A PROOF OF THE GENERAL THETA TRANSFORMATION FORMULA

Transcription

1 A PROOF OF THE GEERAL THETA TRASFORATIO FORULA BRUCE C. BERDT, CHADWICK GUGG, SARACHAI KOGSIRIWOG, AD JOHA THIEL. Introduction In Ramanujan s notation for theta functions, set f(a, b : a n(n/2 b n(n/2, ab <. n Arguably, the two most important properties of f(a, b are the Jacobi triple product identity and the modular transformation formula, which can be written in two different ways [3, p. 36, Entry 20], [2, pp , Entry 7]. Theorem.. Let α, β > 0, with αβ π. Let z be any complex number. Then e z2 /4 αf(e α2 izα, e α2 izα βf(e β2 zβ, e β2 zβ (. and e z2 /4 α 2 e α2 n 2 cos(αzn β n 2 e β2 n 2 cosh(βzn. (.2 The most familiar proof of Theorem., especially in the form (.2, is via Poisson s summation formula. That proof is briefly sketched in [2, p. 253]. It should be emphasized that only in special cases is f(a, b a modular form. In these special cases, especially in the case a b q, more proofs are known. In particular, there are three important special cases of Theorem.. For two of them, following Ramanujan, set ϕ(q : f(q, q q n2 and ψ(q : f(q, q 3 q n(n/2, q <. n Then [3, p. 43, Entry 27 (i, (ii] if αβ π with Re α 2, β 2 > 0, αϕ(e α 2 βϕ(e β2 (.3 and n n0 2 αψ(e 2α2 βe α2 /4 ϕ(e β2. (.4 For the third major special instance, in [4], the first author and K. Venkatachaliengar gave a proof of Theorem. for f(q : f(q, q 2 ( n q n(3n/2 e 2πiτ/24 η(τ, q e 2πiτ, Im τ > 0, n (.5 where η(τ is the Dedekind eta-function. If Re α, β > 0, this transformation formula is given by e α/2 4 αf(e 2α e β/2 4 βf(e 2β. (.6

2 2 Bruce C. Berndt, Chadwick Gugg, Sarachai Kongsiriwong, and Johann Thiel There are several known proofs of (.6. A virtue of the proof given in [4] is that it is quite elementary. It is natural to ask if the ideas from [4] can be used to prove the more general theta transformation formula given in Theorem.. In fact, in their doctoral theses [7], [8], the second and third authors of this paper used the methods from [4] to establish (.4 and (.3, respectively. The purpose of this paper is to give a proof of the general transformation formula in the form (. along the lines of those in [4]. This more general proof encounters serious difficulties that are not found in the proofs of (.6, (.3, and (.4. In particular, in order to complete the last step in the proof, we are required to prove that we can invert the order of summation in a certain conditionally convergent double series. We think that this challenging problem is of interest in itself. In Section 2, we give the first part of our proof of Theorem., and in Section 3, we give the second portion of our proof, i.e., justifying that we can indeed invert the order of summation in the double series in question. In an earlier paper [9], Venkatachaliengar sketched short proofs of the transformation formulas (.3 and (.6 for, respectively, the classical theta function ϕ(q and the Dedekind eta-function η(τ. The proof for ϕ(q is similar to that given in Whittaker and Watson s text [, pp ], but Venkatachaliengar s determination of the constant is simpler. In his monograph [0, pp ], Venkatachaliengar sketched another proof of the transformation formula for ϕ(q, with the details more fully worked out later by S. Cooper [5]. 2. First Part of Our Proof It will be slightly convenient to reformulate (.. Let a, b > 0 with ab π 2. Then e z2 /4 a /4 f(e aiz a, e aiz a b /4 f(e bz b, e bz b. (2. We now prove (2.. Using the Jacobi triple product identity, f(a, b ( a(ab n ( b(ab n ( (ab n, ab <, n0 we can rewrite (2. in the form e z2 /4 a /4 n0 n0 ( e (2naiz a ( e (2naiz a ( e 2na b /4 ( e (2nbz b ( e (2nbz b ( e 2nb. (2.2 Taking logarithms on both sides of (2.2, we find that z log a log( e (2naiz a log( e (2naiz a n0 n0 n n log( e 2na 4 log b log( e (2nbz b n n0

3 log( e (2nbz b n0 Theta Transformation Formula 3 log( e 2nb. (2.3 n Using the aclaurin series for log( w, inverting the order of summation, and then summing the resulting geometric series, we easily find that log( e (2naiz a ( m e maizm a m e 2ma. n0 Using this and similar expansions for the five remaining logarithmic series in (2.3, we find that, if z is sufficiently small, 2 ( m m e ma cos(zm a e 2ma ow, from [4, Eq. (2], m e 2ma e 2mb ( m e mb cosh(zm b 2 m e 2mb m e 2ma e 2mb 4 log ba z2 4. (2.4 4 log a b b a 2. (2.5 Thus, using (2.5 in (2.4, we see that we must show that ( m e ma cos(zm a ( m e mb cosh(zm b 2 2 m e 2ma m e 2mb b a 2 z2 4. (2.6 ow from [2, p. 268], we know the partial fraction decomposition cosh(φmy sinh(πmy πmy my π k k 0 ( k cos(kφ k(imy k. (2.7 Set πy a and φ πiz/ a in (2.7. For the remainder of the proof, we will need to assume that z is purely imaginary and, as mentioned above, sufficiently small in modulus. At the end of our proof, we appeal to analytic continuation. Using the relation ab π 2 and simplifying, we find that cos(zm a sinh(ma ma m b ma m b k k 0 ( k cosh(kz b k(ima/π k ( k cosh(kz b. (2.8 k 2 m 2 a/b

4 4 Bruce C. Berndt, Chadwick Gugg, Sarachai Kongsiriwong, and Johann Thiel If we now use (2.8 in (2.6, we find that ( m e ma cos(zm a ( m 2 m e 2ma m b 2 ext, recall that [6, p. 44, Formula.422, no. 3] sinh y y 2y π 2 Using (2.0 and the relation ab π 2, we find that 2 ( m m e mb cosh(zm b ( m e 2mb m ( m m cosh(zm b ( m cosh(zm b 2 b m 2 ma m ( k cosh(zk b ( mk cosh(zk b. (2.9 ( k y 2 /π 2 k 2. (2.0 cosh(zm b sinh(mb mb 2mb ( k b 2 m 2 π 2 k 2 ( mk cosh(zm b. (2. We examine the first sum on the far right-hand side of (2.. Set z ix and recall [, p. 805, Eq ] the Fourier expansion of the second Bernoulli polynomial B 2 (t t 2 t 6 : B 2(t π 2 cos(2πkt k 2, 0 t. (2.2 Thus, from (2.2, if x b/π <, ( m cos(xm b cos(2xm b b m 2 b 4m 2 m0 cos(2xm b cos(xm b b 2 m 2 m 2 ( π 2 x 2 ( b x ( b π 2 b 2 π π 6 cos(x(2m b (2m 2 x 2 ( b 2π x b 2π 6 x2 4 a 2 z2 4 a 2. (2.3

5 Theta Transformation Formula 5 We now employ (2.3 in (2., then use (2.9 and (2. in (2.6, and find that the latter equality now reduces to the new equality ( mk cosh(zk b ( mk cosh(zm b 0. If we invert the roles of the summation variables m and k in the first sum above, we see that we are now required to show that, for x R and ab π 2, ( mk cos(xm b ( mk cos(xm b. (2.4 In other words, our task is to show that we can invert the order of summation in the double sums above. 3. Justifying the Inversion in Order of Summation Denote the right-hand side of (2.4 by A. Since A converges, for every there is an integer such that if l, then <. So, A m 2 m 2 ml ( ( m cos (xm b bm 2 k m 2 k m 2 ( ( k (a/b(k/m. 2 ( The inner sum of the second double sum is a convergent alternating series, and so we can estimate it crudely by ( k ( (a/b(k/m 2 b a( m /m O 2 a,b. ( m 2 k m 2

6 6 Bruce C. Berndt, Chadwick Gugg, Sarachai Kongsiriwong, and Johann Thiel Combining this estimate with the work above yields A m 2 m 2 a,b ( 2 ( a,b ( m 4. (3. ote that the constant implied by the error term of (3. is independent of. ow let B denote the left-hand side of (2.4. Since B converges, for every there is an integer such that if l, then So, B k2 k2 kl ( k ak 2 ( ( mk cos (xm b m k2 m k2 <. ( m cos (xm ( b. (b/a(m/k 2 ( The inner sum of the second double sum is estimated by m k2 ( m cos (xm b (b/a(m/k 2 m k2 ak2 b m k2 O a,b ( (b/a(m/k 2. m 2

7 Putting this estimate back into the work above yields k2 ( mk cos (xm ( b B a,b k2 Theta Transformation Formula 7 ( a,b ( k 2. (3.2 As in (3., the constant implied by the O-symbol in (3.2 does not depend on. ote that (3. and (3.2 also hold for max,. We now switch the roles of m and k in (3.2 and take differences to find that A B m2 m2 m2 m2 m2 m2 k m2 k k k m k/ ( mk cos (xk b We need to estimate the two sums above. First, m2 m2 ( mk cos (xk ( b a,b ( (mk cos (xk b a,b (mk cos (xk ( b a,b m2 m2 m2 ( a,b m2 k b m 2 b m 2. (3.3 m2 k ( a 2 m2 ( O a,b. ( k (a/b(k/m 2 (a/b( /m2

8 8 Bruce C. Berndt, Chadwick Gugg, Sarachai Kongsiriwong, and Johann Thiel Second, k m k/ ( mk cos (xk b k k k O a,b ( m k/ b k 2 ( b k 2. Using these last two estimates in (3.3, we conclude that ( A B O a,b, and so the proof of (2.4 is complete. m k/ ( mk cos (xk b (a/bk 2 ( m (a/b(m/k 2 The authors are grateful to Shaun Cooper for helpful comments. In particular, he informed us of Venkatachaliengar s proofs of theta function transformation formulas delineated at the end of Section. References []. Abramowitz and I.A. Stegun, eds., Handbook of athematical Functions, Dover, ew York, 965. [2] B.C. Berndt, Ramanujan s otebooks, Part II, Springer-Verlag, ew York, 989. [3] B.C. Berndt, Ramanujan s otebooks, Part III, Springer-Verlag, ew York, 99. [4] B.C. Berndt and K. Venkatachaliengar, On the transformation formula for the Dedekind etafunction, in Symbolic Computation, umber Theory, Special Functions, Physics and Combinatorics, F.G. Garvan and.e.h. Ismail, eds., Kluwer, Dordrecht, 200, pp [5] S. Cooper, Cubic theta functions, J. Comp. Appl. ath. 60 (2003, [6] I.S. Gradshteyn and I.. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, San Diego, 994. [7] C. Gugg, odular Identities for the Rogers-Ramanujan Functions and Analogues, Ph.D. Thesis, University of Illinois at Urbana Champaign, 200. [8] S. Kongsiriwong, Theta Functions and Related Infinite Series, Ph.D. Thesis, University of Illinois at Urbana Champaign, [9] K. Venkatachaliengar, Elliptic modular functions and Picard s theorem, in Hyperbolic Complex Analysis (Proc. All India Sem., Ramanujan Inst., Univ. adras, adras, 977, pp. 2 4; Publ. Ramanujan Inst., 4, Univ. adras, adras, 979. [0] K. Venkatachaliengar, Development of Elliptic Functions According to Ramanujan, Technical Report, o. 2, Department of athematics, adurai Kamaraj University, adurai, 988. [] E.T. Whittaker and G.. Watson, A Course of odern Analysis, 4th ed., Cambridge University Press, Cambridge, 966.

9 Theta Transformation Formula 9 Department of athematics, University of Illinois at Urbana-Champaign, 409 West Green Street, Urbana, IL 680, USA address: berndt@illinois.edu Department of athematics, University of Toledo, 280 West Bancroft Street, Toledo, Ohio USA address: Chadwick.Gugg@utoledo.edu Department of athematics, Prince of Songkla University, Hat Yai, Songkhla, Thailand 902 address: sarachai.k@psu.ac.th Department of athematics, University of Illinois at Urbana-Champaign, 409 West Green Street, Urbana, IL 680, USA address: jthiel2@math.uiuc.edu