4.1. Definitions and Main Properties of Elliptic Functions
Elliptic functions, which are much studied because of their high importance in mathematics and physics (see, e.g., [
10,
11,
12,
13,
14,
15,
16]; whenever possible, below we will mostly cite “encyclopedia-like” Ref. [
10]), are a fertile ground for the approach based on the generalized Littlewood theorem. First, we need to give necessary definitions and specify a notation used, especially because, unfortunately, different notations and conventions still co-exist in the research field of elliptic functions. We define four theta functions as follows:
Here,
(it is named a nome), and
. As functions of
z for any fixed
, they are entire and
periodic, and they are quasiperiodic on the lattice formed by the points
, in the sense that the following relation holds:
And similar relations exist for other theta functions. Here and below, unless specifically stated to the contrary, , but we will not always repeat this statement. The notations and are used on equal footing.
These properties guaranty that for large |
z|, the theta function is at most
O(
exp(
C|z|2)) with some constant
C; hence, in the limit of infinitely large contours, we have the disappearance of the contour integrals
for
j = 1, 2, 3, 4 and
n = 3, 4… The location of zeroes
, which are all simple for these functions, is also well known. Namely, the functions
for
j = 1, 2, 3, 4 have zeroes at the points
,
,
and
, respectively [
10]. Taylor expansions of the theta functions are the following [
10]:
,
,
, and
(that is,
), where:
The order of summation is important for these sums if
j = 1; see additional discussion in our previous paper [
7].
From these formulae, we easily obtain the Taylor expansions of the logarithms of theta functions in some vicinity of zero, like
or
. Note, also useful for the future relation is
which immediately follows from the Taylor series of
given above.
The Weierstrass
function is defined as [
10]
where
is a lattice formed by points
, with again
, and
. We have by definition
,
,
. Below we will use the notation
,
, etc. if an arbitrary lattice is used; if
, which is usually the case, we shall write
,
, etc.
The following Laurent expansion is well known (and evident from the above definition) [
10]:
where
are the Eisenstein series defined by Equation (13a). To simplify the notation, below we may omit
, writing
simply as
. Thus,
and
. Of course, we have
which is often written as
where evidently
.
Jacobi elliptical functions
sn(
z,
k),
cn(
z,
k),
dn(
z,
k) and other similar, are properly scaled ratios of the appropriate theta functions [
10]. They will not be considered here.
4.2. Equalities between Different Elliptic Functions
It is well known that all double-periodic meromorphic functions can be expressed via Weierstrass elliptical functions and their derivatives, and numerous concrete examples of the corresponding expressions/representations are commonly found in the literature [
10,
11,
12,
13,
14,
15,
16]. For this reason, here, we do not aim to consider many corresponding examples. Our purpose is to illustrate the principle and to concentrate on the cases which, for some reasons, are seemingly not widely indicated or even not known in all generality.
Our first example is the Weierstrass sigma function
. From its definition, it is clear that this is an entire function having simple zeroes at the lattice points. Thus, the Weierstrass sigma function
and the elliptic theta function
are both entire functions that have simple zeroes at
. We know the Taylor expansion of the
; see Equation (14). Due to the Lemma 1, the Taylor expansion of the
must be exactly the same starting from the
O(
z4) term:
. Directly from Equation (16), we see that
a = b = 0; hence
Scaling for an arbitrary
immediately gives the “standard” formula [
10]:
where
. Here and below, it might be useful to exploit [
10]
Remark 1. In light of the abovesaid, the Taylor expansion of the function is given by , sf. [10,13]. To fully restore the known results, we need to add the formulae expressing for j = 4, 5, 6… via and , which readily follow from, e.g., the differential equation [10]by equating the Taylor series coefficients obtained by differentiating of Equation (18). Usually, these formulae are written in the form including the coefficients dk of Equation (18a): (Differentiating Equation (18a), we see that the coefficient in front of the term in the Taylor expansion of l.h.s of Equation (22) is , while in the r.h.s it is ).
Our next example of the inverse application of the generalized Littlewood theorem to elliptic functions is the Weierstrass
functions. These functions have second-order poles at the same points where simple zeroes of
function are located, which means that the function
has the same poles as the Weierstrass
function. Here,
and
; again, we use this notation to be consistent with the data presented in [
10].
But contrary to the entire theta function in the power of −1, the Weierstrass ℘ functions have zeroes. The situation is easiest with the functions
, where
,
and
. By construction, these functions have zeroes at points, coinciding with the “demi-lattice” points
,
or
, respectively, and it is clear that these are second-order zeroes. It is also known that there are no other zeroes for functions
. Thus, in particular, for the function
, the positions of second-order zeroes coincide with those of the entire function
, and Lemma 1 gives
, so that
. Coefficients
and
must be equal to zero since other coefficients are incompatible with the double periodicity of both sides of the equality. (The ratio of the squares of theta functions in the r.h.s. here is proportional to the square of Jacobi elliptic function
; see, e.g., [
10] for definitions). Analysis for the value of
z = 0 equating the coefficients in front of the 1/
z2 term in the Taylor expansions shows
. Then, we can use the known equality [
10]:
to write the “standard” form:
Similar equalities can be in the same fashion established for other
functions, which we will not do here; see, e.g., [
10] for the list of formulae. We also will not exercise ourselves on the expression of the ratio of theta functions occurring in these formulae via appropriate Jacobi elliptic functions; see again [
10]. Instead (see our motivation discussed above), we want to underline again that there is nothing special with the functions
. The following theorem, which seems to almost never appear in the monographs devoted to the elliptic functions, holds true.
Theorem 2. Let a be an arbitrary complex number not equal to defined above, and numbers be such that the equality , j = 1, 2, holds true. Let also the difference . Then,where , , Proof. First, we note that due to the argument principle, double-periodic function , having in each fundamental parallelogram one pole of the second order, has in this parallelogram exactly two zeroes taking into account their multiplicity. (Exactly for a = ej, j = 1, 2, 3, excluded by the theorem conditions, this function has one double zero inside the fundamental parallelogram). Thus, for any a, we can find the numbers requested by the theorem. Clearly, the functions and have the same poles of the second order at the lattice points. Function additionally has simple zeroes at the points , and the product is an entire function also having simple zeroes exactly at these points. The functions involved are both double periodic; hence, the Lemma 1 is applicable already for n = 2. The double integration of the obtained equality of the second derivatives of the corresponding logarithms gives . (Again, the possible factor due to the double periodicity of the functions involved). The coefficient C is established by analyzing the case when z tends to zero by equating the terms proportional to 1/z2 in Laurent expansions. We have , and the theorem is proven. □
Corollary 1. Applying (26), (27) for a = 0, we have the following presentation of the Werierstrass ℘(z) function via elliptical functions:where are some solutions of the equation such that . In particular, we can take these solutions as discussed (for ) in Refs. [17,18] to be equal to ; here, belongs to the first fundamental parallelogram. Then, . See also our previous paper [10] for some additional considerations of zeroes and sums over inverse powers of the zeroes of the Weierstrass function. Remark 2. It is instructive to see what happens in the limit when tends to . The smallest in the module solutions of ℘(z) = 0 are then given by the complex numbers —the values which are still not easy to work with. Thus, for even greater simplicity, let us consider the more transparent case of, e.g., . We have: , and this gives the aforementioned presentation for the case.
The derivative of the Weierstrass
function
has third-order poles at
and zeroes at
,
,
[
10]. Thus, the lattice, formed by the irregular points of the logarithm (i.e., poles and zeroes) of the
, coincides with the zeroes of the function
. We can convert the third-order poles of the
function into simple zeroes by multiplying it by the
function: thus, both functions
and
are entire and have the same simple zeroes. Using our method, we can then establish the well-known equality
Of course, there are also infinitely many other possibilities of such types. For instance, we see that the entire function
has simple zeroes at
points, thus at the same points as the entire function
, so that
. Omitting clearly from the abovesaid details, we arrive at
Similar equalities hold for the functions
,
; see, e.g., [
10].
The next example is apparently also not quite common. We observe that
and
are both entire and have the same simple zeroes. Thus, we have
and
. Equating when
z tends to zero, we obtain
; see Equation (13) for definitions. In Ref. [
7], we used the generalized Littlewood theorem in the direct way showing that
, thus
. For the first derivatives
, so that trivially
C2 = 0. Thus
. Equating functions at zero, we have
and with Equation (25),
; hence,
As usual,
can be expressed via the derivatives ratio using Equation (15). Now note that by Equation (19)
, so that we have the following expression
This, of course, can also be established by our approach in the “direct way”. See the discussion of the transformation rules for the elliptical functions in the next subsection.
Analogously—and this is our final example in this sub-section—we see that
and
have the same poles and zeroes. Omitting standard clearly from the abovesaid details, we obtain
, and equating at zero, we have
. Hence,
and
Here in the r.h.s. the expressions of
,
and
via Jacobi elliptical functions
, respectively (see [
10] for details), can be used.
4.3. n-Tuple Products
Let us consider the entire theta function
with the integer
n = 2, 3, 4… It has zeroes at the points
, i.e.,
(as usual,
), and thus, has the same zeroes as the product
. We illustrate this coincidence in
Figure 1 below.
This coincidence of zeroes, together with the asymptotic of the functions at hand for large values of
|z|, immediately enable to write, by applying Lemma 1:
.
and
are clearly zero because other values are incompatible with the
z-periodicity of the functions involved. (Purely imaginary
with integer
k might be compatible (for we do not have
double periodicity now), but such a value cannot be obtained by comparison of the real Taylor expansions at
). The constant
should be determined from this equality applied at any particular value of
z. Using for
z tending to zero
, we have
; hence,
In particular, and .
Remark 3. For unclear reasons, this relation is seemingly never presented in such a (general) form and, if presented for some particular case, is given in somewhat “artificial” form like, for example, the following Landen transformations [10]: , or . (This looks understandable, because Landen transformations historically came from the manipulations of the elliptical integrals achieved mainly with variable changes [10,12,13,14,15]). Of course, the equivalence of the transformations can be shown using formulae connecting the values of the elliptical functions and their derivatives at certain values (see [10,11,12,13,14,15,16], especially [14]), but this is actually not easy. Alternatively, the comparison of these formulae can be seen as a proof of certain relations like, e.g., . Further, let us note the following. From Equation (11a) and the definition of nome, it follows that
for any finite z. From the same, we have Applying this (properly scaled) limit to both sides of Equation (34), we obtain Equation (8).
There is also a quite different possibility. We can consider not but instead and obtain the following theorem.
Theorem 3. For odd 2l + 1 = 3, 5, 7…, we have:with For even 2l = 2, 4, 6…, we havewith Proof. We see that the functions
and
are both entire and have the same zeroes (a simple “two-dimensional” generalization of
Figure 1 suffices), but to find an exponential factor we need to “symmetrize” this product.
Let first
n be odd,
n = 2
l + 1. Then,
has the same zeroes and (no) poles as the “symmetric” product
. Thus, by the Lemma 1 we obtain
As usual, the coefficient C2 should be equal to zero due to the z-periodicity. The coefficient C1 is equal to zero due to the even nature of the and functions. This is seen as follows: is even, and all factors in the come in pairs with if , . For such pairs, we have so that this product is an even function. If k or j are equal to zero, this is also valid. If both k = j = 0 simultaneously, the factor is even. Thus, comparing the O(z) terms of the Taylor development of the logarithms of both sides of Equation (39), we obtain C1 = 0. Coefficient C is obtained equating the values at zero: .
What does happen if
n = 2
l is even? Then, in passing from
to the symmetrized form, we obtain not the fully symmetric product
. The situation with
is the same as before: this is an even function, but now we obtain also the “unpaired” factors at
k = l and
j = l. These are the functions
and
, respectively. They are respectively equal to
and
, where the constant
A easily follows from [
10]
but for our current purposes itis only important that it does not depend on
z. The functions
with
, when paired as
, again are even functions. The function
, corresponding to
j = 0, is also even. Thus, among the functions corresponding to
k = l, only
rests unpaired. It is equal to
[
10], where
B does not depend on
z; the function
is even.
In the second group of functions (corresponding to j = l), the functions for are also paired: . For k = 0, we have with an even function ; the case k = j = l has already been considered just before.
Thus, we see that the product of all these unpaired factors, related with the theta functions at k = l and j = l, is , where is an even function. Thus, comparing the O(z) terms of Taylor expansions of the logarithms of both sides of the equation , we obtain .
We might finish the consideration at this point, but it seems preferable to give the purely real form using Equation (40) again:
. (Note, that for
m = l in the first product,
[
12], we used this earlier). Coefficient
is obtained equating the values of both sides of the above equation at zero:
. □
Remark 4. 1. The appearance of the factor during the derivation of the n-tuple relation for in the case of even n = 2l might look unexpected. However, let us see how we obtain it already for n = 2. Directly from the Lemma 1, we prove . From our Theorem 3, we have , and this is consistent because , , .
2. Along the same lines, relations similar to Equations (35)–(38) can be established for the Weierstrass sigma functions , . These same relations can also be obtained from the above relations for the first theta function, supplemented with Equation (19) or Equation (20). The present author, however, was able to find only the relation analogous to Equation (35) for the Weierstrass function, formulae 23.10.13–23.10.16 in [10], (with a misprint in 23.10.14: instead of it is written ), but not for the theta functions. 4.4. Fundamental Modular Transformations and Jacobi’s Triple Product
Let us briefly discuss the fundamental modular transformations. What is presented below are the known results, but now they are obtained in a very transparent and easy way.
Analogously to what has been done above, we establish : the possible exponential factor . (Any is incompatible with the z-periodicity, while B = 0 simply due to the evenness of the functions ).
Similarly,
. Here, in the possible factor
,
B = 0 for the same reason as in the previous relation, while to establish the coefficient
A we used the Taylor expansion (13a). But this calculation requires certain caution! Clearly,
where
is defined by Equation (13a), so that one might decide that
and
A = 0 which is clearly impossible:
and
have different incompatible periods. However, the
order of the summation in the r.h.s of Equation (41) is different from that of Equation (13a), so we need to recall the Eisenstein relation
(see the discussion and further references in our recent paper [
7]), and in such a way we obtain
.
Of course,
and
. To move further, the use of the original definitions Equation (11) is necessary to obtain [
10]:
Finally, let us look at the Jacobi triple product. We note that a whole function, having zeroes at
, that is coinciding with those of the function
, can be written as
, and similar expressions can be obtained for other theta functions. (This might not look too familiar because the more standard terminology/notation in the field is the use of
,
(or
and the nome
) and
; then we write that
, etc.). Application of Lemma 1 immediately enables us to write (in somewhat unusual notation):
, where the possible quadratic in
z term is again excluded by periodicity. The exact value of
C1 is calculated comparing the values of the derivatives of the logarithms at
z = 0. Trivially,
, while for the r.h.s we have
, and further
, together with
. Finally,
, so that
. Using the well-known value
—see Equation (21) but reminding that now
—we find
and thus restore (cf. e.g., [
19])
where
. Similar relations can be established for other theta functions, including the best known [
10]:
reminding that now
and
[
10].