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Some theta identities and their implications.

[section]1. Introduction

Assume throughout this paper that q = [e.sup.[pi]x[tau]], where ℑ[tau]>0. As usual, the classical Jacobi theta functions are defined as follows,

[[theta].sub.1](z|[tau]) = -i[q.sup.1/4][[infinity].summation over (n = -[infinity])][(-1).sup.n][q.sup.n(n + 1)][e.sup.(2n + 1)iz] = 2[q.sup.1/4][[infinity].summation over (n = 0)][(-1).sup.n][q.sup.n(n + 1)]sin(2n + 1)z, (1)

[[theta].sub.2](z|[tau]) = [q.sup.1/4][[infinity].summation over (n = -[infinity])][(-1).sup.n][q.sup.n(n + 1)][e.sup.(2n + 1)iz] = 2[q.sup.1/4][[infinity].summation over (n = 0)][(-1).sup.n][q.sup.n(n + 1)]cos(2n + 1)z, (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The q-shifed factorial is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and sometimes is written as

[([a.sub.1], [a.sup.2], ..., [a.sub.m];q).sub.[infinity]] = [([a.sub.1];q).sub.[infinity]][([a.sup.2];q).sub.[infinity]]... ([a.sub.m];q)

With above notation, the celebrated Jacobi triple product identity can be expressed as follow

[[infinity].summation over (n = -[infinity])][(-1).sup.n][q.sup.n(n - 1)/2][z.sup.n] = [(q;q).sub.[infinity]][(z;q).sub.[infinity]][(q/z;q).sub.[infinity]]. (5)

Employing the Jacobi triple product identity, we can derive the infinite product expressions for theta function.

Proposition 1.1 (Infinite product representations for theta functions)

[[theta].sub.1](z|[tau]) = 2[q.sup.1/4] sinz[([q.sup.2];[q.sup.2]).sub.[infinity]][([q.sup.2][e.sup.2iz];[q.sup.2]).sub.[infinity]][([q.sup.2][e.sup.- 2iz];[q.sup.2]).sub.[infinity]], (6)

[[theta].sub.2](z|[tau]) = 2[q.sup.1/4][([q.sup.2];[q.sup.2]).sub.[infinity]] [(-[q.sup.2][e.sup.2iz];[q.sup.2]).sub.[infinity]][(-[q.sup.2][e.sup.-2iz];[q.sup.2]).sub.[infinity]], (7)

[[theta].sub.3](z|[tau]) = [([q.sup.2];[q.sup.2]).sub.[infinity]][(-q[e.sup.2iz];[q.sup.2]).sub.[infinity]] [(-q[e.sup.-2iz];[q.sup.2]).sub.[infinity]], (8)

[[theta].sub.4](z|[tau]) = [([q.sup.2];[q.sup.2]).sub.[infinity]][(q[e.sup.2iz];[q.sup.2]).sub.[infinity]] [(q[e.sup.-2iz];[q.sup.2]).sub.[infinity]]. (9)

When there is no confusion, we will use [[theta].sub.i](z) for [[theta].sub.i](z|[tau]), [[theta]'.sub.i] for [[theta]'.sub.i](z|[tau]) to denote the partial derivative with respect to the variable z, and [[theta].sub.i] for [[theta].sub.i](0|[tau]), i = 1,2,3,4. From the above equations, the following facts are obvious

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

With respect to the (quasi) period [pi] and [pi][tau], Jacobi theta functions [[theta].sub.i], i = 1,2,3,4, satisfy the following relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where M = [q.sup.-1/4][e.sup.-iz]

The following trigonometric series expressions for the logarithmic derivative with respect to z of Jacobi Theta functions will be very useful in this paper,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

One may prove the Jacobi imaginary transformation formulas of the the theta functions employing the Poisson summation formula [?, pp.7-11]

Lemma 1.1. If z is complex, Tz > 0, and [square root of (-i[tau])] = 1 for [tau] = i, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

In particular, by setting z = 0 in above formulas we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Suppose that [E.sub.2k]([tau]) is the normalized Eisenstein series of weight 2k defined as

[E.sub.2k] = 1 - 4k/[B.sub.2k] [[infinity].summation over (n = 1)][n.sup.2k - 1][q.sup.2n]/1 - [q.sup.2n]

where [B.sub.2k] is Bernoulli numbers defined as the coefficients in the power series

z/[e.sup.z]-1 = [[infinity].summation over (k - 0)][B.sub.k] [z.sup.k]/k!

And it is easy to show that [B.sub.2k + 1] = 0 for k [greater than or equal to] 1, and the first few nonzero values of [B.sub.k] are given by [B.sub.0] = 1, [B.sub.1] = -1/2, [B.sub.2] = 1/6, [B.sub.4] = -1/30, [B.sub.6] = 1/42. Then near z = 0, [[theta].sub.1]'/[[theta].sub.1](z|[tau]) has the Laurent expansion formula

[[theta].sub.1]'/[[theta].sub.1](z|[tau]) = [[infinity].summation over (k = 0)] [(-1).sup.k] [2.sup.2k][B.sub.2k]/(2k)! [E.sub.2k]([tau])[z.sup.2k-1]

Let [[sigma].sub.k](n) denote the sum of the kth powers of divisors of n, namely, [[sigma].sub.k](n) = [[summation].sub.d|n][d.su7p.k] Then the first three Eisenstein series [E.sub.2k] are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Theorem 1.1. The sum of all the residues of an elliptic function in the period parallelogram is zero.

[section]2. Main theorem and proofs

Theorem 2.1. For x, y,

[[theta]'.sub.4]/[[theta].sub.4](x) + [[theta]'.sub.4]/[[theta].sub.4] (y) - [[theta]'.sub.4]/[[theta].sub.4] (x + y) = [[theta].sub.2][[theta].sub.3] [[theta].sub.1](x)[[theta].sub.1](y)[[theta].sub.1](x + y)/[[theta].sub.4](x)[[theta].sub.4](y)[[theta].sub.4](x + y). (17)

Proof. We consider the following function

f(z) = [[theta].sub.4](z + x)[[theta].sub.4](z + y)[[theta].sub.4](z-x-y)[[theta].sub.1]/2(z)[[theta].sub.4](z),

by the definition of [[theta].sub.i](z|[tau]), we can readily verify that f(z) is an elliptic function with periods [pi] and [pi][tau]. The only poles of f(z) are 0 and [pi][tau]/2. Furthermore, [pi][tau]/2 is its simple pole and 0 is its pole with order two. By virtue of the residue theorem of elliptic functions, we have

Res(f;[pi][tau]/2) + Res(f;0) = 0. (18)

And applying relation of [[theta].sub.1] and [[theta].sub.4] in (12) and L Hopitall rules, we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Next we compute Res(f;0),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

From theorem 1.1, substituting (19) and (20) into (18), by performing a little reduction we can complete the proof of theorem.

Corollary 2.1.

([[theta]'.sub.4]/[[theta].sub.4]) (y) - ([[theta]'.sub.4]/[[theta].sub.4]) 7(x) = [([[theta]'.sub.1]).sup.2] [[theta].sub.1](x + y)[[theta].sub.1](x - y)/[[theta].sub.4]2(x)[[theta].sub.4]2(y). (21)

Proof. We differentiate (17) with respect to y, and then set y = 0, then

([[theta]'.sub.4]/[[theta].sub.4])' (0) - ([[theta].sub.4]'/[[theta].sub.4])'(x) = [([[theta].sub.2][[theta].sub.3]).sup.2][([[theta].sub.1](x)/[[theta].sub.4](x)).sup.2] (22)

Now we combine with another elementary identity [7, p.467],

[[theta].sup.2.sub.1](x)[[theta].sup.2.sub.4](y) - [[theta].sup.2.sub.1](y)[[theta].sup.2.sub.4](x) = [[theta].sup.2.sub.4][[theta].sub.1](x + y)[[theta].sub.1](x - y). (23)

From (22) and (23), we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof of corollary 2.1.

Remark. The identity (21) is often written in terms of the Weierstrass elliptic and sigma functions as [7, p.451],

p(y) - p(x) = [sigma](x + y)[sigma](x - y)/[[sigma].sup.2](x)[[sigma].sup.2](y).

Theorem 2.2. (Ramanujan's modular identity)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Proof. We recall (13) for [[theta]'.sub.4]/[[theta].sub.4] (z|[tau]),then

([[theta]'.sub.4]/[[theta].sub.4])'(z|[tau]) = 8[[infinity].summation over (n = 1)] cos2nz = 4 [[infinity].summation over (n = 1)] n[q.sup.n]([e.sup.2inz] + [e.sup.-2inz])/1 - [q.sup.2n], (25)

in (21), we replace [tau] by |[tau], and choose y = 3[pi][tau]/4 and x = [pi][tau]/4, using the Jacobi triple product identities for the theta functions. We can see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Then from (21), (25) and (26), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof of the theorem.

From above procedure, we can rewrite theorem as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

From Jacobi imaginary transformation lemma 1.1, we have

([[theta]'.sub.4]/[[theta].sub.4])'(z/[tau]| - 1/[tau]) = [[tau].sup.2]([[theta]'.sub.2]/[[theta].sub.2])'(z|[tau]) + 2i[tau]/[pi] (28)

Then we have the following corollary:

Corollary 2.2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Proof. We can replace [tau] by -1/[tau] in (27) and applying to (28), then (27) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then replace [tau] by 5[tau]/4, it becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

From (13), we can obtain

([[theta]'.sub.2]/[[theta].sub.2])'(z|[tau]) = -[sec.sup.2]z + [[infinity].summation over (n = 1)] [(-1).sup.n]n[q.sup.2n]cos2nz/1 - [q.sup.2n]. (31)

Next we can deduced (29) from (30) and (31) directly.

Therefore, This complete the corollary 2.2.

It should be remark that Bailey [1] also derived (24) and (29) in a similar fashion. However the key difference between our approach is that he derived corollary 2.1 from 6[PSI]6 which is much less elementary than (17). Moreover by recasting these identities (27) and (30) in terms of theta function identity, we can easily discover and deduce other companion identities from the basic properties of the theta functions.

Corollary 2.3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. We recall the identity (21), and replace x by x + [pi]/2 and y by y + [pi]/2, then we obtain

([[theta]'.sub.3]/[[theta].sub.3])'(y) - ([[theta]'.sub.3]/[[theta].sub.3])'(x) = [([[theta]'.sub.1]).sup.2] [[theta].sub.1](x + y)[[theta].sub.1](x - y)/[[theta].sup.2.sub.3](x)[[theta].sup.2.sub.3](y)

Now replacing [tau] by 5/2 [tau], and choosing y = 3[pi][tau]/4 and x = [pi][tau]/4, we can deduce the corollary

[section]3. Implications for modular identity of Ramanujan

Clearly the identity (21) can generate unlimited number of similar identities. Since the proofs are identical and straightforward, we only list a few without providing the details of proofs here.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

By acting the imaginary transformation on above three formulas (32), (33) and (34), we can obtain, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Theorem 3.1. In [6], Ewell derived the following identity in virtue of the quintuple product identiy

[[theta].sup.3.sub.4](0|[tau])/[[theta].sub.4](0|3[tau]) = 1 - 6[[infinity].summation over (n = 0)]([q.sup.3n + 1]/1 + [q.sup.3n + 1] - [q.sup.3n + 2]/1 + [q.sup.3n + 1]), (38)

to study the sums of three squares. The identity (36) is the Lambert series for the square of (38)

On the other hand, in [5], authors define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

to study some cubic modular identities of Ramanujan. It is interesting to observe that the identities (32) and (35) are the Lambert series for 1/9c(q)c([q.sup.2]) and b(q)b([q.sup.2]) respectively.

[section]4. Extension and generalization

It is surprising that hidden with (32), (33) and (34) is the following beautiful modular identity of Ramanujan,

[([[theta].sub.2](0|[tau]/3)/[[theta].sub.2](0|3[tau]) - 1).sup.3] = [([[theta].sub.2](0|[tau])/[[theta].sub.2](0|3[tau])).sup.4] - 1. (39)

To see this, we note that the identity (32) is exactly the difference of (33) and (34), therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

On the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

Clearly, the identity (39) now follows from (40) and (41).

We note that if we apply the imaginary transformation to (39), then we obtain

[(3[[theta].sub.4](0|[tau])/[[theta].sub.4](0|[tau]) - 1).sup.3] = [9(3[[theta].sub.4](0|3[tau])/[[theta].sub.4](0|[tau])).sup.4] - 1. (42)

And using the familiar facts that [[theta].sub.4](0|[tau] + 1) = [[theta].sub.3](0|[tau]), [[theta].sub.3](0|-1/[tau]) = [square root of (-i[tau])][[theta].sub.3](0|[tau]) and [[theta].sub.4](0| - 1/[tau]) = [square root of (-i[tau])][[theta].sub.2](0|[tau]), it is easy to derive the general formula

[([[theta].sub.i](0|[tau])/[[theta].sub.i](0|9[tau]) - 1).sup.3] = [[theta].sup.4.sub.i](0|3[tau])/[[theta].sup.4.sub.i](0|9[tau]) - 1, [(3[[theta].sub.i](0|9[tau])/[[theta].sub.i](0|[tau]) - 1).sup.3] = 9[[theta].sup.4.sub.i](0|3[tau])/[[theta].sup.4.sub.i](0|[tau]) - 1.

for i = 2,3,4. (see a different proof of reference to [4, p.143].

Finally we end this paper by extending this (17) and applying this generalization to derive a few well-known identities.

The identity (17) together with the following identity [17,p.324]

[[theta].sub.1](y)[[theta].sub.4](x)[[theta].sub.1](x + y + z)[[theta].sub.4](z) + [[theta].sub.1](x)[[theta].sub.1](z)[[theta].sub.4](y)[[theta].sub.4](x + y + z) = [[theta].sub.4][[theta].sub.4](x + z)[[theta].sub.1](y + z)[[theta].sub.1](x + y)

can yields an identity which is an extension of (17),

[[theta]'.sub.4]/[[theta].sub.4](x) + [[theta]'.sub.4]/[[theta].sub.4](y) + [[theta]'.sub.4]/[[theta].sub.4](z) - [[theta]'.sub.4]/[[theta].sub.4](x + y + z) = ([[theta]'.sub.1]) [[theta].sub.1](x + y)[[theta].sub.1](y + z)[[theta].sub.1](x + y)/[[theta].sub.4](x)[[theta].sub.4](y)[[theta].sub.4](z)[[theta].sub.4](x + y + z).

We can see that this identity can re-derive many identities of Ramanujan. For example, if replacing [tau] by 11[tau]/2 and setting x = [pi][tau]/4, y = 3[pi][tau]/4, z = 5[pi][tau]/4, then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Again, we applied the imaginary transformation to (17), then we have

[[theta]'.sub.2]/[[theta].sub.2](x) + [[theta]'.sub.2]/[[theta].sub.2](y) - [[theta]'.sub.2]/[[theta].sub.2](x + y) = [[theta].sub.3][[theta].sub.4] [[theta].sub.1](x)[[theta].sub.1](y)[[theta].sub.1](x + y)/[[theta].sub.2](x)[[theta].sub.2](y)[[theta].sub.2](x + y). (43)

In (17) and (43), replacing [tau] by 7[tau] and setting x = [pi][tau], y = 2[pi][tau], we can obtain, respectively,

1/4[[theta].sub.2](0|[tau]/2)[[theta].sub.2](0|7[tau]/2) = [[summation].sup.[infinity].sub.n = 1][q.sup.n] - [q.sup.3n] - [q.sup.5n] + [q.sup.9n] + [q.sup.11n]-[q.sup.13n]/1 - [q.sup.14n],

and

[[theta].sub.4](0|[tau])[[theta].sub.4](0|7[tau]) = 1 - 2[[infinity].summation over (n = 1)][(-1).sup.n]([q.sup.n] + [q.sup.2n] - [q.sup.3n] + [q.sup.4n] - [q.sup.5n] - [q.sup.6n]/1 - [q.sup.7n]).

Both identities are known to Ramanujan in [3,p.302].

Acknowledgements. The author express many thanks to the referees for valuable suggestions and comments. This paper was partially supported by The Educational sciences funding project of Shanxi Province(12JK0881) and the natural sciences funding of project Weinan city(2012KY-IO).

References

[1] W. N. Bailey, A further note on two of Ramanujan's formulae, Q. J. Math. (Oxford) 3(1952), 158-160.

[2] R. Bellman, A brief introduction to the theta functions, Holt Rinehart and Winston, New York(1961).

[3] B. C. Berndt, Ramanujan's Notebooks III, Springer-Verlag, New York (1991).

[4] J. M. Borwein and P. B. Borwein, Pi and the AGM-A Study in Analytic Number Theory and Computational Complexity, Wiley, N. Y., (1987).

[5] J. M. Borwein, P. B. Borwein and F. G. Garvan, Some cubic modular Indentities of Ramanujan, Trans. of the Amer. Math. Soci., 343(1994), No. 1, 35-47.

[6] J. A. Ewell, On the enumerator for sums of three squares, Fibon. Quart., 24(1986), 151-153.

[7] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4-th ed. Cambridge Univ. Press, (1966).

[8] Li-Chien Shen, On the Additive Formulae of the Theta Functions and a Collection of Lambert Series Pertaining to the Modular Equations of Degree 5, Trans. of the Amer. Math. Soci., 345(1994), No. 1, 323-345.

Hai-Long Li ([dagger]) and Qian-Li Yang ([double dagger])

Department of Mathematics, Weinan Teachers University, Weinan City, Shaanxi Province.

E-mails: lihailong@wntc.edu.cn, darren2004@126.com

(1) The author was supported partially by the Science Foundation of Shaanxi province
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Date:Sep 1, 2013
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