Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

Transcription

1 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series James Mc Laughlin Abstract The bilateral series corresponding to many of the third- fifth- sixth- and eighth order moc theta functions may be derived as special cases of ψ series n ac;q) n z n. bd;q) n Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these moc theta functions and the corresponding bilateral series. New and existing summation formulae for these bilateral series are also used to mae explicit in a number of cases the fact that for a moc theta function say χq) and a root of unity in a certain class say ζ that there is a theta function θ χ q) such that lim q ζ χq) θ χq)) exists as q ζ from within the unit circle. 1 Introduction The moc theta functions were introduced to the world by Ramanujan in his last letter to G.H. Hardy [4 pp ] [9 pp. 0 3]) in which he also gave examples of moc theta functions of orders three five and seven. Ramanujan did not explain precisely what he meant by a moc theta function and Ramanujan s James Mc Laughlin Mathematics Department 5 University Avenue West Chester University West Chester PA jmclaughlin@wcupa.edu This wor was partially supported by a grant from the Simons Foundation #09175 to James Mc Laughlin). 1

2 James Mc Laughlin statements were interpreted by Andrews and Hicerson [5] to mean a function f q) defined by a q-series which converges for q < 1 and which satisfies the following two conditions: 0) For every root of unity ζ there is a θ-function θ ζ q) such that the difference f q) θ ζ q) is bounded as q ζ radially. 1) There is no single θ-function which wors for all ζ ; i.e. for every θ-function θq) there is some root of unity ζ for which f q) θq) is unbounded as q ζ radially. A similar definition was given by Gordon and McIntosh [15] [17] where they also distinguish between a moc theta function and a strong moc theta function. The modern view of moc theta functions is based on the wor of Zwegers [930] who showed that the moc theta functions are holomorphic parts of certain harmonic wea Maass forms. In relation to the results in the present paper we recall two areas of investigation in the subject of moc theta functions. Firstly as regards condition 0) above Folsom Ono and Rhoades [13] mae this condition explicit for the third order moc theta function f q) in that they found a formula for the θ-function θ ζ q) and an expression for the limit of the difference f q) θ ζ q) as q ζ radially where ζ is a primitive even-order root of unity see Theorem 3 below). Secondly there is the subject of basic hypergeometric transformations of moc theta functions and summation formula for sums/differences of moc theta functions. Several identities of these types were stated by Ramanujan [4] and were subsequently investigated by Watson [6] and later wor was carried out by Andrews [1] [] and more recently by Gordon and McIntosh [16] [17]. The starting point for the investigation in the present paper is the observation that many of the moc theta functions are special cases of one side n 0 or n < 0) of certain general bilateral series bilateral series which in turn derive from the ψ series n These include: ac;q) n z n bd;q) n ac;q) n z n + bd;q) n n1 ) q/bq/d;q) n bd n. q/aq/c;q) n acz Third order - all 9 Ramanujan Watson Gordon and McIntosh); Fifth order - 8 of 10 Ramanujan); Sixth order - 8 Ramanujan); Eighth order - 4 of 8 Gordon and McIntosh). A number of transformations and summation formula for the ψ series due to Bailey [6] are combined with the representation of these moc theta functions in terms of the ψ series together with other existing summation and transformation formulae for q-series to derive new representations for the moc theta functions and other q-series identities.

3 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 3 Results in the present paper include: 1) radial limit results for a number of third- fifth- sixth- and eighth order moc theta functions similar to that of Folsom Ono and Rhoades [13] alluded to above ) new summation formulae for the bilateral series associated with some of these order moc theta functions 3) new transformation formulae for some of these moc theta functions deriving from these general bilateral transformations 4) a number of other summation formulae. One example of a new summation formula is the following identity for the third order moc theta function φq): φq) + r1 1;q ) r q r n q n q ;q q qq ;q ) ) n q q ;q. ) This formula in turn implies that if ζ is a primitive even-order 4 root of unity then as q approaches ζ radially within the unit dis lim φq) q q q;q ) ) q ζ q q;q ) ζ )1 + ζ 4 ) ζ n )ζ n+1. For the third order moc theta function ψq) there is the transformation ψq) q;q ) n 1) n 1 + q ;q ) q r +r 4r + 1) 1) r. r Note: See the remar at the end of the proof of Theorem 1 about the convergence of the first series on the right. As an example of one of the new summation formulae there is the following: 10r + 1)q 5r +r)/ r 4qq 4 q 16 q 0 ;q 0 ) q ;q 4 + q q 3 q 5 ;q 5 ) ) q;q). ) q;q) q;q) A number of results of a similar nature may be found throughout the paper. Remar: The first version of the present paper was written in 014 and subsequently the author attention was directed my thans to the anonymous referee) to a number of recent papers containing similar results of which the present author was previously unaware. In [1] Mortenson derived several identities involving the Appell-Lerch sum mxqz) : 1 jz;q) r 1) r q rr 1)/ z r 1 q r 1 1) xz

4 4 James Mc Laughlin here jz;q) zq/zq;q) ) and the universal moc theta function gxq) see 9)) and some of these were used in [] to derive explicit radial limits for moc theta functions. As well deriving such radial limits for several particular moc theta functions in [] the author also derives a general result for gxq) a result which permits an explicit radial limit to be derived for any even-order moc theta function that may be expressed in terms of gxq). We will compare results in the present paper with those in [1 ] in several places throughout the paper. For example by applying a formula of Mortenson [ Eq. 6.10)]) a different radial limit result is obtained for the eighth order moc theta function S 0 q) see 9)). Subsequent to writing the first draft of the present paper the author was also directed to the recent paper [7] in which the authors also derive explicit radial limits for all of Ramanujan s third and fifth order moc theta functions as well as giving the level and weight information for the theta functions which are modular forms). The authors in [7] also state without proof explicit radial limits for many of the even-order moc theta functions. In the present paper we also derive these explicit radial limits using somewhat different methods but in addition also derive many identities that come from the aforementioned connections with the ψ series. Some required basic hypergeometric formulae To prove some of the results in the present paper it is necessary to use a number of transformation- and summation formulae for basic hypergeometric series. a;q) n : a;q) 1 aq n ;q) aq n q/a)n q nn 1)/ ) ;q) n q/a;q) n z) n q n zqq/zq ;q ). 3) n n n ac;q) n z n azczqb/aczqd/acz;q) 4) bd;q) n bdq/aq/c;q) ) acz/bacz/d;q) n bd n. azcz;q) n acz n ac;q) n z n 5) bd;q) n b/ad/cazqb/acz;q) ) aacz/b;q) n b n bq/czbd/caz;q). n azd;q) n a

5 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 5 ) e f ;q) n qa n q/cq/daq/eaq/ f ;q) n aq/caq/d;q) n e f aqq/aaq/cdaq/e f ;q) 1 aq n ) )cde f ;q) n qa 3 n q n. 6) n 1 a)aq/caq/daq/eaq/ f ;q) n cde f n q a q abcde;q) n qa a aaq/baq/caq/daq/e;q) n bcde aqaq/bcaq/bdaq/beaq/cdaq/ceaq/deqq/a;q) aq/baq/caq/daq/eq/bq/cq/dq/eqa /bcde;q). 7) ) n n bc;q) n aq/baq/c;q) n ) qa n bc aq/bc;q) aq /b aq /c q aqq/a;q ) aq/baq/cq/bq/c qa/bc;q). 8) The identity at 3) is the famous Jacobi triple product identity. The bilateral transformations at 4) 5) and 6) are all due to Bailey [6]. The identity at 7) Bailey s 6ψ 6 summation formula and 8) is a special case of this see [14 Eq. II.30) p. 357]). 3 Moc theta functions of the third order The third order moc theta functions stated by Ramanujan [4 pp ] [9 pp. 0 3]) are the following basic hypergeometric series: f q) χq) q n q q;q) n φq) q n q;q) n q 3 ;q 3 ψq) ) n n1 q n q ;q ) n q n q;q ) n. All of the third order moc theta functions of Ramanujan as well as those stated later by Watson [6] and Gordon and McIntosh [16] may be expressed in terms of the function gxq) where gxq) : q n +n xq/x;q) n+1 x q n x;q) n+1 q/x;q) n ). 9)

6 6 James Mc Laughlin This was shown by Hicerson and Mortenson [18 Eqs. 5.4) )] this function was also defined by Gordon and McIntosh [17] where it was labelled g 3 xq) ). For completeness we consider a generalization namely the series G 3 stq) : 1 + n1 s n t n q n 10) sqtq;q) n which was defined in [10 Eq. 7)] and state a number of transformation formulae for this function. Note that the connection with the third order moc theta functions is that G 3 xq/xq) 1 x)1 q/x)gxq). 11) Proposition 1. Let G 3 stq) be as defined at 10) above. Then G 3 stq) r1 r1 r1 s 1 t 1 ;q) r q r + q/sq/t;q) sqtq;q) st;q) r q r ; 1) r s 1 t 1 ;q) r q r + q/t;q) sqq;q) t;q) r s) r q rr+1)/ ; 13) r tq;q) r s 1 t 1 ;q) r q r 14) + q/sq/t;q) 1 stq stqq/st)q;q) r )st;q) r st) r q r. r 1 st)sqtq;q) r Proof. The transformations at 1) and 13) will follow as special cases of two more general identities. Replace z with zq/ac let ac and set b sq and d tq in respectively 4) and 5) to get that n z n q n sq/ztq/z;q) sqtq;q) n sqtq;q) ) stq r z/sz/t;q) r 15) r z qs/z;q) z/s;q) r s) sqstq/z;q) r q rr+1)/. 16) r tq;q) r Lastly replace z with st and use ) on the terms of negative index in the new series on the left sides. We also prove a generalization of the transformation at 14) first by letting e f in 6) and then replacing a with z c with z/s and d with z/t to get n z n q n sqtq;q) n sq/ztq/z;q) 1 zq zqq/zstq/z;q) r )z/sz/t;q) r zst) r q r. 17) r 1 z)sqtq;q) r The identity at 14) follows after replacing z with st.

7 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 7 The identities 1) - 14) may be more concisely expressed using the function as follows G 3stq) : n s n t n q n 18) sqtq;q) n G 3stq) q/sq/t;q) sqtq;q) G 3s 1 t 1 q) 19) q/t;q) t;q) r s) sqq;q) r q rr+1)/ ; 0) r tq;q) r q/sq/t;q) 1 stq stqq/st)q;q) r )st;q) r st) r q r. 1) r 1 st)sqtq;q) r The identity at 1) or 0)) was also proved by Choi [10 Theorem 4] and stated previously by Ramanujan see [4 Entry 3.4.7]). We will employ 13) to derive some results on explicit radial limits as mentioned earlier. Before coming to that we remar that other transformations listed above may be used to derive some new transformations for three of the third order moc theta functions of Ramanujan and one of the third order moc theta functions of Watson similar results may be derived for the other third order moc theta functions of Watson [6] and those of Gordon and McIntosh [16]). Before stating the next theorem we recall Watson s [6] third order moc theta function νq) where Theorem 1. If q < 1 then f q) φq) n1 n1 νq) r0 q n +n q;q ) n ;q) n q n + 4 q;q) q q;q) r +r 4rq r + 1) 3 r 1 + q r ). ) 1;q ) n q n + 4 q ;q ) q q;q) r +r rq r + 1) 3 r 1 + q r ). 3) νq) + 4 q;q ) q;q) 3 ψq) q;q ) n q n q r +r r + 1) r 1 + q r+1 ) ) q;q q;q) 3 q 4 q 1 q 16 ;q 16 ). 4) q;q ) n 1) n 1 + q ;q ) q r +r 4r + 1) 1) r. 5) r

8 8 James Mc Laughlin Proof. For ) replace z with z s and t with z in 17) and then let z 1. A similar application of 17) again with z replaced with z s replaced with iz and t replaced with iz and once again letting z 1 leads to 3). For 4) in 17) again replace z with z and then replaces with iz t with iz let z q and divide through by 1 + q. Finally the transformation at 5) follows similarly from 17) this time with z replaced with z s replaced with z/ q and t replaced with z/ q and once again letting z 1. Note that convergence of the first series on the right of 5) is in the Cesàro sense. As Watson pointed out in [7 Section 7] certain bilateral series related to fifth order moc theta functions which are essentially the sums of pairs of fifth order moc theta functions are expressible as theta functions or combinations of infinite q-products. It seems less well nown that the bilateral series associated with two of Ramanujan s third order moc theta functions are also expressible as infinite products. We also give similar statement for Watson s [6] third order moc theta function νq). Theorem. If q < 1 then φq) + νq) + ψq) + r0 r1 r0 1;q ) r q r q;q ) r q r q;q ) r 1) r n r n q n q ;q q qq ;q ) ) n q q ;q ; 6) ) q n +n q;q ) n+1 q q ;q ) q 4 ;q 4 ) ; 7) q n q;q q qq ;q ) ) n q q ;q. 8) ) Proof. From 8) replace q with q set b z/t a z and let c ) z/t;q) r t r q rr+1)/ t q /z zq q/zq ;q ) r tq;q) r tq tq/z;q) and from 16) with s t) n z n q n t q ;q tq/z;q) ) n tq t q/z;q) Together these equations imply that n z/t;q) r t) r q rr+1)/. r tq;q) r z n q n t q ;q zq q/zq ;q ) ) n t q t q/z;q. 9) ) The identity at 6) is now immediate upon setting z 1 and t 1 and that at 8) results similarly upon setting z 1 and t 1/q. The identity at 7) follows

9 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 9 upon setting z q t q multiplying the resulting product by 1/1 + q) and finally performing some elementary q-product manipulations. Note that the convergence of the sum added to ψq) on the left side of 8) is in the Cesàro sense. Note also that comparison of the infinite products on the right sides of 6) and 8) yields the rather curious identity n q n q ;q ) n n q n q;q ) n 30) where by the previous comment convergence of the part of the bilateral series on the right consisting of terms of negative index is again in the Cesàro sense. The summation formulae in the preceding theorem have some interesting implications. Firstly they allow condition 0) above to be made explicit for some of the third order moc theta functions. We recall the recent result for f q) in [13]. Theorem 3. Folsom Ono and Rhoades [13]) If ζ is a primitive even-order root of unity then as q approaches ζ radially within the unit dis we have that Here lim f q) q ζ 1) bq)) ζ ) 1 + ζ ) ζ n ) ζ n+1. 31) bq) q;q) q;q). The infinite product representation of bq) was not stated in [13] but was stated by Rhoades in [5]. Note that Theorem 3 was also proved recently by Zudilin [8]. The following results are immediate upon rearranging the identities in Theorem and letting q tend radially to the specified root of unity from within the unit circle since the other series accompanying each of the moc theta functions in the bilateral sums terminates the interchange of summation and limit in each of the corresponding series on the right is justified by the absolute convergence of each of these series). Corollary 1. i) If ζ is a primitive even-order 4 root of unity then as q approaches ζ radially within the unit dis we have that lim φq) q q q;q ) ) q ζ q q;q ) ζ )1 + ζ 4 ) ζ n )ζ n+1. 3) ii) If ζ is a primitive even-order 4 + root of unity then as q approaches ζ radially within the unit dis we have that lim νq) q ;q ) q 4 ;q 4 ) ) q ζ 1+ζ )1+ζ 3 )...1+ζ n 1 )ζ n. 33)

10 10 James Mc Laughlin iii) If ζ is a primitive odd-order + 1 root of unity then as q approaches ζ radially within the unit dis we have that lim ψq) q q q;q ) ) q ζ q q;q ) 1 ζ )1 ζ 3 )...1 ζ n 1 ) 1) n. Remar: The results in Corollary 1 were also proved in [7] using somewhat similar arguments as were the results in Corollary 3 below. The second implication is that they imply some summation formulae for some of the bilateral series appearing in Theorem 1. Corollary. If q < 1 then 34) q r +r rq r + 1) r 1 + q r ) q;q ) 4 q;q) 4 4 q r +r r + 1) r 1 + q r+1 ) q ;q ) q;q) 3 q 4 ;q 4 ) q;q + q4 q 1 q 16 ;q 16 ) ) 35) 36) Proof. The first identity 35) follows from combining the results at 3) and 6) and then replacing q with q. The identity at 36) follows directly from comparing the identities 4) and 7). 4 Moc theta functions of the fifth order Ramanujan s fifth order moc theta functions are the following: f 0 q) F 0 q) φ 0 q) ψ 0 q) χ 0 q) q n q;q) n f 1 q) q n q;q ) n F 1 q) q n q;q ) n φ 1 q) q n+1)n+)/ q;q) n ψ 1 q) q n q;q) n χ 1 q) q;q) n q nn+1) q;q) n q nn+1) q;q ) n+1 q n+1) q;q ) n q nn+1)/ q;q) n q n q;q) n. q;q) n+1 Of interest here is the fact that that certain combinations of pairs of moc theta functions of order five may be expressed as single bilateral series and hence in terms of theta products as was described by Watson in section 7 of [7] see also

11 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 11 the forthcoming boo [0] where the proofs of these identities are possibly more transparent than those of Watson [7]). We state these identities directly in terms of q-products rather than employing the Ramanujan functions Gq) and Hq) as Watson did. Proposition. The following identities hold. n n q r f 0 q) + ψ 0 q) 4q q4 q 16 q 0 ;q 0 ) q;q) r q ;q 4 + q q 3 q 5 ;q 5 ). 37) ) q;q) q r +r f 1 q) + ψ 1 q) 4 q8 q 1 q 0 ;q 0 ) q;q) r q ;q 4 qq4 q 5 ;q 5 ). 38) ) q;q) n n q 4r q ;q 4 ) r F 0 q ) + φ 0 q ) 1 q q4 q 16 q 0 ;q 0 ) q ;q 4 ) + q q 3 q 5 ;q 5 ) q;q). 39) q 4r +4r q ;q 4 F 1 q ) φ 1 q ) ) r+1 q q8 q 1 q 0 ;q 0 ) qq ;q 4 ) qq4 q 5 ;q 5 ) q q;q). We note that these summation formulae may be rearranged and used to give explicit radial limits for the difference of certain fifth order moc theta functions and certain corresponding theta functions as q tends to certain roots of unity from within the unit circle in a manner similar to the result of Folsom Ono and Rhoades [13] for the third order moc theta function f q) stated at 31) above or to the results stated for the third order moc theta functions φq) νq) and ψq) stated in Corollary 1. For ease of notation the statements for F 0 q) and F 1 q) are written in terms of q instead of q. Corollary 3. i) If ζ is a primitive even-order root of unity then as q approaches ζ radially within the unit dis we have that [ lim f 0 q) 4q q4 q 16 q 0 ;q 0 ) q ζ q ;q 4 + q q 3 q 5 ;q 5 ]) ) ) q;q) 40) ζ )1 + ζ ) ζ n )ζ n+1)n+)/. 41) ii) If ζ is a primitive even-order root of unity then as q approaches ζ radially within the unit dis we have that [ lim f 1 q) 4 q8 q 1 q 0 ;q 0 ) q ζ q ;q 4 qq4 q 5 ;q 5 ]) ) ) q;q)

12 1 James Mc Laughlin ζ )1 + ζ ) ζ n )ζ nn+1)/. 4) iii) If ζ is a primitive even-order 4 + root of unity then as q approaches ζ radially within the unit dis we have that [ lim F 0 q ) q q4 q 16 q 0 ;q 0 ) q ζ q ;q 4 + q q 3 q 5 ;q 5 ]) ) ) q;q) n1 1 ζ )1 ζ 6 )...1 ζ 4n ) 1) n ζ n. 43) iv) If ζ is a primitive even-order 4 + root of unity then as q approaches ζ radially within the unit dis we have that [ q lim F 1 q 8 q 1 q 0 ;q 0 ) ) q ζ qq ;q 4 qq4 q 5 ;q 5 ]) ) ) q q;q) 1 ζ )1 ζ 6 )...1 ζ 4n ) 1) n ζ n +4n. 44) There are no nown unilateral transformations for moc theta functions of the fifth order similar to those that exist for moc theta functions of the third order. However there are bilateral transformations that may be applied to the bilateral series in Proposition. Here we consider the series G 5 wyq) n w n q n. 45) y;q) n Identities for this function are not so plentiful as those for G 3 stq) and G 3 stq) but two such are given in the next theorem. Theorem 4. Let G 5 wyq) be as defined at 45). Then G 5 wyq) y/w;q) y;q) wq/y;q) r y) r q rr 1)/ 46) r y/w;q) 1 wq wqq/w;q) r )wq/y;q) r yw ) r q 5r 3r)/. 47) r 1 w)y;q) r Proof. In 4) or 5)) replace z with z/ac then let a c and d 0. Then replace z with wq and b with y and 46) follows. For 46) let de f in 6) and then set a w and c wq/y. Remar: For G 5 wyq) to represent a sum of fifth order moc theta function it necessary to have w 1 or w q and in those cases 46) does not provide any non-trivial results for w 1 the right side is just the series in reverse order).

13 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 13 The identity at 47) could be used to derive new expressions for the sums of fifth order moc theta functions found in Corollary. However we instead use it to derive four identities for bilateral series similar to those in Corollary. Corollary 4. The following identities hold for q < 1: 10r + 1)q 5r +r)/ r 4qq 4 q 16 q 0 ;q 0 ) q ;q 4 + q q 3 q 5 ;q 5 ) ) q;q) ) q;q) q;q) 10r + 3)q 5r +3r)/ r 4q 8 q 1 q 0 ;q 0 ) q ;q 4 qq4 q 5 ;q 5 ) ) q;q) ) q;q) q;q) 5r + 1) 1) r q 10r +4r r qq 4 q 16 q 0 ;q 0 ) q ;q 4 + q q 3 q 5 ;q 5 ) ) q 4 ;q 4 ) ) q;q) q ;q 4 ) 5r + ) 1) r q 10r +8r r 48) 49) 50) 51) q 8 q 1 q 0 ;q 0 ) q ;q 4 qq4 q 5 ;q 5 ) ) q 4 ;q 4 ) ) q;q) q ;q 4. ) Proof. For 48) in 47) replace w with w set y wq and simplify the resulting right side to get n w n q n wq;q) n q/w;q) w qq/w ;q) 1 wq r )w 5r q 5r r)/ r 1 w) q/w;q) w qq /w;q) ) 1 wq 1 + r )w 5r q 5r r)/ + 1 wq r )w 5r q 5r +r)/ 1 w r1 q/w;q) w qq /w;q) ) w 1 + 5r q 5r r)/ 1 wq r )w 10r + 1 wq r )q r ) 1 w r1 q/w;q) w qq /w;q)

14 14 James Mc Laughlin 1 + w 5r q 5r r)/ w 1 ) ) w10r 1 1 w10r+1 + qr. r1 1 w 1 w Now let w 1 noting that the left side above tends to the left side of 37) and hence to the right side of 37). After using L Hospital s rule on the terms in the last series on the right side this series becomes 1 + r1 r q 5r r)/ 10r 1) + q r 10r + 1)) 10rq 5r +r)/ + r q 5r r)/. The result now follows. To obtain 49) in 47) replace w with w q set y wq and simplify the resulting right side to get n w n q n +n wq;q) n 1/w;q) w ) w qq/w ;q) 1/w;q) w q 1/w ;q) 1 w q r+1 )w 5r q 5r +3r)/ r0 1 w + ) 1/w;q) w ) w qq/w ;q) w 5r q 5r +3r)/ 1 w 10r 3 ) 1 w ) r0 r0 1 w q r+1 )w 5r q 5r +3r)/ r 1 w q) 1 w q r 1 )w 5r 5 q 5r +7r+)/ 1 w ) w q r+1 1 ) w 10r 7 ) 1 w ) where the second series in the second right side came from taing the terms of negative index in the series on the first right side and replacing r with r 1. The identity at 49) now follows as previously upon letting w 1 this time noting that the left side tends to 38). For 50) and 51) in 47) replace wyq) with w wq q 4 ) and w q 4 wq 6 q 4 ) respectively in the case of 51) after maing the replacements in 47) multiply both sides by 1/1 wq )). The details are similar to those in the proofs of 48) and 49) and are omitted. 5 Moc theta functions of the sixth order The sixth order moc theta functions which concern us here are

15 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 15 φq) ρq) λq) φ q) n1 1) n q n q;q ) n ψq) q;q) n q nn+1)/ q;q) n q;q σq) ) n+1 1) n q n q;q ) n µq) q;q) n q n q;q) n 1 q;q ψ q) ) n 1) n q n+1) q;q ) n q;q) n+1 q n+1)n+)/ q;q) n q;q ) n+1 n1 1) n q;q ) n q;q) n q n q;q) n q;q ) n. The series stated for µq) does not converge but the sequence of even-indexed partial sums and the sequence of odd-indexed partial sums do converge and µq) is defined to be the average of these two values. Andrews and Hicerson [5] proved a number of identities for the sixth order moc theta functions stated by Ramanujan in the Lost Noteboo [3]. Berndt and Chan [8] proved a number of similar identities. The proofs in both of these papers were quite involved employing both Bailey pairs and the constant term method and simpler proofs were later given by Lovejoy [19] for four of the identities proved by Andrews and Hicerson [5]. These four identities are listed in the following theorem. Theorem 5. The following identities hold for q < 1. q 1 ψq ) + ρq) q;q ) q q 5 q 6 ;q 6 ) 5) φq ) + σq) q;q ) q 3 q 3 q 6 ;q 6 ) 53) φq ) µ q) q;q ) q 3 q 3 q 6 ;q 6 ) 54) q 1 ψq ) + λ q) q;q ) q q 5 q 6 ;q 6 ) 55) To maintain uniformity we show that as with the third order- and fifth order moc theta functions the bilateral transformations of Bailey at 4) 5) and 6) may be used to express sums of sixth order moc theta functions as theta functions and that these identities in turn may liewise be used to examine the limiting behavior of some of these sixth order moc theta functions as q tends to certain classes of roots of unity from within the unit circle. As above we begin by stating a number of general bilateral transformations. Theorem 6. i) If q bd/azq < 1 then G 6 abdzq) : n a;q ) r z r q r bd;q ) r n zq qb/az qd/az;q ) bdq /a;q ) azq/b azq/d;q ) r zq;q ) r ) bd r. 56) azq

16 16 James Mc Laughlin ii) If q bd/azq b/a < 1 then a;q ) r z r q r n bd;q b/a qb/az;q ) ) r b bd/azq;q ) iii) If q bd/azq < 1 then n a;q ) r z r q r n bd;q bq/az dq/az qz;q ) ) r bd/aqz q 3 /az aqz;q ) n azq/ba;q ) r d;q ) r ) b r. 57) a 1 + azq 4r 1 )a azq/b azq/d;q ) r bdz) r q 3r 4r 1 + az/q)bd zq;q ) r. 58) Proof. For 56) and 57) replace q with q z with zq/c and then let c in 4) and 5) respectively. The identity 58) is a consequence of replacing q with q in 6) and then replacing c with aq /b and d with aq /d letting f and replacing in turn a with ze/q and finally e with a. The sums of various pairs of sixth order moc theta functions may be expressed in terms of G 6 abdzq) and the above theorem may be used to derive some alternative expressions for these sums. Corollary 5. The following identities hold for q < 1. 4σq) + µq) q;q ) q;q) 1) r 6r + 1)q r3r+1)/ 59) r φq) + φ q) q;q) q ;q 4 ) 60) [ q ;q 4 ) q 6 q 6 q 1 ;q 1 ) q ;q 4 ) q 6 q 6 q 1 ;q 1 ) ] ψq) + ψ q) 3q q;q) q 6 q 6 q 6 ;q 6 ) q ;q ) ρq) + λq) 3q;q ) q 3 q 3 q 3 ;q 3 ) q;q) 61) 6) Proof. In 58) replace z with zq 3 and set a zq b zq 3 and d zq 3 to get zq ;q ) r z) r q r +3r r zq 3 zq 3 ;q 1/qz 1/qzzq4 ;q ) ) r 11/q z q 6 z ;q ) r q 3r +5r z 3r 1 z q 4r+4) q z;q ) r 1 q 4 z )q 4 z;q ) r

17 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 17 q /z ;q 4 ) zq 4 ;q ) z q 1 1/z) q q /z q 6 z ;q ) r3r+5) z 3r 1 + zq r+) r 1 + q z)1 + z) Now multiply both sides by q /1 z q ) and let z 1 noting that the left side tends to σq ) + µq )/ using the definitions above and ). On the right side replace r with r 1 and rewrite the resulting series as 1 + zq r )z 3r q 3r r r 1 + z r r z 3r q 3r r + z 3r+1 q 3r +r 1 + z z 3r q 3r r + z 3r+1 q 3r r 1 + z r z 3r q 3r r 1 + z 6r z where the second equality follows from reversing the order of summation for the second terms in the sum. Now let z 1 to arrive at σq ) + µq ) q ;q 4 ) 4q ;q ) 6r + 1) 1) r q 3r +r. r The identity at 59) now follows upon multiplying this last identity by 4 and replacing q with q 1/. Note that the expression for 4σq) + µq) deriving from 53) and 59) together with 53) imply that 6r + 1) 1) r q 3r +r)/ q;q) r q;q ) ] [ q;q ) q 3 q 3 q 6 ;q 6 ) q;q ) q 3 q 3 q 6 ;q 6 ). 63) Remar: It may be of interest to compare the identity above with that of Fine [1 p.83]: 6r + 1)q r3r+1)/ q;q) 3 q;q ). 64) r For 60) set a zq b zq and d zq in 58) to get after simplifying the right side zq;q ) r z r q r r zqzq ;q q/zq /z zq;q ) ) r qq /z q z ;q ) r 1 + zq r )z 3r q 3r r 1 + z Let z 1 on the left side to get once again using the definitions above and ) φq) + φ q) q;q) q ;q ) 6r + 1) 1) r q 3r +r. r

18 18 James Mc Laughlin An application of 63) with q replaced with q gives the result. Similarly for 61) replace z with zq a with zq b with zq and d with zq 3 in 58) to get after once again simplifying the right side that zq;q ) r z r q r +r r zq zq 3 ;q 1/zq/z zq3 ;q ) ) r q1/z q 4 z ;q ) z 3r q 3r +3r r 1/zq/z zq3 ;q ) q 6 z 3 1/z 3 q 6 ;q 6 ) q1/z q 4 z ;q ) where the Jacobi triple product identity 3) has been used at the last step. The result now follows after multiplying both sides by q/1+q) and letting z 1 as before. The details of the proof of 6) are omitted. Briefly replace z with zq a with zq b with zq 3 and d with zq 3 in 58) simplify and sum the right side using the Jacobi triple product identity 3) let z 1 multiply both sides by /1 q ) and finally replace q with q 1/. Remar: Choi [10 p. 370] also gave expressions for each of the sums of sixth order moc theta functions in Corollary 5 but with different combinations of theta functions on the right sides. Yet another version of 60) was stated by Ramanujan [3 p. 6 and p. 16] see also [11 p. 1740]). Different proofs of 61) and 6) were given by Choi and Kim [11 Theorem 1.4 p. 174]. The identities in Corollary 5 also follow from expressions for the sixth order moc theta functions in terms of the function mxqz) see 1)) proved by Hicerson and Mortenson in [18] and nown results about mxqz). We note that the identities in Corollary 5 may be used to describe the asymptotic behavior of each of the two sixth order moc theta functions on the left side of each identity at particular classes of roots of unity. For example 59) may be used in conjunction with 63) to mae condition 0) in the interpretation by Andrews and Hicerson of a moc theta function explicit for both σq) at primitive roots of unity of odd order and for µq) at primitive roots of unity of even order. We state the result for just one of each pair of moc theta functions and leave the result for the other moc theta function of each pair to the reader. Corollary 6. i) If ζ is a primitive odd-order + 1 root of unity then as q approaches ζ radially within the unit dis we have that lim σq) 1 ] [ q;q ) ) q ζ 4 q 3 q 3 q 6 ;q 6 ) q;q ) q 3 q 3 q 6 ;q 6 ) 1 1 ζ )1 ζ 3 )...1 ζ n 1 ) 1 + ζ )1 + ζ ) ζ n ) 1)n. 65) ii) If ζ is a primitive even-order root of unity then as q approaches ζ radially within the unit dis we have that

19 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 19 lim φq) q;q) q ζ q ;q 4 ) [ q ;q 4 ) q 6 q 6 q 1 ;q 1 ) q ;q 4 ) q ]) 6 q 6 q 1 ;q 1 ) n1 1 + ζ )1 + ζ ) ζ n 1 ) 1 ζ )1 ζ 3 )...1 ζ n 1 ) ζ n. 66) iii) If ζ is a primitive even-order root of unity then as q approaches ζ radially within the unit dis we have that lim ψq) 3q q;q) q 6 q 6 q 6 ;q 6 ) ) q ζ q ;q ) n1 1 + ζ )1 + ζ ) ζ n ) 1 ζ )1 ζ 3 )...1 ζ n 1 ) ζ n. 67) iv) If ζ is a primitive odd-order +1 root of unity then as q approaches ζ radially within the unit dis we have that lim ρq) 3q;q ) q 3 q 3 q 3 ;q 3 ) ) q ζ q;q) 1 1 ζ )1 ζ 3 )...1 ζ n 1 ) 1 + ζ )1 + ζ ) ζ n ) ζ )n. 68) Before considering eighth order moc theta functions we compare the results in the present paper with those implied by an identity of Mortenson [ Eq. 6.10)]): q nn+1)/ q;q) n + x;q) n+1 q/x) n+1 1 q n q/x;q) n x;q) n q;q) n jx;q) g 3 x;q) J J 3 + J 1 jx ;q ) + 1 J 10 j x ;q ) x J1 4J4 4 jx ;q ) j qx ;q ) 1 J4 jx;q) x j x;q) j qx ;q ) 69) where the first series is g xq) the universal moc theta function of Gordon and McIntosh [17 Eq. 4.11)] g 3 x;q) is as defined at 9) and jx;q) : xq/xq;q) J am : jq a ;q m ) J am : j q a ;q m ) J m : J m3m q m ;q m ). As Mortenson indicated in [] if a moc theta function is expressible in terms of g xq) and combinations of infinite products then it may be possible to derive a radial limits result for certain classes of roots of unity and indeed Mortenson derives such results for a second order moc theta function and one of tenth order and states that there are many other cases where 69) may be applied.

20 0 James Mc Laughlin As one way of deriving explicit radial limits one might hope after substituting for g xq) in 69) so that this expression now contains a moc theta function that there is then a class of roots of unity such that as q approaches one of these roots of unity say ζ from within the unit circle the moc theta function becomes unbounded the term involving g 3 x;q) vanishes and the second series on the left terminates. In this case 69) may then be rearranged to give an identity of the form lim moc theta function theta function) finite q-series in ζ q ζ which is the typical form of a radial limits result. For example if q is replaced with q 6 and x with q 3 in 69) and the second identity at [17 Eq. 5.10)] namely ψq 4 ) q3 J J 4J 4 J 1 J 3 J 8 q 3 g q 3 q 6 ) 70) is used to substitute for g q 3 q 6 ) then after some q-product manipulation we get ψq 4 ) + q3 J 5 1 J J17 1 4J6 8 J4 3 J7 1 J8 4 4J6 6 q3 J J 4J4 J4 4 J 1 J 3 J8 q3 J3 J 6 J 1 q 6nn+1) q 3 ;q 6 ) n+1 q 6n+3 q 3 q 3 ;q 6 ) n q 6 ;q 6 ) n. 71) If q ζ where ζ is a primitive even-order root of unity then both the series on the right of 71) and the last term on the left become unbounded and there is no radial limit. Unfortunately for producing explicit radial limits when q ζ where ζ is a primitive odd-order root of unity while the series on the right of 71) terminates and the last term on the left vanishes the series for ψq 4 ) also terminates. After eliminating terms that vanish when q ζ where ζ is a primitive + 1-th root of unity one gets that q 3 J1 5 lim q ζ J6 4 + J17 1 4J6 8 q3 J J 4J ) 4 J8 4 J 1 J 3 J8 r0 ζ 6r+3 ζ 3 ζ 3 ;ζ 6 ) r ζ 6 ;ζ 6 ) r r0 1) r ζ 4r+1) ζ 4 ;ζ 8) r ζ 4 ;ζ 4 ) r+1. 7) Curiously what experiment suggests is that each side is identically zero and in particular that if ζ is a primitive + 1-th root of unity then ζ 6r+3 ζ 3 ζ 3 ;ζ 6 ) r r0 ζ 6 ;ζ 6 ) r r0 1) r ζ 4r+1) ζ 4 ;ζ 8) r ζ 4 ;ζ 4 ) r+1 ψζ 4 )) 73) holds for all 0. Note that equality in 73) does not hold if ζ is replaced with an arbitrary value of q inside the unit circle since the difference is a non-zero function

21 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 1 of q: q 6r+3 q 3 q 3 ;q 6 ) r r0 q 6 ;q 6 ) r r0 1) r q 4r+1) q 4 ;q 8) r q 4 ;q 4 ) r+1 q3 q4 + q 8 + q9 q1 + q15 + q16 3q 0... Note also that each of the three individual terms on the left side of 7) diverges to as q ζ even though the combination converges to zero. Of course 73) will hold if q 3 J 5 1 J J17 1 4J6 8 q3 J J 4J4 J8 4 J 1 J 3 J8 q;q ) θq) where θq) is a function of q that remains bounded as q approaches any primitive odd-order root of unity from within the unit circle. We have not attempted to prove this nor 73). If Ramanujan s identity see [17 Eq. 5.8)]) q 1 ψq ) + λ q) q;q ) j qq 6 ) with q replaced with q is used to replace ψq 4 ) in 71) and a similar analysis of radial limits is attempted what experiment also appears to indicate is that ζ 6r+1 ζ 3 ζ 3 ;ζ 6 ) r r0 ζ 6 ;ζ 6 ) r r0 1) r ζ r ζ ;ζ 4) r ζ ;ζ ) r λζ )) 74) holds where ζ is a + 1-th primitive root of unity and again 74) does not hold if ζ is replaced with an arbitrary q inside the unit circle). Similar results may be obtained from 69) for other sixth order moc theta functions. 6 Moc theta functions of the eighth order We consider four of the eight moc theta functions of order eight introduced by Gordon and McIntosh [15]: S 0 q) T 0 q) q n q;q ) n q ;q S 1 q) ) n q n+1)n+) q ;q ) n q;q T 1 q) ) n+1 q nn+) q;q ) n q ;q ) n 75) q nn+1) q ;q ) n q;q ) n+1. 76) As with moc theta functions of other orders certain sums of eighth order moc theta functions may be written as single bilateral series and it is a straightforward

22 James Mc Laughlin consequence of the definitions and ) that S 0 q) + T 0 q) S 1 q) + T 1 q) r r q;q ) r q r q ;q ) r 77) q;q ) r q r +r q ;q ) r. 78) In fact following the method of the authors in [15] it will be shown that each of these sums has an expression in terms of infinite products. We include the proof here since in [15] the authors omitted the final step of explicitly stating the form of the infinite products although these expressions were stated by them in [17 Eq. 5.1)] and these expressions with further details of the proof were given by them in [16 Section 4]). As with the identities in Corollary 5 the identities in Theorem 7 may also be shown to follow from identities proved by Hicerson and Mortenson in [18]. Theorem 7. If q < 1 then S 0 q ) + T 0 q ) q ;q [ ) q;q ) 3 + q;q ) 3 ] q ;q 79) ) S 1 q ) + T 1 q ) q ;q [ ) q;q ) 3 q;q ) 3 ] q q ;q. 80) ) Proof. Let R 0 q) and R 1 q) denote the series on the right side of 77) and 78) respectively. Next in 7) replace q with q set a q b iq and c iq and then let de. This leads to R 0 q ) qr 1 q ) 1 q r 1 q 4r+1 ) q ;q 4 ) r q r 1 q) q 4 ;q 4 ) r q 3 qq q;q ) iq iq iq iq;q ) q ;q ) q;q ) 3 1 q) q ;q ). Multiply through by 1 q and then replace q with q to get an expression for R 0 q ) + qr 1 q ). The pair of equations may then be solved for in turn R 0 q ) and R 1 q ) to give the results. To mae use of the above identities we consider bilateral sums related to the eighth order moc theta functions the eight order equivalent of Theorem 6). Theorem 8. i) If q < 1 then

23 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 3 G 8 abdzq) : n a;q ) r z r q r b;q ) r zq qb/az;q ) bq /a;q ) n azq/b;q ) r zq;q ) r b) r q r r. 81) ii) If q b/a < 1 then a;q ) r z r q r n b;q b/a qb/az;q ) ) r b;q ) iii) If q bd/azq < 1 then a;q ) r z r q r n b;q bq/az qz;q ) ) r q 3 /az aqz;q ) n Proof. Let d 0 in 56) 57) and 58) respectively. n azq/ba;q ) r b a ) r. 8) 1 + azq 4r 1 )a azq/b;q ) r baz ) r q 4r 4r 1 + az/q)b zq;q ) r. 83) The following identities are a consequence of combining the results in Theorems 8 and 7. Corollary 7. If q < 1 then r r r q 4r q ;q q ;q ) ) r q 4r +4r q ;q q ;q ) ) r+1 q 8r + 1)q 8r +r q ;q ) 3 8r + 3)q 8r +6r q ;q ) 3 r q [ q;q ) 3 + q;q ) 3 ] 84) [ q;q ) 3 q;q ) 3 ] 85) [ q;q ) 3 + q;q ) 3 ] 86) [ q;q ) 3 q;q ) 3 ] 87) Proof. The identity at 84) follows upon setting a q b q and z 1 in 8) reversing the order of summation in the resulting series on the right replacing q with q and using 77) in conjunction with 79). The identity at 85) follows similarly except that z q and 78) is used in conjunction with 80). The identities at 86) and 87) follow similarly from 83). For 86) replace a with zq b with zq and tae the limits as z 1. For 87) replace z with zq set a zq b zq and again tae the limits as z 1. The details are omitted. The identities in Theorem 7 also contain implications for the limiting behaviour of each of the four eighth order moc theta functions that appear in these identities

24 4 James Mc Laughlin as q tends to certain classes of roots of unity from within the unit circle. We state these for S 0 q) and S 1 q) as those for T 0 q) and T 1 q) are equally easily derived. To avoid fractional exponents we state the results for q instead of q. Corollary 8. i) If ζ is a primitive even-order 8 root of unity then as q approaches ζ radially within the unit dis we have that lim q ζ S 0 q ) q ;q [ ) q;q ) 3 + q;q ) 3 ]) q ;q ) ζ 4 )1 + ζ 8 ) ζ 4n ) 1 + ζ )1 + ζ 6 ) ζ 4n+ ) ζ n +6n+4. 88) ii) If ζ is a primitive even-order 8 root of unity then as q approaches ζ radially within the unit dis we have that lim q ζ S 1 q ) q ;q [ ) q;q ) 3 q;q ) 3 ]) q q ;q ) ζ 4 )1 + ζ 8 ) ζ 4n ) 1 + ζ )1 + ζ 6 ) ζ 4n+ ) ζ n +n. 89) Finally as was done for sixth order moc theta functions we compare the results in the present paper for eighth order moc theta functions with those implied by Mortenson s identity at 69). In that identity if q is replaced q 8 x is set equal to q and the identity of Gordon and McIntosh [17 p. 15] is used to replace g qq 8 ) then S 0 q ) S 0 q ) j qq ) jq 6 q 16 ) jq q 8 ) q 8n+1 qq 7 ;q 8) n q 8 ;q 8 ) n qj 18 J 16 qg qq 8 ) 90) q 8nn+1) q q 7 ;q 8 ) n+1 qj J 163 J 18 J 10 J 16 J J 816 J 16 J 18 J 1016 J8 4J4 3 J 16J + 1 J ) 1016 J 8 As with sixth order moc theta functions when q tends to a primitive root of unity of even order the second series on the left side of 91) does not terminate so that the usual ind of explicit radial limit is not obtained. However when ζ is a primitive root of a certain order what is obtained is a convergent infinite series thus leading to another type of explicit radial limit. For example if + i ζ 8 e πi/8 16

25 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 5 a primitive eighth root of unity then it follows from 91) that [ lim S 0 q ) qj3 16 q ζ 8 J 816 J 16 ζ8 8n+1 J16 10 J 16 J8 4J4 3 J 16J + ]) J 1 J J 8 1 ζ 8 )1 ζ 8 ) ζ8 ζ8 7;q8) n ζ 8 8 ;ζ8 8 ) ζ 8 n ζ 8 Note how this compares with the radial limit given by 88): lim S 0 q ) q ζ 8 iq; q ) 3 + iq; q ) 3 q ) ; q ) q ; q ) ) n ) n 1 + i. 9) 1 + i. 93) While the limits are the same the two theta functions subtracted from S 0 q ) are not equal as functions of q. Here also as with the sixth order moc theta functions ψq) and λq) a radial limit is not obtained q tends to a primitive root of unity of odd order. What is true is that if ζ is a primitive root of unity of order + 1 then 1) n ζ n ζ ;ζ 4) n ζ 4 ;ζ 4 ) n lim q ζ ζ 8n+1 ζ ζ 7 ;ζ 8) n ζ 8 ;ζ 8 ) n J 1 J 616 qj3 16 J 8 J 816 J 16 J16 10 J 16 J 1016 J 4 8 J4 3 J 16 ) 94) and once again experiment seems to suggest that each side is identically zero when ζ is any primitive root of unity of odd order. As with the hypotheses suggested by experiment for moc theta functions of sixth order we have not attempted to prove these assertions. Results similar to those described above may be derived for other eighth order moc theta functions. References 1. Andrews G. E. On basic hypergeometric series moc theta functions and partitions. I. Quart. J. Math. Oxford Ser. ) ) Andrews G. E. On basic hypergeometric series moc theta functions and partitions. II. Quart. J. Math. Oxford Ser. ) )

26 6 James Mc Laughlin 3. Andrews G. E. Asey R. and Roy R. Special functions. Encyclopedia of Mathematics and its Applications 71. Cambridge University Press Cambridge 1999) xvi+664 pp. 4. Andrews G. E.; Berndt B. C. Ramanujans Lost Noteboo Part II Springer New Yor009) xii+418 pp. 5. Andrews G. E.; Hicerson D. Ramanujans lost noteboo: the sixth order moc theta functions Adv. Math ) Bailey W. N. On the basic bilateral hypergeometric series ψ. Quart. J. Math. Oxford Ser. ) ) Bajpai J.; Kimport S.; Liang J.; Ma D.; Ricci J. Bilateral series and Ramanujan s radial limits. Proc. Amer. Math. Soc ) Berndt B. C.; Chan S. H. Sixth order moc theta functions. Adv. Math ) no Berndt B.C.; Ranin R.A. Ramanujan: Letters and Commentary History of Mathematics 9 American Mathematical Society Providence RI London Mathematical Society London 1995) xiv+347 pp. 10. Choi Y-S. The basic bilateral hypergeometric series and the moc theta functions. Ramanujan J ) no Choi Y-S.; Kim B. Partition identities from third and sixth order moc theta functions. European J. Combin ) no Fine N. J. Basic hypergeometric series and applications. Mathematical Surveys and Monographs 7. American Mathematical Society Providence RI 1988) xvi+14 pp. 13. Folsom A.; Ono K.; Rhoades R. C. Moc theta functions and quantum modular forms. Forum Math. Pi 1 013) e 7 pp. 14. Gasper G. and Rahman M. Basic hypergeometric series. With a foreword by Richard Asey. Second edition. Encyclopedia of Mathematics and its Applications 96. Cambridge University Press Cambridge 004) xxvi+48 pp. 15. Gordon B.; McIntosh R. J. Some eighth order moc theta functions. J. London Math. Soc. ) 6 000) no Gordon B.; McIntosh R. J. Modular transformations of Ramanujan s fifth and seventh order moc theta functions. Ranin memorial issues. Ramanujan J ) no Gordon B.; McIntosh R. J. A survey of classical moc theta functions. Partitions q-series and modular forms Dev. Math. 3 01) Springer New Yor. 18. Hicerson D. R.; Mortenson E. T. Hece-type double sums Appell-Lerch sums and moc theta functions I. Proc. Lond. Math. Soc. 3) ) no Lovejoy J. On identities involving the sixth order moc theta functions. Proc. Amer. Math. Soc ) no Mc Laughlin J. Topics and Methods in q-series. - Forthcoming. 1. Mortenson E. T. On the dual nature of partial theta functions and Appell-Lerch sums. Adv. Math ) Mortenson E. T. Ramanujans radial limits and mixed moc modular bilateral q - hypergeometric series. Proceedings of the Edinburgh Mathematical Society to appear. 3. Ramanujan S.1988) The Lost Noteboo and Other Unpublished Papers. With an introduction by George E. Andrews. Springer-Verlag Berlin; Narosa Publishing House New Delhi 1988). xxviii+419 pp. 4. Ramanujan S. Collected papers of Srinivasa Ramanujan. Edited by G. H. Hardy P. V. Seshu Aiyar and B. M. Wilson. Third printing of the 197 original. With a new preface and commentary by Bruce C. Berndt. AMS Chelsea Publishing Providence RI 000). xxxviii+46 pp. 5. Rhoades R. C. On Ramanujan s definition of moc theta function. Proc. Natl. Acad. Sci. USA ) no Watson G. N. The final problem: An account of the moc theta functions J. London Math. Soc ) Watson G. N. The moc theta functions ) Proc. London Math. Soc ) Zudilin W. On three theorems of Folsom Ono and Rhoades. Proc. Amer. Math. Soc ) no

27 Moc Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series 7 9. Zwegers S. Moc θ-functions and real analytic modular forms q-series with applications to combinatorics number theory and physics. Contemp Math ) Zwegers S. Moc theta functions. PhD thesis 00) Univ of Utrecht Utrecht The Netherlands).