention

[TABLE]

*In particular, for any non zero constant $A$, we have *

[TABLE]

(6)* (Fourier expansions) p107 in [5]*

[TABLE]

(4)* (Derivations)*

[TABLE]

http://functions.wolfram.com/EllipticFunctions/JacobiCS/20/01/01/

http://functions.wolfram.com/EllipticFunctions/JacobiDS/20/01/01/

http://functions.wolfram.com/EllipticFunctions/JacobiNS/20/01/01/

(5)* (Relations between the Weierstrass $\wp$ function)*

[TABLE]

Here, $\wp(z,\tau)$ is the Weierstrass $\wp$ function defined by

[TABLE]

http://functions.wolfram.com/EllipticFunctions/JacobiCS/27/02/07/

http://functions.wolfram.com/EllipticFunctions/JacobiDS/27/02/07/

http://functions.wolfram.com/EllipticFunctions/JacobiNS/27/02/07/

(5)* (trigonometric degenerations)*

[TABLE]

Here, $\lfloor w\rfloor$ denotes the greatest integer not exceeding $w$.

For convenience, we put

[TABLE]

According to these notations, we have the following expressions of parity (2.1) - (2.3), periodicity (2.4) - (2.6), Laurent expansions at $z=0$ (2.7) - (2.9), partial fraction expansions (2.10) - (2.12), residues at simple poles (2.13) - (2.15), derivations (2.19) - (2.21) and relations between the Weierstrass $\wp$ function (2.22) - (2.24) respectively.

[TABLE]

Here indices of $f_{i,j}(z,\tau)$ are regarded as elements in $\mathbb{Z}/2\mathbb{Z}$.

Remark 2**.**

(1) Fukuhara-Yui use

[TABLE]

instead of $2K\mathrm{cs}\,(2Kz,k)$. However, Fukuhara-Yui did not mention that $\varphi(\tau,z)$ is the Jacobi elliptic function $2K\mathrm{cs}\,(2Kz,k)$ exactly.

(2) If we use Mumford’s notations [4]

[TABLE]

then our $f_{i,j}(z,\tau)$ is written by

[TABLE]

3 Main results

Under the following we assume $a$ and $b$ are relatively prime positive numbers and $i,j,m,n\in\{0,1\}$. We mention and prove the main theorem.

Theorem 3**.**

*If $ia+mb$ or $ja+nb$ is odd, then *

[TABLE]

Proof.

We put

[TABLE]

and

[TABLE]

Under the condition $2\not|\,\,ia+mb$ or $2\not|\,\,ja+nb$, we claim that

[TABLE]

First we show that $\Phi_{(i,j),(m,n)}((a,b),z,\tau)$ and $\Psi_{(i,j),(m,n)}((a,b),z,\tau)$ have same periodicity. From periodicity (2.32), for any integers $M$ and $N$ we have

[TABLE]

Similarly, for $\Phi_{(i,j),(m,n)}((a,b),z,\tau)$ we have

[TABLE]

Thus we obtain double periodicity of $U_{(i,j),(m,n)}((a,b),z,\tau)$

[TABLE]

Next we consider all the poles of $U_{(i,j),(m,n)}((a,b),z,\tau)$ and their Laurent expansions. We remark that $\Phi_{(i,j),(m,n)}((a,b),z,\tau)$, $\Psi_{(i,j),(m,n)}((a,b),z,\tau)$ and $U_{(i,j),(m,n)}((a,b),z,\tau)$ are holomorphic at $z=0$. Actually, since $a$ and $b$ are relatively prime positive integers, the Laurent expansions of $\Phi_{(i,j),(m,n)}((a,b),z,\tau)$ at $z=0$ is the following.

[TABLE]

On the other hand, from (2.36) and (2.33), we have

[TABLE]

Then we obtain the Laurent expansion of $U_{(i,j),(m,n)}((a,b),z,\tau)$ at $z=0$

[TABLE]

Hence we investigate other poles. By the definition or partial fractional expansion (2.34) of $f_{i,j}(z,\tau)$, all other poles of $\Phi_{(i,j),(m,n)}((a,b),z,\tau)$ and $\Psi_{(i,j),(m,n)}((a,b),z,\tau)$ are

[TABLE]

or

[TABLE]

Furthermore, all the poles are simple and the residues at these poles are equal. Actually, from (2.35), we have

[TABLE]

and

[TABLE]

Thus for $M,N\in\mathbb{Z}$, $\mu,\nu\in\{0,1,\ldots,a-1\}$ and $(\mu,\nu)\not=(0,0)$,

[TABLE]

Similarly, for $M,N\in\mathbb{Z}$, $\mu,\nu\in\{0,1,\ldots,b-1\}$ and $(\mu,\nu)\not=(0,0)$ we have

[TABLE]

Therefore $U_{(i,j),(m,n)}((a,b),z,\tau)$ is an entire function.

Summarizing the above discussion, $U_{(i,j),(m,n)}((a,b),z,\tau)$ is a doubly periodic entire function on $\mathbb{C}$. Then by the well-known Liouville theorem, there exists a constant $c_{(i,j),(m,n)}((a,b),\tau)$ such that

[TABLE]

If $ia+mb$ is odd, changing the variable from $z$ to $z+\tau$, we have

[TABLE]

Here the second equality follows from double periodicity (3.2). If $ia+mb$ is even, from the assumption of theorem, then $ja+nb$ is odd. Hence, changing the variable from $z$ to $z+1$, we have

[TABLE]

Therefore $c_{(i,j),(m,n)}((a,b),\tau)\equiv 0$ in any cases and we obtain the conclusion. ∎

Expanding both side of (3.1) and comparing coefficients of $z^{2N}$, we obtain reciprocity laws for elliptic Dedekind sums, which is a natural generalization of Fukuhara-Yui’s main result (Theorem 2.2 (1) in [3]).

Theorem 4** (Reciprocity laws for elliptic Dedekind sums).**

If $ia+mb$ or $ja+nb$ is odd, then

[TABLE]

where $f_{ia+mb,ja+nb}^{(2N)}(z)$ is the $2N$-th derivative of the $f_{ia+mb,ja+nb}(z)$, and $C_{i,j}(s,\tau)$ is the coefficients of the Laurent expansions of $f_{i,j}(z,\tau)$ at $z=0$

[TABLE]

In particular, considering the case of $N=0$ of (3.4) or taking the limit $z\to 0$ in (3.3), we obtain reciprocity laws for elliptic Dedekind sums.

Corollary 5**.**

If $ia+mb$ or $ja+nb$ is odd, then

[TABLE]

4 All the examples of (3.1) and (3.5)

In this section, we give all the examples of (3.1) and (3.5) up to the constant factor $(2K)^{2}$ explicitly.

4.1 $(i,j)=(1,0)$, $(m,n)=(1,0)$

4.1.1 $2\not|\,\,a+b$

In this case, (3.1) and (3.5) are Fukuhara-Yui’s results (1.11) and (1.12) respectively.

4.2 $(i,j)=(1,1)$, $(m,n)=(1,0)$

4.2.1 $2\not|\,\,a+b$, $2\not|\,\,a$

[TABLE]

By taking the limit $\tau\to\sqrt{-1}\infty$ and (2.26) - (2.30), (4.1) and (4.2) degenerate to (1.3) and (1.8) respectively.

4.2.2 $2\not|\,\,a+b$, $2\,\,|\,\,a$

[TABLE]

[TABLE]

By taking the limit $\tau\to\sqrt{-1}\infty$ and (2.26) - (2.30), we have

[TABLE]

Since $a$ is even and the third term in (4.5) vanishes, (4.3) and (4.4) degenerate to (1.2) and (1.7) respectively.

4.2.3 $2\,\,|\,\,a+b$, $2\not|\,\,a$

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.6) and (4.7) degenerate to (1.3) and (1.8) respectively.

4.3 $(i,j)=(0,1)$, $(m,n)=(1,0)$

4.3.1 $2\not|\,\,b$, $2\not|\,\,a$

[TABLE]

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.8) and (4.9) degenerate to (1.3) and (1.8) respectively.

4.3.2 $2\not|\,\,b$, $2\,\,|\,\,a$

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.10) and (4.11) degenerate to (1.2) and (1.7) respectively.

4.3.3 $2\,\,|\,\,b$, $2\not|\,\,a$

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.12) and (4.13) degenerate to (1.3) and (1.8) respectively.

4.4 $(i,j)=(1,1)$, $(m,n)=(1,1)$

4.4.1 $2\not|\,\,a+b$

[TABLE]

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.14) and (4.15) degenerate to (1.5) and (1.10) respectively.

4.5 $(i,j)=(0,1)$, $(m,n)=(1,1)$

4.5.1 $2\not|\,\,b$, $2\not|\,\,a+b$

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.16) and (4.17) degenerate to (1.5) and (1.10) respectively.

4.5.2 $2\not|\,\,b$, $2\,\,|\,\,a+b$

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.18) and (4.19) degenerate to (1.4) and (1.9) respectively.

4.5.3 $2\,\,|\,\,b$, $2\not|\,\,a+b$

[TABLE]

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.20) and (4.21) degenerate to (1.5) and (1.10) respectively.

4.6 $(i,j)=(0,1)$, $(m,n)=(0,1)$

4.6.1 $2\not|\,\,a+b$

[TABLE]

Taking the limit $\tau\to\sqrt{-1}\infty$, (4.22) and (4.23) degenerate to (1.5) and (1.10) respectively.