2 The n-ary elliptic function of Du Val

Let p be a Weierstrass elliptic $\wp $ -function and write $\Lambda _p \subseteq \mathbb {C}$ for its period lattice. Let $(2 \omega , 2 \omega ')$ be an integral basis for $\Lambda _p$ . Thus, $(2 \omega , 2 \omega ')$ is a fundamental pair of periods for p and we may write $p = \wp (-; \omega , \omega ')$ . As is often the custom, we shall suppose that the necessarily nonreal ratio $\omega ' / \omega $ lies in the upper halfplane $\mathcal {H} = \{ z \in \mathbb {C} : \mathrm {Im}\,z> 0 \}$ . Associated to p are the corresponding zeta function $\zeta = \zeta (-; \omega , \omega ')$ and sigma function $\sigma = \sigma (-; \omega , \omega ')$ : the former may be defined by

$$ \begin{align*} \zeta ' = - p \quad \mathrm{and} \quad \lim_{z \to 0} \left\{ \zeta(z) - \frac{1}{z} \right\} = 0; \end{align*} $$

the latter by

$$ \begin{align*} \frac{\sigma'}{\sigma} = \zeta \quad \mathrm{and} \quad \lim_{z \to 0} \frac{\sigma(z)}{z} = 1. \end{align*} $$

Having chosen the fundamental pair $(2 \omega , 2 \omega ')$ of periods for the $\wp $ -function p, we now fix an integer n greater than unity and focus on the nth period (or nth division point)

$$ \begin{align*} \alpha : = 2 \omega / n. \end{align*} $$

Further, let $P = \wp (-; \omega , n \omega ')$ be the Weierstrass $\wp $ -function with $(2 \omega , 2 n \omega ')$ as a fundamental pair of periods, so that its period lattice $\Lambda _P$ comprises all linear combinations of $2 \omega $ and $2 n \omega '$ with integer coefficients. Associated to P are the corresponding zeta function $Z = \zeta (-; \omega , n \omega ')$ and sigma function $\Sigma = \sigma (-; \omega , n \omega ')$ . When convenient, we may instead write the chosen half-periods of P as $\Omega = \omega $ and $\Omega ' = n \omega '$ .

Now, to each choice of constants $a_1, \ldots , a_{n - 1}$ we associate the function

(2.1) $$ \begin{align} \phi : = \frac{(-1)^n}{(n - 1)!}\, p_{n - 2} + a_1 p_{n-3} + \cdots + a_{n - 2} p_0 + a_{n - 1} \end{align} $$

where we abbreviate to $p_k$ the kth derivative of p. Note that for each such choice of constants, $\phi $ is an elliptic function with $\Lambda _p$ as both its period lattice and its pole lattice, each pole having order n; note also that the coefficient of $p_{n - 2}$ in the expansion (2.1) secures the behaviour $z^n \phi (z) \to 1$ as $z \to 0$ .

In the $n = 3$ case the following result is presented by Du Val in [Reference Du Val3, page 141] and again in [Reference Du Val4, page 129] as the basis for one of his constructions (the second) of a ternary elliptic function.

Proof. For convenience, let $\phi _k$ denote the kth derivative of $\phi $ and let us write $\lambda = (-1)^n / (n - 1)!$ . The requirement that $\phi $ have $\alpha $ as a zero is

(2.2) $$ \begin{align} a_{n - 1} + p_0 (\alpha) a_{n - 2} + \cdots + p_{n-3} (\alpha) a_1 = - \lambda p_{n-2} (\alpha); \end{align} $$

the requirement that the zero $\alpha $ of $\phi $ have order at least equal to n augments (2.2) by

(2.3) $$ \begin{align} p_{k} (\alpha) a_{n-2} + \cdots + p_{n - 3 + k} (\alpha) a_1 = - \lambda p_{n-2+k} (\alpha) \end{align} $$

for $1 \leqslant k < n$ . For k any positive integer, let $\mathbb {P}_k$ be the (Hankel) matrix-valued function

$$ \begin{align*} \mathbb{P}_k = \begin{bmatrix} p_1 & p_2 & \cdots & p_{k}\\ p_2 & p_3 & \cdots & p_{k+1}\\ \vdots & \vdots & \vdots & \vdots \\ p_{k} & p_{k+1} & \cdots & p_{2 k - 1}\ \end{bmatrix} .\end{align*} $$

The system comprising Equations (2.3) for $1 \leqslant k < n-1$ then assumes matrix form

(2.4) $$ \begin{align} \mathbb{P}_{n-2} (\alpha) \, [a_{n-2}, \ldots , a_1]^T = - \lambda [p_{n-1} (\alpha), \ldots , p_{2n - 4} (\alpha)]^T. \end{align} $$

On the one hand, $(n - 1) \alpha $ is not a period of p, so a classical theorem of Kiepert (which we recall after our proof) implies that the coefficient matrix $\mathbb {P}_{n-2} (\alpha )$ here is nonsingular; in consequence, there is a unique vector $[a_{n-2}, \ldots , a_1]^T$ satisfying the system (2.4). On the other hand, as $n \alpha $ is a period of p, the same theorem of Kiepert implies that the matrix $\mathbb {P}_{n-1} (\alpha )$ is singular; the one-dimensionality of its null space ensures that the very same constants $a_1, \ldots , a_{n-2}$ further satisfy the system

$$ \begin{align*} \mathbb{P}_{n-1} (\alpha) \, [a_{n-2}, \ldots , a_1, \lambda]^T = 0 \end{align*} $$

so that

$$ \begin{align*} \phi_1 (\alpha) = \cdots = \phi_{n-1}(\alpha) = 0. \end{align*} $$

Precisely one value of the constant $a_{n-1}$ now guarantees that $\phi (\alpha ) = \phi _0 (\alpha ) = 0$ also, whence $\alpha $ is a zero of $\phi $ having order at least n. Finally, as $\phi $ is elliptic of order n, the zeros of $\phi $ have order at most n.

The classical theorem of Kiepert [Reference Kiepert7] to which we refer in our proof of Theorem 2.1 asserts the following:

(2.5) $$ \begin{align} \mathrm{Det} \, \mathbb{P}_{k - 1} (z) = (-1)^{k-1} \ [1! \, 2! \, \cdots \, (k - 1)! ]^2 \frac{\sigma (k z)}{\sigma (z)^{k^2}}, \end{align} $$

with the notation established in our proof. Our application of this classical theorem rests on the fact that the zeros of $\sigma $ are precisely the periods of p.

Henceforth, we choose the constants $a_1, \ldots , a_{n-1}$ that are singled out by Theorem 2.1. The $\Lambda _p$ -periodic meromorphic function $\phi $ then has not only poles of order n but also zeros of order n. Accordingly, $\phi $ then has globally meromorphic nth roots; these nth roots are n in number and may be distinguished by their residues at the origin, these being the nth roots of unity since $z^n \phi (z) \to 1$ as $z \to 0$ . We define the n-ary elliptic function $\psi $ to be the meromorphic nth root of $\phi $ whose residue at the origin is unity itself; thus,

$$ \begin{align*} \psi^n = \phi \quad \mathrm{and} \quad \mathrm{Res}_{\, 0} \psi = 1. \end{align*} $$

Note that $\phi $ has $\Lambda _p$ as its pole lattice, and its zeros are precisely the points congruent to $\alpha $ modulo this lattice. Naturally, the poles and zeros of $\psi $ coincide with those of $\phi $ but are all simple. It is at once evident that the meromorphic function $\psi $ is indeed elliptic as claimed: addition of $2 \omega $ or $2 \omega '$ fixes $\psi ^n = \phi $ and therefore multiplies $\psi $ by an nth root of unity, whence $\psi $ has $2 n \omega $ and $2 n \omega '$ as periods; however, these periods do not constitute a fundamental pair.

Theorem 2.2. The n-ary elliptic function $\psi $ satisfies the identity

$$ \begin{align*} \psi(z) \psi(z + \alpha) \, \cdots \, \psi(z + (n - 1) \alpha) = \psi ' (\alpha) \psi(2 \alpha) \, \cdots \, \psi( (n - 1) \alpha). \end{align*} $$

Proof. Consider the n-fold product on the left as a function of z. The poles and zeros of the n factors cancel one with another, in cyclic fashion, whence the product is constant. Evaluation of this constant may be effected by passage to the limit as $z \to 0$ , since then

$$ \begin{align*} \psi(z) \psi(z + \alpha) = \{\psi(z) z \} \left\{ \frac{\psi(z + \alpha) - \psi(\alpha)}{z} \right\} \to \psi'(\alpha).\\[-40pt] \end{align*} $$

Immediately, we see that $2 \omega $ is a period of $\psi $ : simply observe the effect of replacing z by $z + \alpha $ in the identity of Theorem 2.2.

Proof. Already, $2 \omega $ and $2 n \omega '$ are periods of $\psi $ . Let $\Pi _n$ be the parallelogram with $0$ , $2 \omega $ , $2 n \omega '$ , $2 \omega + 2 n \omega '$ as its vertices (but translated slightly, so as to make $0$ an interior point). In $\Pi _n$ the elliptic function $\psi $ has (simple) poles at the points

$$ \begin{align*} 0, \, 2 \omega', \, \ldots , \, 2(n-1) \omega' \end{align*} $$

and (simple) zeros at the points

$$ \begin{align*} \alpha, \, \alpha + 2 \omega', \, \ldots , \, \alpha + 2 (n-1) \omega'. \end{align*} $$

Now let m be a positive divisor of n. In the (similarly shifted) parallelogram $\Pi _m$ with $0$ , $2 \omega $ , $2 m \omega '$ , $2 \omega + 2 m \omega '$ as its vertices, $\psi $ has pole sum

$$ \begin{align*} 0 + 2 \omega' + \cdots + 2(m-1) \omega' = m (m - 1) \omega' \end{align*} $$

and zero sum

$$ \begin{align*} \alpha + (\alpha + 2 \omega') + \cdots + (\alpha + 2(m-1) \alpha) = m \alpha + m (m-1) \omega', \end{align*} $$

whence

$$ \begin{align*} \mathrm{(zero \; sum)} - \mathrm{(pole \; sum)} = m \alpha = \frac{m}{n} \cdot 2 \omega. \end{align*} $$

If $\Pi _m$ is a period parallelogram for $\psi $ then this difference is a period of $\psi $ and therefore of $\phi $ ; thus, if $m < n$ then $\Pi _m$ is not a period parallelogram and so the period pair $(2 \omega , 2 n \omega ')$ is indeed fundamental.

From this, it follows that addition of $2 \omega '$ to its argument multiplies $\psi $ by a primitive nth root of unity. In fact, we can identify this nth root of unity explicitly: it is precisely the ‘first’ nth root of unity, namely

$$ \begin{align*} \varepsilon : = \exp \{ 2 \pi i / n \}. \end{align*} $$

Theorem 2.4. The n-ary elliptic function $\psi $ satisfies the identity

$$ \begin{align*} \psi(z + 2 \omega') = \varepsilon \cdot \psi(z). \end{align*} $$

Proof. Almost as in the proof of Theorem 2.3, we take the following full sets of $\Lambda _P$ -incongruent zeros and poles of $\psi $ , all of them simple: for the zeros

$$ \begin{align*} a_0 = \alpha, \, a_1 = \alpha + 2 \omega', \, \ldots , \, a_{n-1} = \alpha + 2 (n-1) \omega' \end{align*} $$

and for the poles

$$ \begin{align*} b_0 = 2 \omega, \, b_1 = 2 \omega', \, \ldots , \, b_{n-1} = 2(n-1) \omega'. \end{align*} $$

As the zero sum and pole sum agree, an elliptic function f may be defined by

$$ \begin{align*} f (z) = \prod_{j = 0}^{n - 1} \frac{\Sigma (z - a_j)}{\Sigma (z - b_j)} \end{align*} $$

where $\Sigma $ is the sigma function associated to the lattice $\Lambda _P$ . As $\psi $ and f share a period lattice and have precisely the same poles and zeros to matching order, there exists $\mu = \mathrm {Res}_{\, 0}f \in \mathbb {C}$ such that

$$ \begin{align*} \mu \, \psi = f. \end{align*} $$

The precise value of $\mu $ is immaterial for our purposes: its mere existence is sufficient, as it implies that identically

$$ \begin{align*} \frac{\psi(z + 2 \omega')}{\psi(z)} = \frac{ f (z + 2 \omega')}{f (z)}. \end{align*} $$

The last-mentioned ratio is readily calculated using appropriate care and the quasiperiodicity of $\Sigma $ , according to which

$$ \begin{align*} \Sigma (z + 2 \Omega) &= - \exp \{ 2 Z(\Omega) [z + \Omega] \} \Sigma (z),\\ \Sigma (z + 2 \Omega') &= - \exp \{ 2 Z(\Omega') [z + \Omega'] \} \Sigma (z), \end{align*} $$

and is found to have the value

$$ \begin{align*} \exp \left\{ \frac{4}{n} [Z(\Omega) \Omega' - Z(\Omega') \Omega ] \right\} \end{align*} $$

where we recall that $\Omega = \omega $ and $\Omega ' = n \omega '$ . Finally, as $\Omega ' / \Omega $ lies in the upper halfplane, the Legendre relation

$$ \begin{align*} Z(\Omega) \Omega' - Z(\Omega') \Omega = \tfrac{1}{2} \pi i \end{align*} $$

ends the proof.

Note that this is a consequence of our requiring that the ratio $\omega ' / \omega $ be in the upper halfplane; were we to assume instead that $\omega ' / \omega $ has negative imaginary part, then $\varepsilon $ would be replaced by its reciprocal.

The ternary elliptic functions of Du Val enjoy a variety of interesting properties, many of which are explored in [Reference Du Val3, Reference Du Val4]. The same is true for the n-ary elliptic functions; here we offer but a couple of examples. Thus, $\psi $ satisfies the reflection identity

$$ \begin{align*} \psi( \alpha - z) \psi(z) = - \psi ' (\alpha). \end{align*} $$

Viewing the left-hand side as a function of z, the poles and zeros of the one factor cancel the zeros and poles of the other, so the elliptic product is constant; its value appears in the limit as $z \to 0$ since $\psi $ has unit residue at the origin. Also, $\psi $ satisfies the identity

$$ \begin{align*} \psi(z) \psi (- z) = p(\alpha) - p(z). \end{align*} $$

The elliptic functions of z on the two sides of this identity have the same poles (double, at points of the lattice $\Lambda _p$ ) and the same zeros (simple, at points congruent to $\pm \alpha $ modulo $\Lambda _p$ ); their ratio is therefore a constant, whose value is unity since $\mathrm {Res}_{\, 0} \psi = 1$ and $z^2 p(z) \to 1$ as $z \to 0$ .

Our account thus far has already involved the fact that, up to a multiplicative constant, an elliptic function with given periods is determined by its poles and zeros, via a sigma function expansion. Our further discussion will make use of the fact that, up to an additive constant, an elliptic function with given periods is determined by its principal parts at its poles, via a zeta function expansion.

Note that $P = \wp (-; \omega , n \omega ')$ is the $\wp $ -function coperiodic with the n-ary elliptic function $\psi $ , which has $\Lambda _P$ as its period lattice. Recalling that $\psi $ has its poles (all of them simple) at the points of $\Lambda _p$ , we may take $0, 2 \omega ', \ldots , 2 (n - 1) \omega '$ as a full set of poles, incongruent modulo periods. As $\psi $ has unit residue at the origin, Theorem 2.4 implies that $\psi $ has residue $\varepsilon ^j$ at each pole $2 j \omega '$ , these residues having zero sum. Now consider the function $\psi _0$ defined by

$$ \begin{align*} \psi_0 (z) = \sum_{j = 0}^{n - 1} \varepsilon^j \, Z(z - 2 j \omega') \end{align*} $$

where Z is the zeta function associated to the lattice $\Lambda _P$ . The elliptic functions $\psi $ and $\psi _0$ share the period lattice $\Lambda _P$ and have the very same poles, at which they have the very same principal parts; as remarked above, it follows that $\psi $ and $\psi _0$ differ by an additive constant, which we identify to produce the following expression for $\psi $ .

Theorem 2.5. The n-ary elliptic function $\psi $ is given by

$$ \begin{align*} \psi(z) = \frac{2 Z(n \omega')}{\varepsilon - 1} + \sum_{j = 0}^{n - 1} \varepsilon^j \, Z(z - 2 j \omega'). \end{align*} $$

Proof. Say $\psi = c + \psi _0$ with $c \in \mathbb {C}$ constant. Recall that the zeta function Z is quasiperiodic: in particular, it satisfies the identity

$$ \begin{align*} Z (z + 2 \Omega') = Z(z) + 2 Z(\Omega'). \end{align*} $$

It follows easily from this that $\psi _0$ satisfies the identity

$$ \begin{align*} \psi_0 (z + 2 \omega') - \varepsilon \cdot \psi_0 (z) = 2 Z (\Omega'), \end{align*} $$

whence from $\psi = c + \psi _0$ we deduce that

$$ \begin{align*} \psi(z + 2 \omega') - \varepsilon \cdot \psi(z) = (1 - \varepsilon) c + 2 Z (\Omega'), \end{align*} $$

and reference to Theorem 2.4 concludes the proof.

In the $n = 3$ case this result is tantamount to the first construction of a ternary elliptic function by Du Val (see [Reference Du Val3, page 136] and [Reference Du Val4, page 119]).

Incidentally, we note that Theorem 2.5 implies a cyclotomic decomposition of the zeta function at a half-period, on account of the fact that $\psi (\alpha )$ vanishes:

$$ \begin{align*} 2 Z (\Omega') = (1 - \varepsilon) \, \sum_{j = 0}^{n - 1} \varepsilon^j \, Z(\alpha - 2 j \omega'). \end{align*} $$

We may reformulate this identity in more familiar terms by renaming $\Omega $ and $\Omega '$ as $\omega $ and $\omega '$ : this results in

$$ \begin{align*} 2 \zeta (\omega') = (1 - \varepsilon) \, \sum_{j = 0}^{n - 1} \varepsilon^j \, \zeta ( [2 \omega - 2 j \omega' ] / n). \end{align*} $$

3 The very remarkable function of Halphen

Here we offer an entirely different approach to the n-ary elliptic function, prompted by an inspection of [Reference Halphen6] by Halphen. There, his ‘very remarkable’ function appears at Equation (24) on page 224, in a section devoted to ‘Fonctions analogues à sn u, cn u, dn u’, and emerges from his prior discussion involving the classical results of Kiepert, Frobenius and Stickelberger. Here, we do not propose to reproduce the genesis of this function in [Reference Halphen6], whose perusal we recommend to the reader; instead, we present the ‘very remarkable’ function as an important object of study in its own right, by means of which to represent and investigate an n-ary elliptic function.

Recall that the n-ary elliptic function $\psi $ has simple poles at the points of the lattice $\Lambda _p$ and has simple zeros at the points congruent to $\alpha $ modulo $\Lambda _p$ . As a function of z, the ratio $\sigma (z - \alpha ) / \sigma (z)$ plainly has these poles and zeros, but it fails to be elliptic. Taking into account the fact that quasiperiodicity of the sigma function $\sigma $ introduces exponentials when its argument is shifted by elements of $\Lambda _p$ , we modify this ratio and consider (with Halphen) the function $\psi _*$ defined by

$$ \begin{align*} \psi_*(z) = \frac{\sigma (z - \alpha)}{\sigma (z)} \, \exp \{ \beta z \} \end{align*} $$

where $\beta \in \mathbb {C}$ is a constant whose value will be chosen shortly. The aforementioned quasiperiodicity of $\sigma $ leads directly to the identities

$$ \begin{align*} \psi_* (z + 2 \omega) &= \psi_*(z) \, \exp \{ 2 [ \beta \omega - \alpha \eta] \},\\ \psi_* (z + 2 \omega ') &= \psi_*(z) \, \exp \{ 2 [\beta \omega' - \alpha \eta']\}, \end{align*} $$

where $\eta = \zeta (\omega )$ and $\eta ' = \zeta (\omega ')$ as is customary. We choose

$$ \begin{align*} \beta = \frac{\alpha}{\omega} \, \eta = \frac{2 \eta}{n} \end{align*} $$

so that $\psi _*$ has $2 \omega $ as a period. It further follows that

$$ \begin{align*} \beta \omega' - \alpha \eta' = \frac{2 \eta}{n} \omega' - \frac{2 \omega}{n} \eta' = \frac{2}{n} [ \eta \omega' - \omega \eta'] \end{align*} $$

which the Legendre relation $ \eta \omega ' - \omega \eta ' = \tfrac {1}{2} \pi i$ converts to

$$ \begin{align*} \beta \omega' - \alpha \eta' = \frac{\pi i}{n} \end{align*} $$

and results in

$$ \begin{align*} \psi_* (z + 2 \omega') = \psi_* (z) \, \exp \left\{ \frac{2 \pi i}{n} \right\} = \psi_*(z) \cdot \varepsilon \end{align*} $$

so that $\psi _*$ has $2 n \omega '$ as a period.

The following representation of the n-ary elliptic function $\psi $ is now almost immediate.

Theorem 3.1. The n-ary elliptic function $\psi $ is given by

$$ \begin{align*} \psi(z) = \frac{\sigma (\alpha - z)}{\sigma(\alpha) \sigma(z)} \exp \{ \beta z\} \end{align*} $$

where $\alpha = 2 \omega / n$ and $\beta = 2 \zeta (\omega ) / n$ .

Proof. The functions $\psi _*$ and $\psi $ have precisely the same poles and zeros by design; as we have just seen, they also share $2 \omega $ and $2 n \omega '$ as periods. It follows that $\psi _*$ is a constant multiple of $\psi $ : namely, $\psi _* = (\mathrm {Res}_{\, 0} \psi _*) \, \psi $ where

$$ \begin{align*} \mathrm{Res}_{\, 0} \psi_* = \lim_{z \to 0} \left\{ \frac{z}{\sigma(z)} \sigma (z - \alpha) \exp \{ \beta z \} \right\} = \sigma (- \alpha) = - \sigma (\alpha) \end{align*} $$

whence the claim in the theorem.

This elegant expression for the n-ary elliptic function $\psi $ naturally facilitates alternative proofs of its various properties. Thus, the reflection property $\psi (\alpha - z) \psi (z) = - \psi ' (\alpha )$ may be established as follows: on the one hand, simple cancellations yield

$$ \begin{align*} \psi(\alpha - z) \psi(z) = \frac{\exp \{ \beta \alpha \} }{\sigma(\alpha)^2}; \end{align*} $$

on the other hand,

$$ \begin{align*} \psi ' (\alpha) = \lim_{z \to \alpha} \frac{\psi(z)}{z - \alpha} = \lim_{z \to \alpha} \left\{ \frac{\sigma (\alpha - z)}{z - \alpha} \frac{\exp \{ \beta z\}}{\sigma (\alpha) \sigma (z)} \right\} = - \frac{\exp \{ \beta \alpha \} }{\sigma(\alpha)^2}. \end{align*} $$

Also, the property $\psi (z) \psi (-z) = p(\alpha ) - p(z)$ amounts to a familiar classical relation between the $\wp $ -function p and the sigma function $\sigma $ : indeed,

$$ \begin{align*} \psi(z) \psi(-z) = \frac{\sigma (\alpha - z)}{\sigma (\alpha) \sigma (z)} \exp \{\beta z\} \frac{\sigma (\alpha + z)}{\sigma (\alpha) \sigma (-z)} \exp \{- \beta z\} = - \frac{\sigma (\alpha - z) \sigma (\alpha + z)}{\sigma (\alpha)^2 \sigma(z)^2}. \end{align*} $$

As a further example, Theorem 3.1 leads to an evaluation of the (constant) product displayed in Theorem 2.2 as follows:

$$ \begin{align*} \psi(z + \alpha) \cdot \, \cdots \, \cdot \psi(z + n \alpha) = \frac{(-1)^{n-1}}{\sigma(\alpha)^n} \, \exp \left\{ \frac{2}{n} \omega \, \zeta(\omega) \right\}. \end{align*} $$

In the $n = 2$ case the elliptic function of Theorem 3.1 is entirely classical, as Halphen notes in [Reference Halphen6]. To make contact with the relevant notation, let the $\wp $ -function p have $\omega _1$ and $\omega _2$ as a fundamental pair of half-periods and write $\omega _3 = - \omega _1 - \omega _2$ for symmetry. For $j = 1, 2, 3$ , let $\eta _j = \zeta (\omega _j)$ and define the sigma function $\sigma _j$ by

$$ \begin{align*} \sigma_j (z) = \exp \{ - \eta_j z\} \, \frac{\sigma (z + \omega_j)}{\sigma (\omega_j)} \end{align*} $$

(see [Reference Whittaker and Watson11, page 448], [Reference Du Val4, page 110] or [Reference Lawden8, page 151]). The elliptic ratio $\sigma _j / \sigma $ is then a square root of ${p - p(\omega _j)}$ and hence a Jacobian function (see [Reference Whittaker and Watson11, page 451], [Reference Du Val4, page 110] or [Reference Lawden8, page 158]). Quasiperiodicity of $\sigma $ reveals this ratio to be the very function that appears in Theorem 3.1.

4 The elliptic functions of Dixon

In [Reference Dixon2] Dixon introduces and quite thoroughly analyses a family of third-order elliptic functions that effectively lay dormant for more than a century, until their connections with combinatorics, with geometry, and more recently with generalised trigonometry, were exposed. His elliptic functions $s = \mathrm {sm} (-, \kappa )$ and ${c = \mathrm {cm} (-, \kappa )}$ of complex modulus $\kappa $ , not a cube root of $-1$ , are the solutions to the initial value problem

$$ \begin{align*} s ' = c^2 - \kappa \, s, \quad s(0) = 0, \end{align*} $$

$$ \begin{align*} c ' = \kappa \, c - s^2, \quad c(0) = 1. \end{align*} $$

In [Reference Robinson10] we show that the ternary elliptic functions of Du Val are intimately related to the classical elliptic functions sm and cm of Dixon: indeed, they are so related that either can be obtained from the other by elementary shifts. Our purpose in this section is to present a brief account of this relationship, taking advantage of the realisation that the ternary elliptic function is the very remarkable function of Halphen in the $n = 3$ case.

We begin by deriving a first-order differential equation that is satisfied by $\psi $ . This differential equation appears first in [Reference Du Val3, Section 20] and then as [Reference Robinson10, Theorem 8] but derived by a very different line of argument.

Theorem 4.1. The ternary elliptic function $\psi $ satisfies the differential equation

$$ \begin{align*} \psi ' (z) = A \, \psi(z) - \psi (-z)^2 \end{align*} $$

where the constant A is given by

$$ \begin{align*} A = 2 \zeta (\alpha) - \frac{4}{3} \zeta (\omega) = - \frac{1}{3}\frac{p "(\alpha)}{p '(\alpha)}. \end{align*} $$

Proof. Logarithmic differentiation of the Halphen expression for $\psi $ in Theorem 3.1 yields

$$ \begin{align*} \psi '(z) = \{ \beta - \zeta (\alpha - z) - \zeta(z) \} \psi (z) \end{align*} $$

with $\beta = 2 \eta / 3$ and $\eta = \zeta (\omega )$ as usual. Further application of Theorem 3.1 then shows that

$$ \begin{align*} A(z) : = \frac{\psi '(z) + \psi(-z)^2}{\psi (z)} = \{\beta - \zeta (\alpha - z) - \zeta (z) \} + \frac{\sigma (\alpha + z)^2}{\sigma (\alpha) \sigma (\alpha - z) \sigma (z)} \, \exp \{-3 \beta z \} \end{align*} $$

and we claim that this function A is constant. Note that

$$ \begin{align*} \zeta (2 \alpha) = \zeta (- \alpha + 2 \omega) = \zeta (- \alpha) + 2 \eta = 2 \eta - \zeta (\alpha), \end{align*} $$

whence the vanishing of $\sigma $ at $0$ implies that

$$ \begin{align*} A ( - \alpha) = \beta - \zeta (2 \alpha) + \zeta (\alpha) = \beta + 2 \zeta (\alpha) - 2 \eta = 2 \zeta (\alpha) - 4 \zeta (\omega) / 3. \end{align*} $$

Accordingly, we claim that this is the constant value of A; equivalently, we claim that identically

$$ \begin{align*} \frac{\sigma (\alpha + z)^2}{\sigma (\alpha) \sigma (\alpha - z) \sigma (z)} \, \exp \{- 2 \eta z \} = 2 \zeta (\alpha) + \zeta (\alpha - z) + \zeta (z) - 2 \eta. \end{align*} $$

As a function of z, each side of this equation is elliptic, with $2 \omega $ and $2 \omega '$ as periods: on the left-hand side, this follows from (multiplicative) quasiperiodicity of $\sigma $ ; on the right-hand side, it follows from (additive) quasiperiodicity of $\zeta $ . Further, it is readily confirmed that each side has simple poles of residue $+1$ at the points of the lattice $\Lambda _p$ and simple poles of residue $-1$ at points of the coset $\alpha + \Lambda _p$ . Thus, the two sides share both periods and poles along with their principal parts, so the two sides differ by a constant, which vanishes by evaluation at $z = - \alpha $ . Finally, the identification of ${2 \zeta (\alpha ) - 4 \zeta (\omega ) / 3}$ as $- p " (\alpha ) / 3 p'(\alpha )$ is an easy consequence of the zeta function duplication law, according to which

$$ \begin{align*} 2 \zeta (2 z) - 4 \zeta (z) = p"(z) / p'(z). \\[-32pt] \end{align*} $$

Before we proceed further, it is convenient to observe here the fact that

$$ \begin{align*} \psi(- \alpha)^3 = p'(\alpha). \end{align*} $$

To see this, note first that

$$ \begin{align*} \sigma (2 \alpha) = \sigma (- \alpha + 2 \omega) = \sigma (\alpha) \exp \{ \eta \alpha \} \end{align*} $$

by quasiperiodicity. Now

$$ \begin{align*} \psi(- \alpha)^3 = \left\{\frac{\sigma (2 \alpha)}{\sigma(\alpha) \sigma(- \alpha)} \exp \{ - \beta \alpha \} \right\}^3 = - \frac{\exp \{ \eta \alpha \}}{\sigma(\alpha)^3} = - \frac{\sigma( 2 \alpha)}{\sigma(\alpha)^4} = p ' (\alpha) \end{align*} $$

where the last equation instances the sigma function duplication law.

As a consequence of the preceding theorem we have the next, which appears first in [Reference Du Val3, Section 19], then in [Reference Du Val4, Section 59] and subsequently as [Reference Robinson10, Theorem 6].

Theorem 4.2. The ternary elliptic function $\psi $ satisfies the cubic identity

$$ \begin{align*} \psi(z)^3 + \psi(-z)^3 - 3 A \psi(z) \psi(-z) = p ' (\alpha). \end{align*} $$

The following theorem is established in [Reference Robinson10] but without the assistance of Halphen. Here, we use the convenient abbreviation

$$ \begin{align*} \mu : = \psi(- \alpha). \end{align*} $$

Theorem 4.3. The functions s and c defined by

$$ \begin{align*} s(z) = \frac{1}{\mu} \, \psi \left(\alpha - \frac{z}{\mu}\right) \quad and \quad c(z) = \frac{1}{\mu} \, \psi \left(\frac{z}{\mu} - \alpha \right) \end{align*} $$

are the elliptic functions of Dixon with modulus

$$ \begin{align*} \kappa = - \frac{1}{3 \mu} \, \frac{p"(\alpha)}{p'(\alpha)} = - \frac{1}{3} \, \frac{p"(\alpha)}{\psi (- \alpha)^4}. \end{align*} $$

Proof. Plainly, $s(0) = 0$ since $\psi (\alpha ) = 0$ and $c(0) = 1$ by definition of $\mu $ . Further, direct application of Theorem 4.1 shows that

$$ \begin{align*} s' = c^2 - \kappa s \quad \mathrm{and} \quad c' = \kappa c - s^2 \end{align*} $$

with the modulus $\kappa = A / \mu $ as advertised.

We remark here that the stated modulus $\kappa $ is not a cube root of $-1$ ; we see this as follows. First, recall that the $\wp $ -function p satisfies the ‘eliminant’ equation ${(p')^2 = 4 p^3 - g_2 p - g_3}$ and the derived equation $p " = 6 p^2 - \tfrac {1}{2} g_2$ along with the resulting $p "' = 12 p p'$ ; here, the constants $g_2$ and $g_3$ are the invariants of p. As $\alpha $ is a third period of p, the Kiepert formula (2.5) informs us that

$$ \begin{align*} 0 = p'(\alpha) p"'(\alpha) - p"(\alpha)^2 = 12 p(\alpha) p'(\alpha)^2 - p"(\alpha)^2. \end{align*} $$

If possible, suppose $\kappa ^3 = -1$ : that is, suppose $p" (\alpha )^3 = 27 \psi (- \alpha )^{12} = 27 p'(\alpha )^4$ . By elimination of the derivatives between the equations here assembled, it may be deduced that $g_2 = 4 p(\alpha )^2 / 3$ and $g_3 = 8 p(\alpha )^3 / 27$ and therefore that $g_2^3 = 27 g_3^2$ ; for a $\wp $ -function, this is impossible.

We refer to [Reference Robinson10] for a proof of the ‘converse’ fact that for any $- \kappa $ other than a cube root of unity, there exist a $\wp $ -function p and third period $\alpha $ with associated ternary function $\psi $ such that the formulas in Theorem 4.3 recover the elliptic functions $s = \mathrm {sm} (-, \kappa )$ and $c = \mathrm {cm} (-, \kappa )$ of Dixon; in [Reference Robinson10] it is further shown that we may insist upon $\mu : = \psi (- \alpha ) = 1$ and so simplify the formulas in Theorem 4.3.

Incidentally, Theorems 4.2 and 4.3 combine to show that s and c give an elliptic parametrisation of the cubic to which Dixon refers in the title of [Reference Dixon2].

Finally, we draw attention to the fact that Cooper presents an alternative, q-theoretic construction of the Dixonian functions sm and cm in Chapter 1 of his recent comprehensive text [Reference Cooper1] on Ramanujan’s theta functions. Of course, once his q-analogue of the sine function is expressed in terms of the Weierstrass sigma function, the echo of the Halphen function in his formulas for sm and cm is promoted to an exact agreement with the formulas for sm and cm that we present here.