UNIFICATION OF MODULAR TRANSFORMATIONS FOR CUBIC THETA FUNCTIONS. (Received April 2003) 67vt

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (2004), 2-27 UNIFICATION OF MODULAR TRANSFORMATIONS FOR CUBIC THETA FUNCTIONS S. B hargava Sye d a N oor Fathim a (Received April 2003) Abstract. We obtain a modular transformation for the function a(q, C, z) = S qn2+nm+m2cn+m zn -m ) namely, a (e~ 2nt, el p, eld) = exp[ ( ^ )] a(e~ 3t~, e t, e $ i ), t\/3 67vt demonstrate how this unifies cubic modular transformations recently found by S. Cooper their precursors established by J.M. Borwein P.B. Bor- wein. Here throughout the paper, unless otherwise stated, it is understood that the summation index or indices range over all integral values. We employ some properties of a(q,, z) its relation with the classical theta function (in Ramanujan s notation) f(a,b ) = a n(n + ) / 26n(n - )/2 simply established by the first author in an earlier publication. Also used are some simple properties of f(a,b ) a well known modular transformation for f(a,b ) as recorded by Ramanujan.. Introduction The function ^ ^ gti2+ n m + m 2^ m + n ^ n m a(q,(,z) (l.i) \q\ <, C, ^ ^ z was introduced by the first author in [2] was shown to have properties which unified generalized several known properties of the Hirschhorn-Garvan-Borwein cubic analogues a(q,z), a'(q,z), b(q,z) c(q,z) of classical theta function [4]: a(q,z) (.2) a'{q, z) b(q, z) \ s n2 +nm+m2 n Z ^ q (.3) \ ' n 2+nm-\-m2,,m n n = / & z» (.4) \ c(q,z) := ' n 2+ n m + m 2+ n + m + ~ n m - z (.5) 2000 A M S Mathematics Subject Classification: 54B20, 54E5, 54E55. Supported by the UGC Grant No. F6-29/200. xthe functions c(q,z) in [4] c(q, z) in [2] each differ from the ones defined here by means of (.5) (.8), by a factor of q/ 3.

2 22 S. BHARGAVA AND SYEDA NOOR FATHIMA where to = e 2. That these functions are indeed obtainable from (.) can easily by seen by putting = in (.) in the following variants of (.). a'(q, C,z) := a(9>cv^ / 2, r / 2zI/2) = (.6) b(q, (,z) := a'(5^ z u, 2) = g " 2+" +mv - " C A (.7) c(q, C, z) := 5/ 3a(5, q(, z) = q n2+ n m + m * + n + m + ± ^ n + r r ^ n -m ^ g) Indeed, a(q,z) = a{q,l,z), a'(q,z) = a'(q,l,z), b(q,z) = b(q,l,z) c(q,z) = c(q,l,z). (.9) In his recent paper [5], S. Cooper established the following elegant modular transformations for a'(q, z), a(q, z), b(q, z) c(q, z) which generalize those of J.M. Borwein P.B. Borwein [3] for a(q) := a{q, ), a'(q) := a'(q, ), b(q) := b(q, ) c(q) c(q, ): af(e~,-e ) = 92 ^ expr 6. f a(e 3t 5e 3t)5 (.0) 27r 7 (.) e2 \ exp - 6 ^ j c(e 3t,e 3t), 2n 6 e 3t,e ). c(e' e,9) = ^ e x p ( - ) b{ (.2) (.3) The main purpose of the present paper is to establish (Section 2) the transformation a(e~27rt, eiip, ew) = W 3 exp ft a(e 3*,e,e^). (-4) Moreover, we show (Section 3) that this is in fact equivalent to the following generalizations of (.0)-(.3), a(e~27rt, eiip,,eid) tv 3 ip >2 Qirt // _2tl 2( 39 + ^> a (e 3,e3t,e 3t ), (-5) a'(e~2nt, ei(p. eie) ty/ 3 exp <p 2 - y e + e 2 67r a(e 3,e^,e 6t<P), (-6) b{e~2nt, eilfi,,eie) tv5 v 2 - ipq + e2 (p 6 nt 3 / _27t j _ 20 v eye 3*,e2t,e et j9 (.7)

3 UNIFICATION OF MODULAR TRANSFORMATIONS 23 e x p ( ^ ) c ( e - 2'r,, e - i*,e il>) = ^ 5 exp ip2 + 3 e2 67rt 2 ir 2i 30 + ifi. x b[e 3,e3t,e at ). (.8) That Cooper s identities (.0) (.3) are special cases of (.5) (.8) is immediate by putting ip = 0 in the latter using (.9). We end the paper by applying (Section 4) the transforms (.5) (.8) to a pair of identities in [2], thereby obtaining new identities involving a'(q, z). 2. The Main Transformation We first note some simple relations between the function (.) the classical theta function (in Ramanujan s notation [Eqn. (8.), Ch. 6,]), /(a, b) := ^ V ( " + i ) / 2bn(n-i)/2? \ab \ < l (2.) that we will need in the course of this paper. It was indeed shown by the first author in [Eqns. (.29) (4.2), 2] that simple series manipulations yield, among other results (A /z: integers), a(q,, z) = g 3* 2+ 3AM+M2 2A +m z h a ^ g3(2a+m)/2 ^ qv/2 ^ ^. 2 ) a(q, C, 2) = / (? C 2 ~, 9 C ' z)/(93cz3.«3c 2 3) + q(z f(q2( z - \ C ' z) / f a V. C *-3 ). (2.3) A special case of (2.2), with A = = /i with changed to q(, that will be of our use is Or, what is the same, on using (.8), a{q,q(,z) = C _ ^ a{q, g _/2C, ql/2 z). c(q,(,z) = z a(q,q~ / 2C,q / 2 z). (2.2') We also need the classical theta function transformation [Entry 20,] y/af(e~a2+na, e_q2~na) = en2/ 4\ /^ /(e _/32+in/3, e- ^2_in/?), (2.4) provided a/3 = 7r ^ ( a 2) > 0. In fact the following two special cases of (2.4) will be used: 7rt+ i6 n t id\ _ f ^ \ / ( e _7rt+, e ) = e x p (^ - J /(c ), (2.5) n-* I irt in- / 2 / 7rt 0 2 \ r. 27T + 2S 27T-20, / ( e - * + -, e-» ) = ^ 7 exp( ) / ( - e., - e. ). (2.6) Indeed, to obtain (2.5) from (2.4), we put a = y /j, P = n To obtain (2.6) from (2.4), we substitute a = /3 n =

4 24 S. BHARGAVA AND SYEDA NOOR FATHIMA We also need the following simple consequences of the definition (2.), noted by Ramanujan (Entries 30(ii) 30(iii), 3 of []): f(a, b) + f( -a, -b) = 2 f(a 3 b,ab3), (2.7) f { a,b ) - f ( - a,- b ) = 2 af Q, ^ a4b4^j. (2.8) (.), Lastly, we need the following identity which easily follows from the definition a ( y, - ) = a(q,y,x ). (2.9) Indeed, we have ( \ / \ n m = ^ 9 n,+nm+m2( ^ ) n+m (I) = xin+2my2m^ (2.0) This at once transforms to (2.9) on putting m = i + j n j using (.) again. We are now in a position to state prove our master formula. T heorem 2.. With a(qx,z) defined by (.), the transformation formula (.4) holds. P ro of. We have from (2.3) repeated use of (2.5)-(2.6) afe ~ 2nt^ei<p^ ei6^ _ j ^ e -2nt+i(<p-6) ^e -2Tvt-i(<p-6)^j^e -6nt+i(ip+30) ^e -6nt-i(ip+30)^ _ _ 6-2 trt+i(<p+0) J ^e -4nt+i(<p-0) ^e ~i(ip-0)^ j ^e ~2nt+i(tp+36) ^e ~i(ip+3 2V3 exp v?2+ 3 e2 6 TTt {a/3 + a'p') (2. where -K+tfi-e Tr-<p + e _7r-H +3 _ TT-y-38 a = f(e 2t ; e 2* )? p - / ( e et, e 64 ) / J., n + ip-e 7r -y + 9. TT+ ip+ 36 _ 7r-y -36 a = f ( - e 2t, e 2* )? = /(~e 6,-e «* ).

5 UNIFICATION OF MODULAR TRANSFORMATIONS 25 On using (2.7) (2.8) repeatedly the trivial identity 2(a/3 + a'/?') = (a + a')(/3 + (3') + (a ct'){/3 - /?'), equation (2.) becomes a(e, et(p, e ) = exp > T t r 2 7 T -e + y X l/(e *, e _2n+6-y 2 n v 2 t t - 3 f l - y )/(e st, e 3* ) 27T T -e + y y + 3 fl 47T + y + 3e + e 3 /(e t }C * )/(e 3t,e 3* )] tvs exp ^2 + 36>2 67ri 2 7r. 9 + /J 3 6, with q e 3t, = e at 52 = e et This is same as the identity o (e -a,rt,e ^,e " ) = ^ = e x p x [ / («3<z3, q3c z - 3 )f(qc.z-\qc z) + 9C */(96C*3. r ^ 3) / ( «2C * ".C " *)] (2-2) y2 + 3fl2 67r / 27T cp 30.. a(e~3*,e ar.e «), (2.3) on employing (2.3) the property a(g,,z) = a(g, C *>2 l) which is an easy consequence of the definition (.). Now, (.4) follows immediately from (2.3) on using (2.9) with x, y e^t. 3. Equivalent Transformations Theorem 3.. The transformation formulas (.5) (.8) are all equivalent to the main transformation (.4). P roof. That (.4) (.5) are equivalent follows immediately on using the equivalence of (.4) with (2.3), the symmetry a'(q,(,z) = a'(q,zx ) (which trivially follows from the series form in (.6)) in the right side of (.5), finally the first part of (.6). Now, (.6) follows from (.5) on putting ~Vf i(2q' (pf) # 2' 6 t' then using the trivial identity a'(q, (, z) = a'(q, C_, z~x) (which again easily follows from the series form in (.6)). To prove (.7), we have from (.7) b(e~27rt,e iip,e ie) = a ' f e - ^ e ^ + ^ e ^ " ^ ). Applying (.6) to this gives, on some simplification b(e~2irt, eiip, eie) exp tvs 67T t 2tt y 2?r Y 27T 27r' <p + t r x a(q,q~/2c,q/2z)

6 26 S. BHARGAVA AND SYEDA NOOR FATHIMA with q e~%t, ( = 2 = e ~ ^ e. However, this at once becomes (.7) on employing (2.2)'. Lastly, (.8) follows from (.7) in exactly the same fashion as (.6) from (.5) but with the substitutions - 2 icp' -i(<p' + 39') t =, (p = ' * 3' 3' We will also need the trivial identity c(q, from the series form in (.8). 4. New Identities Involving a'(q,c,z) z) c(q,, _), which of course follows We now apply the results of the previous section to identities (.3) (-32) in [2], namely 2 ct(q,, z)a(q2, ( 2, z2) = b(q2 )[b(q, C2, C*3) + C 2, r ^3)] + C2c(q, (, z)c(q2, C2, z2) + C~2c(q, C_, 2)c(g2, C"2, z2) (4.) 3z a(q, C, z)f(q3 ( ~,()f{q z~ iz) = / ( 0 C -, C ) / 3 ( g * " \ * ) + b(q)f(q3 z~3, z3)[3 q C 'fi q 9C"3, C3) - /(<?C \ C)]- (4-2) We may clarify that we have indeed taken in (4.2) a version of (.32) of [2] convenient for our purpose here. The equivalence is easily established on repeated use of the triple product identity for the theta function (Entry 9, []), /(a, b) = (-a,a&)oo(-m &)oo(am &)oo (4-3) the product form of b(q) (Equation (.6) of [3]) where, as usual, Theorem a'{q,c2,cz)a'{q2 X 2 X z) = [b(q,(2 Xz)b{q2, ( 2,(z) (q\q)lo faa\ b{q) = v m z (4-4) OO (A; 4)00 := JJ(l-Agn). n 0 + b(q,c 2, C z)b{q2, C 2, C z)} + c(q) x [(2 c(q2, C3, z) + (~ 2 c(q2, C 3, z)\ (4.5) 3 a'(q6, C2, Cz)f(q3C \ q 3C)f(q9 z -\ q 9 z) = m q 9 C 3,q9C3 ) f 3 (q9 z -\ q 9 z) + c(q6 )f(q3 z -\ q 3 z ) [ f ( q C \ q O -f ( q 9 C 3,q9 C3)}- (4-6)

7 UNIFICATION OF MODULAR TRANSFORMATIONS 27 P roof. Put q = e 2nt, ( = 2 = eie in (4.) apply (.5) (.8), including the special case of (.2) with 6 0. Some simplification change of variables (e_^,e^,e into q,, z respectively) gives (4.2). Similarly, applying (.5) (2.6) repeatedly to (4.2) yields (4.6). We omit the details. R em ark 4.2. By putting ( = in (4.5) we obtain an identity of Cooper ((6.3) of [5]), a'{q,z)a'(q2,z) = b{q,z)b(q2,z) + c(q)c(q2,z). R em ark 4.3. We can recast (4.6) into a form analogous to that of (.32) of [2], a'(q2, ( 2 Xz) 6^3 {-q3c 3 ;q6 )oo{-q3 ( 3 ;q6 )oo(-q3z ;q6)200(-q3z',q6)lo(q6',q6) ( - t f C - ; q2)oo{-qci q2)oo(q2; q2) oo q V H -q z-w U i-q zrfu im l ( E <n9 " V 3 - E C3V 2) + ( - 9 C _ ; 9 2 ) o o ( - 9 C ; 9 2 )o o ( z ' ; 9 6 ) o o ( «; 9 6 ) o o ( 9 2 ; 9 2 ) o o One need only make repeated use of (4.3) the dual of (4.5) (Equation (.7) of [3]) ciq) = ^ ^ m. We omit the details. \Qi Q) 00 Acknow ledgem ent. The authors are grateful to the referee for his valuable suggestions which considerably improved the quality of the paper. R eferences. C. Adiga, B.C. Berndt, S. Bhargava G.N. Watson, Chapter 6 of Ramanujan s second notebook: Theta functions q-series, Mem. Am. Math. Soc. 53 no. 35 (985), American Mathematical Society, Providence, S. Bhargava, Unification of the cubic analogues of Jacobian theta functions, J. Math. Anal. Appl. 93 no. 2 (995), J.M. Borwein P.B. Borwein, A Cubic counterpart of Jacobi s identity the A G M, Trans. Amer. Math. Soc. 323 no. 2 (99), M.D. Hirschhorn, F.G. Garvan J.M. Borwein, Cubic analogues of the Jacobian theta functions 9(z,q), Canadian J. Math. 45 no. 4 (993), S. Cooper, Cubic theta functions, J. Computational Applied Math., to appear. 00 S. Bhargava Department of Studies in Mathematics University of Mysore, Manasa Gangotri Mysore INDIA sribhargava@hotmail.com Syeda Noor Fathima Department of Studies in Mathematics University of Mysore, Manasa Gangotri Mysore INDIA srinivasamurthyb@yahoo.com

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