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Reduction formulae for products of theta functions.

In four cases it is already known that the product of two distinct Jacobian theta functions having the same variable z and the same nome q is a multiple of a single Jacobian theta function, with the multiple independent of z. The main purpose of the present note is to show that this property also applies in the remaining two cases.

Key words: Jacobian theta functions, especially products; normalizing factors.

Accepted: September 19, 2012

Published: November 6, 2012

http://dx.doi.org/10.6028/jres.117.017

1. Introduction

The formulae

[[[theta].sub.1](z,q)[[theta].sub.2](z,q)]/[[[theta].sub.1](2z,[q.sup.2])] = [[[theta].sub.3](z,q) [[theta].sub.4](z,q)]/[[[theta].sub.4](2z,[q.sup.2])] = [[theta].sub.4](0,[q.sup.2]), (1)

[[[theta].sub.1]](z,[q.sup.2])[[theta].sub.4](z,[q.sup.2])]/[[[theta].sub.1](z,q)] = [[[theta].sub.2]](z,[q.sup.2])[[theta].sub.3](z,[q.sup.2])]/[[[theta].sub.2](z,q)] = 1/2[[theta].sub.2] (0,q), (2)

occur in the theory of Jacobian theta functions and their applications to Jacobian elliptic functions. For example, Eq. (1) leads to the Landen transformations (Ref. (1), Subsection 20.7(vi) and Section 22.7).

The main purpose of the present paper is to prove that

[[[theta].sub.1]](z,[q.sup.2])[[theta].sub.3](z,[q.sup.2])]/[[[theta].sub.1](z,iq)] = [[[theta].sub.2]](z,[q.sup.2])[[theta].sub.4](z,[q.sup.2])]/[[[theta].sub.2](z,iq)] = [i.sup.-1/4] [square root of ([[[theta].sub.2]](0,[q.sup.2])[[theta].sub.4](0,[q.sup.2])]/2)], (3)

thereby covering the remaining two cases.

However, we shall also prove Eqs. (1) and (2) by the same method, thereby providing new proofs for these equations. Incidentally, since the right-hand sides of Eqs. (1), (2), and (3) are independent of z, we shall refer to them as normalizing factors.

2. Infinite Products

We begin with the well-known expansions of the theta functions as infinite products (Ref. [1], Eqs. (20.5.1)-(20.5.4)):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

In particular,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

The prime in [[theta].sub.1]'(0,q) indicates differentiation with respect to the variable z. In these formulae, z,q are complex and [absolute value of q] < 1 to ensure convergence. Here [q.sup.1/4] denotes the principal value of the root. With q = [e.sup.i[pi][tau]], [tau] is a complex parameter with positive imaginary part.

With this notation we have

1 - 2[q.sup.2n] cos(2z) + [q.sup.4n] = 1 - 2[e.sup.2ni[pi][tau]] cos(2z) + [e.sup.4ni[pi][tau]] = 2[e.sup.2ni[pi][tau]] (cos(2n[pi][tau]) - cos(2z)) = 4[e.sup.2ni[pi][tau]] sin(z - n[pi][tau])sin(z + n[pi][tau]), (6)

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Note that as n [right arrow] [infinity], the n th factor in this infinite product is 1 + O([e.sup.2mi[pi][tau]), which ensures rapid convergence since [Imaginary part][tau] > 0

Repetition of this calculation (details omitted) for [[theta].sub.2], [[theta].sub.3], and [[theta].sub.4] yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The required results now follow on combining these equations in pairs.

For the product [[theta].sub.1][[theta].sub.2] we have from Eqs. (7) and (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where we have used the identity [q.sup.2] = [e.sup.2i[pi][tau]]. Hence

[[[theta].sub.1](z,q)[[theta].sub.2](z,q)]/[[[theta].sub.1](2z,[q.sup.2])] = 1/2 [[[[theta].sub.1]'(0,q)[[theta].sub.2](0,q)]/[[[theta].sub.1]'(0,[q.sup.2])]] (12)

The translation z [right arrow] z + 1/2[tau] changes [[theta].sub.1] [right arrow] [[theta].sub.3] and [[theta].sub.2] [right arrow] [[theta].sub.4]. Hence we have

[[[theta].sub.1](z,q)[[theta].sub.2](z,q)]/[[[theta].sub.1](2z,[q.sup.2])] = [[[theta].sub.3](z,q)[[theta].sub.4](z,q)]/[[[theta].sub.3](2z,[q.sup.2])] = 1/2 [[[[theta].sub.1]'(0,q)[[theta].sub.2](0,q)]/[[[theta].sub.1]'(0,[q.sup.2])]], (13)

which is the first required result, Eq. (1), except for the normalizing factor. We will return to this below and for now concentrate on the quotients of functions of z.

Turning to the product [[theta].sub.1][[theta].sub.4], we have from Eqs. (7) and (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where in the last line we have combined the even (2n) and the odd (2n - 1) terms to obtain the complete sequence. Hence on rearranging we have,

[[[theta].sub.1](z,[q.sup.2)[[theta].sub.4](z,[q.sup.2])]/[[[theta].sub.1](z,q)] = [[[theta].sub.1]'(0,[q.sup.2])[[theta].sub.4](0,[q.sup.2])]/[[[theta].sub.1]'(0,q)], (15)

followed by translation in z by 1/2[pi]:

[[[theta].sub.1](z,[q.sup.2)[[theta].sub.4](z,[q.sup.2])]/[[[theta].sub.1](z,q)] = [[[theta].sub.2](z,[q.sup.2)[[theta].sub.3](z,[q.sup.2])]/[[[theta].sub.2](z,q)] = [[[theta].sub.1]'(0,[q.sup.2])[[theta].sub.4](0,[q.sup.2])]/[[[theta].sub.1]'(0,q)], (16)

which is Eq. (2), except for the normalizing factor.

For the third identity we note that q [right arrow] iq corresponds to [tau] [right arrow] [tau] + 1/2. From Eqs. (7) and (9) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Now in Eq. (18), we separate the terms according to the parity of n.

If n = 2k (k [greater than or equal to] 1) is even, then

Sin(n[pi]([tau] + 1/2) [+ or -] z) = [(-1).sup.k] sin(2k[pi][tau] [+ or -] z), (19)

and

[sin.sup.2](n[pi]([tau] + 1/2)) = [sin.sup.2](2k[pi][tau]), (20)

corresponding to the first quotient on the right-hand side of Eq. (17).

Similarly if n = 2k - 1 (k [greater than or equal to] 1) is odd, then

Sin(n[pi]([tau] + 1/2) [+ or -] z) = sin((2k - 1)[pi]([tau] + 1/2) [+ or -] z) = [(-1).sup.k]cos((2k - 1)[pi][tau] [+ or -] z), (21)

and

[[sin].sup.2]((2k - 1)[pi]([tau] + 1/2)) = [cos.sup.2]((2k - 1)[pi][tau]), (22)

corresponding to the second quotient in Eq. (17). Hence

[[[theta].sub.1](z,[q.sup.2])[[theta].sub.3](z,[q.sup.2])]/[[[theta].sub.1]'(0,[q.sup.2])[[theta].sub.3](0,[q.sup.2])] = [[theta].sub.1](z,iq)/[[theta.sub.1]'(0,iq), (23)

Or

[[[theta].sub.1](z,[q.sup.2])[[theta].sub.3](z,[q.sup.2])]/[[[theta].sub.1](z,iq)] = [[[theta].sub.1]'(z,[q.sup.2])[[theta].sub.3](z,[q.sup.2])]/[[[theta].sub.1]'(z,iq)] (24)

and translation in z by 1/2[pi] gives

[[[theta].sub.1](z,[q.sup.2])[[theta].sub.3](z,[q.sup.2])]/[[[theta].sub.1](z,iq)] = [[[theta].sub.2](z,[q.sup.2])[[theta].sub.4](z,[q.sup.2])]/[[[theta].sub.2](z,iq)] = [[[theta].sub.1]'(0,[q.sup.2])[[theta].sub.3](0,[q.sup.2])]/[[[theta].sub.1]'(0,iq)] (25)

which is Eq. (3) apart from the normalizing factor.

3. Normalizing Factors

If we compare our results (13), (16), and (25), with the desired Eqs. (1), (2), and (3), then we see that we have to verify the following identities:

1/2[[[theta].sub.1]'(0,q)[[theta].sub.2](0,q)]/[[[theta].sub.1]'(0,[q.sup.2)] = [[theta].sub.4](0,[q.sup.2]), (26)

[[[theta].sub.1]'(0,[q.sup.2])[[theta].sub.4](0,[q.sup.2])]/[[[theta].sub.1]'(0,q)] = 1/2[[theta].sub.2](0,q), (27)

and

[[[theta].sub.1]'(0,[q.sup.2])[[theta].sub.3](0,[q.sup.2])]/[[[theta].sub.1]'(0,iq)] = [i.sup.-1/4] [square root of ([[[theta.sub.2](0,[q.sup.2])[[theta].sup.4](0,[q.sup.2])]/2)]. (28)

The following identity, which goes back at least to Euler, is needed. When [absolute value of q]<1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

To verify this, we note that on expanding the product in powers of q, only a finite number of factors are needed to determine the coefficient of each power, so that no limiting process is involved and the terms may be re-ordered without affecting the value of the product. (Alternatively, for an analytic argument, the rearrangement is justified by absolute convergence.) Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

If P(q) is not identically 1, then there will be a least power of q, say [q.sup.k], whose coefficient is not zero. But then [q.sup.2k] will be the least power of q in P([q.sup.2]) whose coefficient is not zero, so P(q) and P([q.sup.2]) cannot be equal, resulting in a contradiction.

To prove Eq. (26) we substitute from Eqs. (5), and see that we require

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

After division by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we see that the last equation is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Replacement of the term (1 - [q.sup.4n]) by (1 - [q.sup.2n])(1 + [q.sup.2n]), and use of Eq. (29) with [q.sup.2] in place of q, completes the proof.

For Eq. (27) we require similarly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

or, on division by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

This completes the proof of Eq. (27).

Finally, for Eq. (28) we begin by calculating the asymptotic values of each side as q [right arrow] or [vector] 0 through positive real values. On the left-hand side we have

{2[([q.sup.2]).sup.1/4]/(2[(iq).sup.1/4])}(1 + O([q.sup.2])) = [(q/i).sup.1/4](1 + O([q.sup.2])). (35)

On the right-hand side we have [i.sup.-1/4] [square root of (2[([q.sup.2]).sup.1/4]/2)](1 + O([q.sup.2]), which is the same. Hence we can simply square both sides and compare the results. For the left-hand side we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

For the right-hand side we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

On comparison of the two sides we see that the result to be proved is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

or equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

which is Eq. (29) with [q.sup.4] in place of q.

4. References

(1.) F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010, print companion to (2).

(2.) NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, online companion to (1) with errata list and other updates.

P. L. Walker

7 Redgrove Park, Cheltenham, GL51 6QY UK

pandjw@hotmail.co.uk

About the author: Peter Walker resides in Cheltenham, UK. Previously, he was a faculty member at the University of Lancaster, UK, and the American University of Sharjah, UAE. He is the joint author, with William Reinhardt of the University of Washington, USA, of Chapters 20, 22, and 23 of Refs. (1) and (2). The National Institute of Standards and Technology is an agency of the U.S. Department of Commerce.
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Author:Walker, P.L.
Publication:Journal of Research of the National Institute of Standards and Technology
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Date:Nov 1, 2012
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