Sigma

In this section we derive an addition formula for the Weiersrass $\sigma$ function. We also derive several different variants for other sigma products and theta functions.

In their 1877 paper [1], Frobenius and Stickelberger start with the ingenious formula:

\begin{equation} \label{eq:fsz} \begin{vmatrix} 0 & 1 & 1 & \ldots & 1 \\ 1 & \zeta(u_1+v_1) & \zeta(u_1+v_2) & \ldots & \zeta(u_1+v_n) \\ 1 & \zeta(u_2+v_1) & \zeta(u_2+v_2) & \ldots & \zeta(u_2+v_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \zeta(u_n+v_1) & \zeta(u_n+v_2) & \ldots & \zeta(u_n+v_n) \\ \end{vmatrix} \space = \space - \frac {\sigma\left(\sum\limits_{i=1}^n (u_i + v_i)\right) \space \prod\limits_{1 \le i \lt j \le n} \space \sigma(u_i - u_j) \sigma(v_i - v_j)} {\prod\limits_{i=1}^n \prod\limits_{j=1}^n \sigma(u_i + v_j)} \end{equation}

From this they obtain their classic addition formula \begin{equation} \label{eq:fsn} \begin{vmatrix} 1 & \wp(u_1) & \wp'(u_1) & \ldots & \wp^{(n-2)}(u_1) \\ 1 & \wp(u_2) & \wp'(u_2) & \ldots & \wp^{(n-2)}(u_2) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \wp(u_n) & \wp'(u_n) & \ldots & \wp^{(n-2)}(u_n) \\ \end{vmatrix} \space = \space (-1)^{n-1} \space \prod\limits_{i=1}^{n-1} i! \space \frac {\sigma\left(\sum\limits_{i=1}^n u_i\right) \space \prod\limits_{1 \le i \lt j \le n} \sigma(u_j-u_i)} {\prod\limits_{i=1}^n \sigma(u_i)^n} \end{equation} essentially by taking the limit of \eqref{eq:fsz} as $v_i \rightarrow 0$. They also obtain the Kiepert formula \begin{equation} \label{eq:fslimit} \begin{vmatrix} \wp'(u) & \wp''(u) & \ldots & \wp^{(n-1)}(u) \\ \wp''(u) & \wp'''(u) & \ldots & \wp^{(n)}(u) \\ \vdots & \vdots & \ddots & \vdots \\ \wp^{(n-1)}(u) & \wp^{(n)}(u) & \ldots & \wp^{(2n-3)}(u) \\ \end{vmatrix} \space = \space (-1)^{n-1} \space \left(\prod\limits_{i=1}^{n-1} i!\right)^2 \space \frac {\sigma(n u)} {\sigma(u)^{n^2}} \end{equation} by taking the limit of \eqref{eq:fsn} as $u_i \rightarrow u$.

COROLLARY

Substituting \eqref{eq:fsn} with $n=2$ and \eqref{eq:fslimit} with $n=2$ \begin{equation*} \wp(u)-\wp(v) \space = \space -\frac {\sigma(u-v)\sigma(u+v)} {\sigma^2(u)\sigma^2(v)} \qquad\qquad\qquad \wp'(u) \space = \space - \frac {\sigma(2u)} {\sigma^4(u)} \end{equation*} into \eqref{eq:fsn} with $n=3$ \begin{equation*} 2 \frac {\sigma(u+v+w)\sigma(v-u)\sigma(w-u)\sigma(w-v)} {\sigma^3(u)\sigma^3(v)\sigma^3(w)} \space = \space \left[\wp(w)-\wp(v)\right]\wp'(u) \space + \space \left[\wp(u)-\wp(w)\right]\wp'(v) \space + \space \left[\wp(v)-\wp(u)\right]\wp'(w) \end{equation*} gives a three variable addition formula for the Weierstrass sigma function \begin{equation*} \sigma(u+v+w) \space = \space \frac 1 2 \left[ \frac {\sigma(v)\sigma(w)\sigma(v+w)} {\sigma(u)\sigma(u-v)\sigma(u-w)} \sigma(2u) \space + \space \space \frac {\sigma(u)\sigma(w)\sigma(u+w)} {\sigma(v)\sigma(v-u)\sigma(v-w)} \sigma(2v) \space + \space \space \frac {\sigma(u)\sigma(v)\sigma(u+v)} {\sigma(w)\sigma(w-u)\sigma(w-v)} \sigma(2w) \right] \end{equation*} Substituting into \eqref{eq:fsn} with $n=4$ \begin{equation*} -12 \frac {\sigma(u+v+w+x)\sigma(v-u)\sigma(w-u)\sigma(x-u)\sigma(w-v)\sigma(x-v)\sigma(x-w)} {\sigma^4(u)\sigma^4(v)\sigma^4(w)\sigma^4(x)} \space = \space 6\left[\wp(w)-\wp(v)\right]\left[\wp(x)-\wp(v)\right]\left[\wp(x)-\wp(w)\right]\wp'(u) \space + \space \ldots \end{equation*} gives a four variable addition formula for the Weierstrass sigma function \begin{equation*} \sigma(u+v+w+x) \space = \space \frac 1 2 \left[ \frac {\sigma(v+x)\sigma(w+x)\sigma(v+w)} {\sigma(u-x)\sigma(u-v)\sigma(u-w)} \sigma(2u) \space + \space \space \frac {\sigma(u+x)\sigma(w+x)\sigma(u+w)} {\sigma(v-x)\sigma(v-u)\sigma(v-w)} \sigma(2v) \space + \space \space \frac {\sigma(u+x)\sigma(v+x)\sigma(u+v)} {\sigma(w-x)\sigma(w-u)\sigma(w-v)} \sigma(2w) \space + \space \space \frac {\sigma(u+v)\sigma(u+w)\sigma(v+w)} {\sigma(x-u)\sigma(x-v)\sigma(x-w)} \sigma(2x) \right] \end{equation*}

Using the $\zeta$ addition formula we can also write \eqref{eq:fsz} in terms of $\wp$ as follows \begin{equation} \label{eq:fszp} (\tfrac 1 2)^{n-1} \begin{vmatrix} 0 & 1 & 1 & \ldots & 1 \\ 1 & \frac {\wp'(u_1) - \wp'(v_1)} {\wp(u_1) - \wp(v_1)} & \frac {\wp'(u_1) - \wp'(v_2)} {\wp(u_1) - \wp(v_2)} & \ldots & \frac {\wp'(u_1) - \wp'(v_n)} {\wp(u_1) - \wp(v_n)} \\ 1 & \frac {\wp'(u_2) - \wp'(v_1)} {\wp(u_2) - \wp(v_1)} & \frac {\wp'(u_2) - \wp'(v_2)} {\wp(u_2) - \wp(v_2)} & \ldots & \frac {\wp'(u_2) - \wp'(v_n)} {\wp(u_2) - \wp(v_n)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \frac {\wp'(u_n) - \wp'(v_1)} {\wp(u_n) - \wp(v_1)} & \frac {\wp'(u_n) - \wp'(v_2)} {\wp(u_n) - \wp(v_2)} & \ldots & \frac {\wp'(u_n) - \wp'(v_n)} {\wp(u_n) - \wp(v_n)} \\ \end{vmatrix} \space = \space - \frac {\sigma\left(\sum\limits_{i=1}^n (u_i + v_i)\right) \space \prod\limits_{1 \le i \lt j \le n} \space \sigma(u_i - u_j) \sigma(v_i - v_j)} {\prod\limits_{i=1}^n \prod\limits_{j=1}^n \sigma(u_i + v_j)} \end{equation}

Extended Frobenius-Stickelberger Formula

The classic Frobenius-Stickelberger formula \eqref{eq:fsn} can be extended to general elliptic functions. The set of elliptic functions with poles of order at most $n_j \gt 0$ at $a_j$ for $j=1\dots k$ form a vector space of dimension $n = \sum n_j$. If $f_1 \dots f_n$ is a basis for that vector space then

\begin{equation} \label{eq:fsx} \begin{vmatrix} f_1(u_1) & f_2(u_1) & \ldots & f_n(u_1) \\ f_1(u_2) & f_2(u_2) & \ldots & f_n(u_2) \\ \vdots & \vdots & \ddots & \vdots \\ f_1(u_n) & f_2(u_n) & \ldots & f_n(u_n) \\ \end{vmatrix} \space = \space K \space \frac {\sigma\left(\sum\limits_{i=1}^n u_i - \sum\limits_{j=1}^k n_j a_j\right) \space \prod\limits_{1 \le i \lt j \le n} \sigma(u_j - u_i)} {\prod\limits_{i=1}^n \prod\limits_{j=1}^k \sigma(u_i - a_j)^{n_j}} \end{equation}

where $K$ is a constant independent of the $u_i$. An explicit formula for $K$ can be obtained as follows. Let $R$ be the matrix of the coefficients of the negative powers in the Laurent expansions of $f_i$ at $a_j$. Each column of $R$ corresponds to one of the $f_i$. Each row of $R$ corresponds to the coefficients of $(z-a_j)^l$ in the Laurent expansions of $f_i(z)$ at one particular point $a_j$ and for one particular $l \lt 0$. The order of the rows is taken as, first the coefficients of $(z-a_1)^{-n_1} \ldots (z-a_1)^{-1}$ at $a_1$, then of $(z-a_2)^{-n_2} \ldots (z-a_2)^{-1}$ at $a_2$ etc. Under this ordering we have

\begin{equation} \label{eq:fsxK} K \space = \space \begin{vmatrix} R_{11} & R_{12} & \ldots & R_{1n} \\ R_{21} & R_{22} & \ldots & R_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ R_{n-1,1} & R_{n-1,2} & \ldots & R_{n-1,n} \\ f_1(u) & f_2(u) & \ldots & f_n(u) \\ \end{vmatrix} \quad \prod\limits_{1 \le i \lt j \le k} \sigma(a_i - a_j)^{n_i n_j} \end{equation}

Let $D$ be the determinant on the LHS of \eqref{eq:fsx}. The following process can be used to convert $D$ to $R$ one row at a time:

Multiply the first row of $D$, and the RHS, by $(u_1 - a_1)^{n_1}$ and then let $u_1 \rightarrow a_1$.
On the LHS this will convert the first row of $D$ to the first row of $R$.
On the RHS it will replace all $u_1$ by $a_1$ except the terms $\sigma(u_1-a_1)^{n_1}$ in the denominator which will go to 1.
Subtract the first row, multiplied by $(u_2 - a_1)^{-n_1}$, from the second. This will remove any powers of $(u_2-a_1)^{-n_1}$ from the Laurent expansion of $f_i(u_2)$ at $a_1$.
Multiply the second row of $D$ and the RHS by $(u_2 - a_1)^{n_1 - 1}$ and then let $u_2 \rightarrow a_1$.
On the LHS this will convert the second row of $D$ to the second row of $R$.
On the RHS it will cancel out the term $\sigma(u_2 - a_1)$, which appeared in the numerator due to the previous row processing, and $n_1$ similar terms in the denominator.
Repeat the process another $n_1 - 2$ times with $u_i \rightarrow a_1$ converting the first $n_1$ rows of $D$ to $R$.
Repeat the process $n_2$ times with $u_i \rightarrow a_2$ converting the next $n_2$ rows of $D$ to $R$.
Repeat for the other $a_i$ up to the second to last row of $D$, converting $n - 1$ rows of $D$ to $R$.
After processing the second to last row, the numerator on the RHS will contain $\sigma(u_n - a_k) \prod_{j=1}^{n - 1} \sigma(u_n - A(u_j))$ where the function $A$ maps each $u_i$ to its corresponding $a_j$. The denominator will contain $\prod_{j=1}^k\sigma(u_n - a_j)^{n_j}$. These terms will cancel each other out, leaving no $u_n$ terms on the RHS!
Replace the $u_n$ in the last row of $D$ by $u$ and we are done, except we need to tally up all the terms on the RHS like $\sigma(a_j-a_i)$.
From the numerator there will be $\prod_{i \lt j}\sigma(a_j - a_i)^{n_i n_j}$. From the denominator there will be a similar term and another with the indices reversed ie. $\prod_{i \lt j}\sigma(a_i - a_j)^{n_i n_j}$. The first two products cancel leaving just the term with reversed indices in the denominator.

As a sanity check we see that if we swap $a_1$ and $a_2$ then, to restore to the original formula, we need to make $n_i n_j$ row swaps in the determinant and $n_i n_j$ sign changes in the product. As expected these sign changes will cancel each other out. Similar if we swap $a_{k-1}$ and $a_k$ but this time we also need to manipulate the row of residues for $a_{k-1}$ to turn them into the residues for $a_k$.

As a further check we see that if we if we make a linear change of basis in \eqref{eq:fsx} both sides of the equation will be multiplied by the same determinant.

Note $\rank(R) = n - 1$ because the sum of the $k$ rows corresponding to the residues of $f_i$ is a row of zeroes. Due to this the last row of residues of $f_i$ at $a_k$, in the formula for $K$, ends up being replaced by the basis functions evaluated at a completely arbitrary point $u$. This may seem surprising but all it is really saying is that a certain linear combination of the basis functions $f_i(u)$ is constant. We may take $u$ to be one of the poles $a_j$ in which case the last row will consist of the coefficients of the $(z-a_j)^0$ terms in the Laurent expansion of $f_i$ at $a_j$.

Nomenclature

I have called \eqref{eq:fsx} the Extended Frobenius-Stickelberger formula because I need a name for it and don't want to confuse it with the original Frobenius-Stickelberger formula \eqref{eq:fsn}. It is more or less equivalent to formula \eqref{eq:fsz} and is a generalisation of formula \eqref{eq:fsn}. Note there are already Generalised Frobenius-Stickelberger formulae in the literature but these refer to the generalisation of \eqref{eq:fsn} to hyperelliptic $\sigma$ functions. For example see [2]. Formula \eqref{eq:fsx} is also closely related to the Vandermonde determinant formula.

COROLLARY

For the basis $2n$-dimensional basis $1,\wp, \dots \wp^n, \wp', \ldots \wp'\wp^{n-2}$ the extended Frobenius-Stickelberger gives, after applying the sub-determinant expansion formula to the left $n+1$ columns \begin{equation*} \sum\limits_{P\in\mathfrak{P}} \left( (-1)^{\order(P)} \prod_{\substack{i \gt j \\ i,j\in P}} \Big(\wp(u_i) - \wp(u_j)\Big) \prod_{\substack{k \gt l \\ k,l\in P'}} \Big(\wp(u_k) - \wp(u_l)\Big) \prod\limits_{m \in P'} \wp'(2u_m) \right) \space = \space 2^{n-1} \frac {\sigma\left(u_1 + u_2 + \ldots u_{2n}\right) \prod\limits_{i \gt j} \sigma(u_i - u_j)} {\prod\limits_{i=1}^{2n} \sigma(u_i)^{2n}} \end{equation*} where $P$ ranges over all subsets $\mathfrak{P}$ of size $n+1$ choosen from the set of integers $1 \ldots 2n$. $P'$ is the complement of $P$ and $\order(P)$ is the number of transpositions required to shuffle the elements of $P$ to the left.

From this the addition formula for the $\sigma$ function can be deduced \begin{equation*} \sigma\left(u_1 + u_2 + \ldots u_{2n}\right) \space = \space \frac 1 {2^{n-1}} \sum\limits_{P\in\mathfrak{P}} \left( \frac {\prod\limits_{\substack{i \lt j \\ i,j\in P}} \sigma(u_i + u_j) \prod\limits_{\substack{k \lt l \\ k,l\in P'}} \sigma(u_k + u_l)} {\prod\limits_{i\in P \space j\in P'} \sigma(u_i - u_j)} \prod\limits_{m \in P'} \sigma(2u_m) \right) \end{equation*} using \begin{equation*} \prod\limits_{i \gt j} \sigma(u_i - u_j) \space = \space (-1)^{\order(P)} \prod\limits_{\substack{i \lt j \\ i,j\in P}} \sigma(u_i - u_j) \prod\limits_{\substack{k \lt l \\ k,l\in P'}} \sigma(u_k - u_l) \prod\limits_{i\in P \space j\in P'} \sigma(u_i - u_j) \end{equation*} Let $u_{2n-1}, u_{2n} \rightarrow 0$. When they are both in $P$ the summand is zero. When they are both in $P'$ the summand is zero. When one is in $P$ and the other is in $P'$ the summands combine to a formula of the form (after judiciously setting a few $u_{2n-1}$ and $u_{2n}$ to zero) \begin{equation} RHS \space = \space \frac 1 2 \sigma(u_1+\ldots u_{2n-2}) \frac {\sigma(2u_{2n-1}) - \sigma(2u_{2n})} {\sigma(u_{2n-1} - u_{2n})} \longrightarrow \sigma(u_1+\ldots u_{2n-2}) \end{equation} verifying the value of the leading coefficient. For example for $n=6$ \begin{equation} \sigma(u_1+u_2+u_3+u_4+u_5+u_6) \space = \space \frac 1 4 \frac {\sigma(u_1+u_2)\sigma(u_3+u_4)\sigma(u_3+u_5)\sigma(u_3+u_6)\sigma(u_4+u_5)\sigma(u_4+u_6)\sigma(u_5+u_6)} {\sigma(u_1-u_3)\sigma(u_2-u_3)\sigma(u_1-u_4)\sigma(u_2-u_4)\sigma(u_1-u_5)\sigma(u_2-u_5)\sigma(u_1-u_6)\sigma(u_2-u_6)} \sigma(2u_1)\sigma(2u_2) \quad + \quad \textsf{14 more terms} \end{equation} Now put $u_1=u_3=0$ to get \begin{equation} \frac 1 4 \frac {\sigma(u_2)\sigma(u_4)\sigma(u_5)\sigma(u_6)\sigma(u_4+u_5)\sigma(u_4+u_6)\sigma(u_5+u_6)} {\sigma(u_2)\sigma(-u_4)\sigma(u_2-u_4)\sigma(-u_5)\sigma(u_2-u_5)\sigma(-u_6)\sigma(u_2-u_6)} \frac {\sigma(2u_1)\sigma(2u_2)} {\sigma(u_1-u_3)} \ldots \end{equation}

EXAMPLE

Extended Kiepert Formula

Letting $u_i \rightarrow u$ in \eqref{eq:fsx} yields an extension of the Kiepert formula:

\begin{equation} \label{eq:fsxlimit} \begin{vmatrix} f_1(u) & f_2(u) & \ldots & f_n(u) \\ f_1'(u) & f_2'(u) & \ldots & f_n'(u) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(u) & f_2^{(n-1)}(u) & \ldots & f_n^{(n-1)}(u) \\ \end{vmatrix} \space = \space K \space \prod\limits_{i=1}^{n-1} {i!} \space \frac {\sigma\left(n u - \sum\limits_{j=1}^k n_j a_j\right)} {\left(\prod\limits_{j=1}^k \sigma(u - a_j)^{n_j}\right)^n} \end{equation}

where $K$ is the constant given in \eqref{eq:fsxK}.

References

[1] F.G. Frobenius and L. Stickelberger Zur Theorie der elliptischen Functionen J. reine angew. Math 83 (1877), 175–179

[2] Yoshihiro Ônishi Determinant Expressions For Hyperelliptic Functions Proceedings of the Edinburgh Mathematical Society, Volume 48, Issue 3, October 2005, pp. 705 - 742

Addition Formula For Sigma And Theta Functions

Nomenclature