MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK

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1 MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK MARIA MONKS AND KEN ONO Abstract Let R(w; q) be Dyson s generating function for partition ranks For roots of unity ζ it is known that R(ζ; q) and R(ζ; /q) are given by harmonic Maass forms Eichler integrals and modular units We show that modular forms arise from G(w; q) the generating function for ranks of partitions into distinct parts in a similar way If D(w; q) : ( + w)g(w; q) + ( w)g( w; q) then for roots of unity ζ ± we show that q D(ζ; q)d(ζ ; q) is a weight modular form Although G(ζ; /q) is not well defined we show that it gives rise to natural sequences of q-series whose limits involve infinite products (and modular forms when ζ ) Our results follow from work of Fine on basic hypergeometric series Introduction and statement of results There are famous examples of q-series which essentially coincide with modular forms when q : e πiτ For example the celebrated Rogers-Ramanujan identities R (q) : R (q) : ( q 5n+ )( q 5n+4 ) + q n ( q)( q ) ( q n ) ( q 5n+ )( q 5n+3 ) + q n +n ( q)( q ) ( q n ) provide series expansions of infinite products which are essentially weight 0 modular forms As another example the partition generating function satisfies P (q) : p(n)q n q + q n n ( q) ( q ) ( q n ) Since q P (q 4 ) /η(4τ) where η(τ) : q /4 ( qn ) is Dedekind s weight / modular form this generating function is another instance of a modular form which is a basic hypergeometric type series Dyson s rank generating function provides two infinite families of basic hypergeometrictype series which are related to automorphic forms To make this precise we first recall 000 Mathematics Subject Classification 05A7 F The authors thank the National Science Foundation for their generous support and the second author thanks the Manasse Family and the Hilldale Foundation

2 MARIA MONKS AND KEN ONO some notation As usual we let { ( a)( aq) ( aq n ) if n () (a; q) n : if n 0 and () (a; q) : ( aq n ) Recall that the rank of a partition is its largest part minus its number of parts The generating function for N(m n) the number of partitions of n with rank m is R(w; q) : + N(m n)w m q n q n + (wq; q) n (w q; q) n The two families R(ζ; q) and R(ζ; /q) where ζ are roots of unity are intimately related to automorphic forms We clearly have that q R(; q 4 ) p(n)q 4n η(4τ) is a weight -/ modular form For ζ a very different phenomenon holds In this case Bringmann and the second author proved that R(ζ; q) is the holomorphic part of a weight / harmonic Maass form [4] For R(w; q) : R(w; /q) another phenomenon occurs A straightforward calculation shows that q (3) R(w; n q) + (wq; q) n (w q; q) n and then a classical identity of Ramanujan [] asserts that (4) R(w; q) ( w) ( ) n w 3n q 3n +n ( w q n+ ) + w ( )n w n q n +n (wq; q) (w q; q) For generic roots of unity w ζ the right hand side of (4) consists of two false theta series which can be thought of as Eichler integrals (for example see page 03 of [8]) and an infinite product which is essentially a modular form (see 7) Here we address the analogous questions for the series (5) G(w; q) n 0 m Z Q(m n)w m q n : + q n +n (wq; q) n This is the generating function for Q(m n) the number of partitions of n into distinct parts with rank m We relate the series G(ζ; q) and Ĝ(ζ; q) : G(ζ; /q) for roots of unity ζ to modular forms and other infinite products

3 MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK 3 The case of G(ζ; q) has been thoroughly investigated for ζ {± ±i} In particular we have that q η(τ) 4 G(; q) η(τ) is a weight 0 modular form For ζ ±i we have the two false theta functions [9] qg(±i; q 4 ) (±i) k q (6k+) + (±i) k q (6k ) k0 Although these two series are not modular they are Eichler integrals of weight 3/ modular forms The case of ζ is more exotic If we define a(n) by qg( ; q 4 ) a(n)qn then the function φ(τ) : y k a(n)e πinx K 0 (πny) is a Maass cusp form [ 5] where K 0 denotes the usual index zero K-Bessel function and τ x + iy with x y R For roots of unity ζ ± it turns out that G(ζ; q) G( ζ; q) G(ζ ; q) and G( ζ ; q) together give a factorization of a certain weight modular form For convenience we define the series (6) D(w; q) : ( + w)g(w; q) + ( w)g( w; q) For roots of unity ζ we require the weight 0 modular form (for example see [7 0]) (7) η(ζ; τ) : q ( ζq n )( ζ q n ) Theorem If ζ ± is a root of unity then we have that q D(ζ; q)d(ζ ; q) is the weight modular form q D(ζ; q)d(ζ η(τ) 4 ; q) 4 η(τ) η(ζ ; τ) Example Here we consider the case where ζ i Since D(i; q) D( i; q) Theorem implies that q 4 D(i; q) is the weight / modular form q 4 D(i; q) η(τ) 3 η(τ)η(4τ) q 4 + q 5 4 q 49 4 Now we turn to the series Ĝ(ζ; q) : G(ζ; /q) A straightforward calculation (see Lemma 3) reveals the formal identity ( w ) n (8) Ĝ(w; q) + (w q; q) n

4 4 MARIA MONKS AND KEN ONO Unfortunately this is not a well defined q-series By that we mean that if m ( w ) n (9) Ĝ m (w; q) : + (w q; q) n then lim m + Ĝ m (w; q) is not well defined However it turns out that if ζ is a root of unity of order m then for each 0 r < m we have that (0) lim Ĝ mn+r (ζ; q) n + is a well defined q-series The next theorem describes these limits and shows that in the case ζ it is the sum of a weight / modular form with a weight 0 modular form Theorem Suppose that ζ is an mth primitive root of unity If 0 r < m then lim Ĝ mn+r (ζ; q) is a well defined q-series and satisfies lim Ĝ mn+r (ζ; q) lim Ĝ mn (ζ; q) + ( ζ ) r ζ + (ζ q; q) Furthermore if we define the sequence b(n) by ( )n b(n)q n ( + qn ) then ( ) and lim Ĝ n (; q) lim Ĝ n+ (; q) ( b(n)q n + (q; q) ) b(n)q n (q; q) Remark Notice that ( ) n b(n)q n 4 q 4 ( + q n+ ) η(τ) η(τ)η(4τ) is a modular form of weight 0 It then follows that q /4 b(n)qn is also a weight 0 modular form Since η(z) is a weight / modular form it follows that q /4 lim Ĝ n (; q) and q /4 lim Ĝ n+ (; q) are both equal to a sum of a weight / modular form and a weight 0 modular form Example To illustrate Theorem in the case ζ we list the first few series in the two convergent sequences: Ĝ (; q) q q q 3 q 4 q 5 q 6 q 7 q 8 q 9 Ĝ 3 (; q) q q q 3 q 4 3q 5 4q 6 5q 7 6q 8 8q 9 Ĝ 5 (; q) q q q 3 q 4 4q 5 5q 6 7q 7 9q 8 3q 9 Ĝ 7 (; q) q q q 3 q 4 4q 5 5q 6 8q 7 0q 8 5q 9 Ĝ 9 (; q) q q q 3 q 4 4q 5 5q 6 8q 7 0q 8 6q 9

5 and MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK 5 Ĝ (; q) + q + q 3 + q 4 + q 5 + 3q 6 + 3q 7 + 4q 8 + 4q 9 + Ĝ 4 (; q) + q + q 3 + 3q 4 + 3q 5 + 5q 6 + 6q 7 + 9q 8 + 0q 9 + Ĝ 6 (; q) + q + q 3 + 3q 4 + 3q 5 + 6q 6 + 7q 7 + q 8 + 3q 9 + Ĝ 8 (; q) + q + q 3 + 3q 4 + 3q 5 + 6q 6 + 7q 7 + q 8 + 4q 9 + Ĝ 0 (; q) + q + q 3 + 3q 4 + 3q 5 + 6q 6 + 7q 7 + q 8 + 4q 9 + We conclude the introduction with the following open problem Problem Assume the notation and hypotheses in Theorem For integers m > determine whether the q-series lim Ĝ mn (ζ; q) n + appears naturally in the theory of automorphic forms In Section we recall some classical identities of Fine and we prove Theorem In Section 3 we prove Theorem Identities of Fine and the proof of Theorem First we recall classical work of Fine on basic hypergeometric series Fine s basic hypergeometric series Theorem follows from the combinatorial properties of Fine s basic hypergeometric function (aq; q) n () F (a b t) : + t n (bq; q) n The next lemma (see (6) (6) and (73) of [6]) gives important closed formulas for certain specializations of this series Lemma Assuming that the following series are well defined: () F (a 0 t) t ( at) n q n +n (tq; q) n (3) F (a t) (atq; q) (t; q) (4) (t; q) n q n (bq; q) n (q; q) n (t; q) (bq; q) (q; q) F (b/t 0 t) The next identity (see (7) of [6]) is the key observation which leads to Theorem

6 6 MARIA MONKS AND KEN ONO Lemma Assume that the series below are well defined Then we have that S F (a 0 t) (b/a; q) H (bq; q) (b/at; q) G (b/at) (b/a; q) n q n (b/at) (bq; q) n (bq/at; q) n F F (a b t) F + G HS Proof of Theorem In Lemma let t ζ and let a /ζ By () we have that S q n +n G(ζ; q) ζ (ζq; q) n ζ Now by letting b we have ( ζ; q) H + ζ ( ζq; q) (q; q) ( ; q) (q; q) ( q; q) G ( ζ; q) n q n (q; q) n ( q; q) n ( ζ q; q) n ζ n F F ( ζ ζ) (q; q) n By (3) we then have that F ζ ( q; q) (ζq; q) Combining these observations with Lemma we can solve for G(ζ; q) to obtain (5) G(ζ; q) + ζ (q; q) ( q; q) ζ (ζq; q) ( ζq; q) + ζ (q; q) ( q; q) ( ζ; q) n q n ( ζq; q) (q; q) n ( q; q) n By (4) where t ζ and b followed by () we find that ( ζ; q) n q n ( ζ; q) F (ζ 0 ζ) (q; q) n ( q; q) n ( q; q) (q; q) ( ζq; q) (q; q) ( q; q) q n +n ( ζq; q) n ( ζq; q) (q; q) ( q; q) G( ζ; q)

7 MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK 7 Plugging this into (5) one easily finds that D(ζ; q) ( + ζ)g(ζ; q) + ( ζ)g( ζ; q) ( q; q) (q; q) (ζ q ; q ) Using (7) and the definition of η(τ) the proof follows easily by multiplying out D(ζ; q)d(ζ ; q) 3 Proof of Theorem To prove Theorem we first prove the following two lemmas Lemma 3 We have that (3) Ĝ(w; q) + Proof We have ( w ) n (w q; q) n as desired Ĝ(w; q) Ĝ(w; /q) + q n(n+)/ ( w/q)( w/q ) ( w/q n ) w q w q + q n(n+)/ ( w q) ( w q ) w q n ( w q n ) ( w ) n + (w q; q) n Lemma 3 Let ζ be an mth root of unity and let k be a positive integer For any positive integer n > k and any positive integer r the coefficient of q k in Ĝmn+r(ζ q) is the same as the coefficient of q k in Ĝ mn (ζ; q) + ( ζ ) r ζ + (ζ q; q) Proof Given any q-series F (q) the coefficient of q k in F (q) is the same as that in F (q) whenever t does not divide k In particular for n > k the coefficient of q k in q t (ζ q;q) n is the same as that in (ζ q;q) Thus for n > k the coefficient of q k in Ĝ mn+r (ζ q) + mn+r t ( ζ ) t (ζ q; q) t Ĝmn + mn+r tmn+ ( ζ ) t (ζ q; q) t

8 8 MARIA MONKS AND KEN ONO is the same as the coefficient of q k in Ĝ mn (ζ q) + and the claim follows mn+r tmn+ ( ζ ) t (ζ q; q) Ĝmn(ζ q) + (ζ q; q) Ĝmn(ζ q) + ( ζ ) r ζ + r ( ζ ) t t (ζ q; q) 3 Proof of Theorem Setting r m in Lemma 3 it follows that for n > k the coefficient of q k in Ĝm(n+)(ζ; q) is the same as that of Ĝ mn (ζ; q) Thus using Lemma 3 repeatedly we see that the coefficient of q k in Ĝm(n+a)+r(ζ; q) is the same as that of Ĝmn+r(ζ; q) for any positive integers a and r Therefore the coefficient of each power of q is eventually constant in the series Ĝmn+r(ζ; q) and so the functions lim Ĝ mn+r (ζ; q) are well-defined q-series Taking the limit as n on both sides of Lemma 3 it follows that (3) lim Ĝ mn+r (ζ; q) lim Ĝ mn (ζ; q) + ( ζ ) r ζ + (ζ q; q) We now examine the case ζ Notice that by (3) we have (33) lim Ĝ n+ (; q) lim Ĝ n (; q) k (q; q) Now define p k (n) to be the number of partitions of n whose largest part is at most k and define p(n k) to be the number of partitions of n whose largest part is equal to k It is clear by inspection that the coefficient of q t in (q;q) k is p k (t) for any positive integers t and k Let c(t) be the coefficient of q t in lim Ĝ n+ (; q) + lim Ĝ n (; q) Suppose t is even Then c(t) equal to the coefficient of q t in t Ĝ t+ (; q) + Ĝt(; ( ) k q) + + (q; q) k (q; q) t+ so c(t) (p 0 (t) p (t) + p t (t) + p t (t)) p t+ (t) since t is even by assumption Let Even(t) denote the number of partitions of t whose largest part is even and let Odd(t) denote the number of partitions of t whose largest part is odd Notice that Even(t) + Odd(t) p(t) p t (t) p t+ (t) p(t) and p k (t) p(t k) + p(t k ) + + p(t ) Therefore c(t) ((p (t) p 0 (t)) + (p 3 (t) p (t)) + + (p t (t) p t (t))) + p(t) (p(t ) + p(t 3) + p(t 5) + + p(t t )) + p(t) Odd(t) + (Even(t) + Odd(t)) Even(t) Odd(t)

9 MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK 9 Now suppose t is an odd integer In this case c(t) (p 0 (t) p (t) + + p t (t) p t (t)) + p t+ (t) ((p (t) p (t)) + + (p t (t) p t (t))) p(t) (p(t ) + p(t 4) + + p(t t )) p(t) Even(t) (Even(t) + Odd(t)) Even(t) Odd(t) Thus c(t) Even(t) Odd(t) for any positive integer t Recall that b(n) is defined by ( )n b(n)q n ( + qn ) so b(n) is equal to ( ) n times the number of partitions of n into distinct odd parts A bijection of Sylvester (cf [3] p 8) shows that b(n) is equal to the difference between the number of partitions of n having an even number of parts and the number of partitions of n having an odd number of parts By taking the conjugates of the partitions of n (interchanging the rows and columns of their Young diagrams) it follows that b(n) Even(n) Odd(n) c(n) for all n Therefore (34) lim Ĝ n+ (; q) + lim Ĝ n (; q) Combining (33) and (34) the conclusion follows References b(n)q n [] G E Andrews An introduction to Ramanujan s lost notebook Ramanujan: essays and surveys Hist Math Amer Math Soc Providence RI 00 pages [] G E Andrews F Dyson and D Hickerson Partitions and indefinite quadratic forms Invent Math 9 (988) pages [3] G E Andrews and K Eriksson Integer Partitions nd Ed Cambridge University Press 004 [4] K Bringmann and K Ono Dyson s ranks and Maass forms Ann of Math to appear [5] H Cohen q-identities for Maass waveforms Invent Math 9 (988) pages [6] N J Fine Basic hypergeometric series and applications Math Surveys and Monographs Vol 7 Amer Math Soc Providence 988 [7] S Lang Elliptic functions Springer-Verlag New York 987 [8] R Lawrence and D Zagier Modular forms and quantum invariants of 3-manifolds Asian J Math 3 (999) pages [9] M Monks Number theoretic properties of generating functions related to Dyson s rank for partitions into distinct parts preprint [0] B Schoeneberg Elliptic modular functions: an introduction Springer-Verlag New York Massachusetts Avenue Cambridge MA 039 address: monks@mitedu Department of Mathematics University of Wisconsin Madison Wisconsin address: ono@mathwiscedu