Eulerian series in q-series and modular forms. Youn Seo Choi

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1 Eulerian series in q-series and modular forms Youn Seo Choi abstrat Eulerian series is very interesting power series even though we do not know any single method to handle the general Eulerian series However, some of Eulerian series are deeply related to q-series, espeially the theory of partitions And some of them are Ramanujan s mok theta funtions S Ramanujan introdued the mok theta funtion in his last letter to G H Hardy Reently, S Zwegers in his thesis shows how to understand the mok theta funtions through the theory of modular forms Later, K Ono, K Bringmann and other mathematiians improved Zwegers idea Still the Bailey s Lemma and the onstant term method are the key parts in the beginning of the most reent understanding of Ramanujan s mok theta funtions We are going to disuss some interesting Eulerian series in this talk In his last letter [3] to G H Hardy, S Ramanujan disussed the Eulerian series Let q e πiτ where τ the upper-half of the omplex plane The following are alled Eulerian series; + q q q m + + q + q + q m The most famous Eulerian series are the Rogers-Ramanujan funtions + + q q q m m0 +m q q q m m0 q 5m+ q 5m+4 q 5m+ q 5m+3 We an understand with the theory of the partition funtion First define that pn : the number of non-inreasing sequenes of natural numbers whose sum is n Sine , we know that p4 5 Combinatorially, we see that pmq m q m + m0 Define Ramanujan s general theta-funtion fa, b by fa, b : a nn+/ b nn /, n q q q m where ab < Note that fa, b a; ab b; ab ab; ab, where ; d n0 dn This is alled the Jaobi triple produt identity Thus, we have f q, q q; q, q; q, ητ :q 4 q; q q 4 f q, q,

2 st R R funtion f q, q 3, nd R R funtion f q, q4 q; q q; q The series and satisfy the following asymptoti behavior When q e t and t 0, t π π exp 6t t + O 4 Similar results at other singularities When q e t and t 0, π π t exp 4t t + O 4 Similar results at other singularities We already know that is the theta funtion However, + q; q q ; q f q, q O at q,q 3,q 5,, q; q q ; q f q, q O at q,q 4,q 6,, O at q,q 3,q 5, Now we an define the mok theta funtion Definition A mok theta funtion is a funtion fq defined by a q-series whih onverges for q < and whih satisfies the following two onditions: 0 For every root of unity ζ, there is a theta funtion θ ζ q suh that the differene fq θ ζ q is bounded as q ζ radially There is no single theta funtion whih works for all ζ: ie, for every theta funtion θq there is some root of unity ζ for whih fq θq is unbounded as q ζ radially Ramanujan alled that is a 3rd order mok theta funtion In Page 9 of Ramanujan s Lost Notebook [0], we an find the four mok theta funtions: φq : Xq : n0 n0 q nn+/ q; q n+, ψq : n q n q; q n, χq : n0 n0 q n+n+/ q; q n+, n q n+ q; q n+ With these funtions, that page ontains eight identities satisfied by these mok theta funtions Basi tools for proving those identities [5, 6, 7, 9] are Bailey s lemma and onstant term method the Jaobi triple produt identity fa, b n ann+/ b nn / and Euler s pentagonal number theorem f q, q q; q transformation formulas for theta funtions transformation formulas for the generalized Lambert series theory of Mordell s integral theory of modular forms Now we disuss Bailey s lemma and onstant term method

3 3 Lemma Bailey s Lemma If for r 0 the sequenes α r and β r are related by β r r α n n0, then for r 0, β r r α n n0, q; q r n aq; q r+n q; q r n aq; q r+n where for any given numbers ρ and ρ, α n ρ ; q n ρ ; q n aq ρ ρ n α n aq ρ ; q n aq, ρ ; q n β r aq ρ ; q r aq ρ ; q r r j0 ρ ; q j ρ ; q j aq ρ ρ ; q r j aq ρ ρ j β j q; q r j Note that a pair of sequenes α n,β n is alled a Bailey pair Let ρ, ρ ρ and a in Bailey s Lemma And let r and ρ tend to Then, from Bailey s Lemma, we see that Let β n q ; q q; q n q nn+/ β n q; q n0 q n ; q n,ifn>0, β 0 0, n α n q 3n n q 4n q j j, j0 n0 q nn+/ α n +q n α n+ q 3n +n q 4n+ q j j n Then, α n, β n form a Bailey pair [] With this Bailey pair, we an prove that ψq sgr r+s+ q r+s +rs+3r+s+ f q, q r,s,sgrsgs Similarly, we have the following Heke type series for φq, Xq and χq: φq sgr r+s q r+s +rs+r+s f q, q Xq χq fq, q 3 q fq, q 3 sgrsgs sgrsgs sgrsgs sgrq r+s +rs+r+s sgrq r+s +rs 3r+s The idea in S Zwegers thesis [] an be applied to these Heke type series In his thesis, he showed that a single vetor-valued mok theta funtion is a ombination of a vetor-valued real-analyti modular form of weight / and a vetor-valued non-holomorphi theta series Then, define Dq, z : z q; q f z, z qf z, z q q; q f z, z q ψq equals the oeffiient of z in the Laurent series expansion of Dq, z Additionally, φq is the oeffiient of z in the Laurent series expansion of Dq, z There are two different representations for Dq, z The first representation is given in terms of theta funtions, generalized Lambert series, and φq and ψq The seond in represented solely in terms of theta funtions Upon equating the oeffiients in the

4 4 two representations for Dq, z, we are able to derive two identities for φq and ψq whih are ψqf z, q0 z q Q z q,q,q5 +a qf z, q0 z +qa z q,q,q5 φqf z, q0 z q Q z q 4,q,q 5 +a qf z, q0 z +qa z q 4,q,q 5, where n q nn+ x n+ z n+ Qz, x, q : q n+, z n Az, x, q : q; q q ; q f zx, z x q f q, qf x, x qf zq, z q, a q : qq5 ; q 5 q 0 ; q 0 f q, q 8 f q, q 4 f q 4, q 6, a q : q5 ; q 5 q 0 ; q 0 f q 4, q 6 f q, q 3 f q, q 8 From the equations above, we are able to derive φq q5 ; q 5 q 0 ; q 0 f q 4, q 6 f q, q 3 f q, q 8 + q f q 5, q 5 n n q 5nn+ q 5n+, ψq qq5 ; q 5 q 0 ; q 0 f q, q 8 q n q 5nn+ f q, q 4 f q 4, q 6 + f q 5, q 5 q 5n+ n Similar, we also derive representations for other mok theta funtions Xq q5 ; q 5 q 0 ; q 0 f q, q 3 q 0nn+/ f q, q 8 f q, q 4 + fq 5,q 5 q 0n+ n χq q q5 ; q 5 q 0 ; q 0 f q, q 4 q 0nn+/ f q 4, q 6 f q, q 3 + fq 5,q 5 q 0n+4 n The right hand side of eah equation above ontains the generalized Lambert series Those series an be transformed to Eulerian series by n q nn+ q; q n q nn+/ f q, q xq n x; q n+ x q; q n+ n fq /,q 3/ n n0 q nn+/ xq n n0 n q / ; q n q n / x; q n+ x q; q n We disuss more about Eulerian series in Ramanujan s lost notebook In the page 8 of RLN, we find two equations for Φq and ψq where q 5n q 5n Φq : q; q 5 n0 n+ q 4 ; q 5, Ψq : n q ; q 5 n0 n+ q 3 ; q 5 n These appear in the mok theta onjetures [] whih are f 0 q q5 ; q 5 q 5 ; q 0 q; q 5 q 4 ; q 5 Φq, f q q5 ; q 5 q 5 ; q 0 q ; q 5 q 3 ; q 5 q Ψq,

5 5 where fifth order mok theta funtions f 0 q and f q are q n q n +n f 0 q : and f q : q; q n0 n q; q n0 n Also, we are able to fine two equations in the page 8 of RLN; q 5 ; q 5 q Ψq3 + q 5 ; q 5 q; q 5 q 4 ; q 5 q 5n q 5n q; q 5 n+ q 4 ; q 5 + n q 4 ; q 5 n+ q ; q 5 n and Φq 3 + n0 n0 n0 q 5n q ; q 5 n+ q 3 ; q 5 q 5n n q q 7 ; q 5 n0 n+ q 8 ; q 5 n q 5 ; q 5 q 5 ; q 5 q ; q 5 q 3 ; q 5 There are the generalization [8] of those equations Theorem For a omplex number q with q <, and x neither 0 nor an integral power of q, q; q q 3n q 3 ; q 3 x; q x q; q x; q 3 n0 n+ x q 3 ; q 3 n x q 3n q 3n xq; q 3 n+ x q ; q 3 + n x q; q 3 n+ xq ; q 3 n n0 n0 There are similar results for tenth order mok theta funtions Theorem q ; q f q, qx ; q x q ; q q; q n q nn+/ q; q n q nn+/ x; q n+ x + q; q n+ x; q n+ x q; q n+ n0 n0 This equation with q q and x q appears in the paper [4] Theorem 3 q ; q fq,q 6 x; q x q ; q n q ; q 4 n q n x; q 4 n+ x q 4 ; q 4 x n q ; q 4 n q n n xq ; q 4 n+ x q ; q 4 n n0 The idea in [4] an be applied to those generalized Lambert series If z x + iy with x, y R, then the weight k hyperboli Laplaian is given by Δ k : y x + y + iky x + i y If v is odd, then define ɛ v by ɛ v : { if v mod 4, i if v 3 mod 4 n0

6 6 Definition A weak Maass form of weight k on a subgroup Γ Γ 0 4 is any smooth funtion f : H C satisfying the following: a b For all A Γ and all z H, we have faz k d d ɛ k d z + d k fz We have that Δ k f 0 3 The funtion fz has at most linear exponential growth at all the usps of Γ, ie, there is a C>0suh that for any usp s Q {} of Γ and γ Γ with γ s, the funtion fz satisfies fz Oe Cv as v uniformly in u, where z u + iv Then, we have the following results Theorem 4 q 96 n0 q n0 q 400 ; q 400 n q 00nn+ q 60 ; q 400 n+ q 40 ; q 400 n+, q 64 n q 00 ; q 400 n q 00n q 80 ; q 400 n+ q 30 ; q 400 n, q 9 n0 n0 q 400 ; q 400 n q 00nn+ q 80 ; q 400 n+ q 30 ; q 400 n+, n q 00 ; q 400 n q 00n q 60 ; q 400 n+ q 40 ; q 400 n are the holomorphi parts of ertain weak Maass forms of weight on Γ 00, where 0 Γ l, 0 8l For0 bd Theorem 5 Let l 8 gd4,bgd4,d a<b, d 4, 4 and d 0, q l d 4 n q l ; q l n q ln ζ a n0 b ql d ; q l n+ ζb aql d ; q l n is the holomorphi part of a weak Maass form of weight on Γ l, and for d 4, 3 4 and d, q l d n q l ; q l 4 n q ln ζb aql d ; q l n+ ζb aql d ; q l n n0 is the holomorphi part of a weak Maass form of weight on Γ l Theorem 6 Let l 8 q l d 4 bd gd4,bgd4,d n0 For0 a<b, d 4, 4 and d 0, n q l ; q l n q ln ζ a n0 b ql d ; q l n+ ζb aql d ; q l n q l d + n q l ; q l 4 n q ln ζb aql d +l ; q l n+ ζb aql d ; q l n is a weakly holomorphi modular form of weight on Γ l Referenes [] G E Andrews: The fifth and seventh order mok theta funtions, Trans of the AMS 93 no 986, 3 34 [] G E Andrews and F Garvan: Ramanujan s Lost Notebook VI:The mok theta onjetures, Adv in Math , 4 55

7 7 [3] B C Berndt and R A Rankin: Ramanujan: Letters and Commentary, Amer Math So, Providene, 995, London Math So, London, 995 [4] K Bringmann, K Ono, and R C Rhoades: Eulerian series as Modular forms, J of the AMS, 008, pages [5] Y-S Choi: Tenth order mok theta funtions in Ramanujan s Lost Notebook Invent Math , [6] Y-S Choi: Tenth order mok theta funtions in Ramanujan s Lost Notebook II Adv Math , [7] Y-S Choi: Tenth order mok theta funtions in Ramanujan s Lost Notebook IV Trans A M S , [8] Y-S Choi: Generalization of two identities in Ramanujan s lost notebook, Ata Arith no 4, [9] Y-S Choi: Tenth order mok theta funtions in Ramanujan s Lost Notebook III Pro Lond Math So , 6 5 [0] S Ramanujan: The Lost Notebook and Other Unpublished papers, Narosa Publishing House, New Delhi, 988 [] S Zwegers: Mok Theta Funtions, Ph D Thesis Korea Institute for Advaned Study, Cheongnyangni -dong, Dongdaemun-gu, Seoul, 30-7, Korea address: y-hoi@kiasrekr