SELF-CONJUGATE VECTOR PARTITIONS AND THE PARITY OF THE SPT-FUNCTION

Transcription

1 SELF-CONJUGATE VECTOR PARTITIONS AND THE PARITY OF THE SPT-FUNCTION GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG Abstract Let sptn denote the total number of appearances of the smallest parts in all the partitions of n Recently we found new combinatorial interpretations of congruences for the spt-function modulo 5 and 7 These interpretations were in terms of a restricted set of weighted vector partitions which we call S-partitions We prove that the number of self-conjugate S-partitions counted with a certain weight is related to the coefficients of a certain mock theta function studied by the first author Dyson and Hickerson As a result we obtain an elementary q-series proof of Ono and Folsom s results for the parity of sptn A number of related generating function identities are also obtained Introduction Let sptn denote the total number of appearances of the smallest parts in the partitions of n The spt-function satisfies three simple congruences 2 3 spt5n mod 5 spt7n mod 7 spt3n mod 3 In a recent paper [8] we found new combinatorial interpretations of the congruences mod 5 and 7 in terms of what we called the spt-crank In this paper we study how the spt-crank is related to the parity of the spt-function Let P denote the set of partitions and D denote the set of partitions into distinct parts Following [2] the set of vector partitions V is defined by the cartesian product V D P P Date: December Mathematics Subject Classification 05A7 05A9 F33 P8 P82 P83 P84 33D5 Key words and phrases spt-function partitions rank crank vector partitions Ramanujan s Lost Notebook congruences basic hypergeometric series mock theta functions The first author was supported in part by NSA Grant H The second author was supported in part by NSA Grant H The third author was supported by the Summer Research Experience for Rising Seniors SRRS program of the University of Florida with funding from the Howard Hughes Medical Institute the Science for Life Program

2 2 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG We call the elements of V vector partitions In [2] new combinatorial interpretations of Ramanujan s partition congruences mod 5 7 and were given in terms of these vector partitions The combinatorial interpretation of the congruences 2 is similar It is in terms of a subset of V S : { π π π 2 π 3 V : sπ < and sπ minsπ 2 sπ 3 } Here sπ as the smallest part in the partition with the convention that s for the empty partition We call the vector partitions in S simply S-partitions For π π π 2 π 3 S we define the weight ω π #π the crank π #π 2 #π 3 and π π + π 2 + π 3 where π j is the sum of the parts of π j and #π j denotes the number of parts of π j The number of S-partitions of n in S with crank m counted according to the weight ω is denoted by N S m n so that 4 N S m n ω π We see that 5 Letting z gives 6 Sz q : π S π n crank πm N S m nz m q n m π S π n ω π m q n q n+ ; q zq n ; q z q n ; q N S m n sptn The number of S-partitions of n with crank congruent to m modulo t counted according to the weight ω is denoted by N S m t n so that 7 N S m t n N S kt + m n ω π k π S π n crank π m mod t The following theorem was our main result in [8] and contains the combinatorial interpretations of 2 Theorem 8 spt5n + 4 N S k 5 5n spt7n + 5 N S k 7 7n for 0 k 4 for 0 k 6

3 The map ι : S S given by PARITY OF THE SPT-FUNCTION 3 ι π ιπ π 2 π 3 ιπ π 3 π 2 is a natural involution An S-partition π π π 2 π 3 is a fixed-point of ι if and only if π 2 π 3 We call these fixed-points self-conjugate S-partitions The number of self-conjugate S-partitions counted according to the weight ω is denoted by N SC n so that 0 N SC n ω π π S π n ι π π Since ι is an an involution that preserves the weight ω we have N SC n sptn mod 2 for all n by 6 A standard argument and some calculation gives 2 SCq : N SC nq n q n qn+ ; q q 2n ; q 2 q; q In Section 2 we prove the following theorem Theorem q; q q n q; q n q n q 2 ; q 2 n q 2n q n q n2 q; q 2 n q n q; q n q n The function on the right side of 4 is a mock theta function studied by the first author Dyson and Hickerson [6] In [6] the arithmetic of the coefficients of the two mock theta functions 5 6 σq σ q Snq n S nq n q nn+/2 q; q n n q n2 q; q 2 n was studied The coefficients Sn and S n are determined by the prime factorization of 24n + and 24n respectively and are connected with the arithmetic of the field Q 6 By 4 and 6 we have 7 8 N SC n S n sptn S n mod 2

4 4 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG By combining this with results of [6] we obtain our main result on self-conjugate S-partitions and the parity of the spt-function Theorem 3 We have the following i N SC n 0 if and only if p e 24n for some prime p ± mod 24 and some odd integer e ii sptn is odd and if and only if 24n p 4a+ m 2 for some prime p 23 mod 24 and some integers a m where p m Remark 4 In ii above we have corrected a statement given by Folsom and Ono [ Theorem 2] on the parity of sptn The details of the proof and discussion of Folsom and Ono s results will be given in Section 2 We note that the proofs of Theorems 3 and 5 in [6] involve only elementary results of arithmetic on Q 6 together with the method of Bailey chains This together with the proof of Theorem 2 constitute an elementary q-series proof of the spt-parity result Theorem 3ii Folsom and Ono s spt-parity result depends on the theory of weak Maas forms and some heavy calculation with modular forms In Section 2 we will also connect the value of sptn mod 4 with another mock theta function Theorem 5 Let Then Ψq q n2 q; q 2 n ψnq n 9 sptn n ψn mod 4 In Section 3 we obtain some results that we discovered in the process of trying to prove Theorem 2 These results include a number of sums of tails identities and generating function identities for the spt-crank and self-conjugate S-partitions 2 Self-conjugate S-partitions the parity of the spt-function and mock theta functions 2 Proof of Theorem 2 q n q; q n q n q 2 ; q 2 n q 2 ; q 2 q n q 2 ; q 2 q n q n q n q n q 2n ; q 2 q n k0 by [4 p9225] q 2nk q 2 ; q 2 k

5 PARITY OF THE SPT-FUNCTION 5 k0 q 2 ; q 2 k q 2 ; q 2 k k0 q n2k+ q n q 2k+ ; q again by [4 p9225] By multiplying by q we have q q n q; q n q q 2 ; q 2 q n q 2 ; q 2 2k q k which simplifies to 3 To prove 4 we need some results from [9] By Theorem of [9] with q q 2 a 0 t q we see that 2 q; q 2 q; q 2 n k0 n q n2 q; q 2 n By Theorem 2 of [9] with q q 2 a b c 0 22 q 2 ; q 2 q 2 ; q 2 n q 2 ; q 2 Hence + q; q 2 q 2n q 2n q 2n q 2n q; q q 2 ; q 2 2n q; q n q; q 2 n q; q 2 + q; q 2 q; q q 2 ; q 2 n q; q 2 n q; q 2 + q; q q 2 ; q 2 q 2 ; q 2 n n q n2 q; q 2 q 2n q; q 2 n q + q; q q 2n 2n q 2 ; q 2 q 2n n q n2 q; q 2 n by 2 and Combinatorial interpretation of Theorem 2 We give a combinatorial interpretation of part of Theorem 2 Definition 2 Let B e n resp B o n denote the number of partitions of n with an odd number of smallest parts and a total number of even resp odd parts

6 6 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG Definition 22 Consider partitions into odd parts without gaps ie if k occurs as a part all the positive odd numbers less than k also occur For j or 3 let C j n denote the number of such partitions of n in which the largest part is congruent to j mod 4 Corollary 23 B e n B o n C 3 n C n Proof B e n B o n q n q n q 3n q 5n q n+ ; q q n q 2n q n+ ; q q; q n q n2 by Theorem 2 q; q 2 n C 3 n C n q n q n q; q n q n as observed on [6 p404] Example 24 n 7 smallest parts Below we list the partitions of 7 with an odd number of π #π We see that B e 7 6 and B o 7 5 There are three partitions of 7 in odd parts with no gaps:

7 PARITY OF THE SPT-FUNCTION 7 π largest part Hence C and C 7 Thus B e 7 B o C 3 7 C 7 23 Proof of Theorem 3 First we need some results from [6] For m mod 24 let T m denote the number of inequivalent solutions of u 2 6v 2 m with u + 3v ± mod 2 minus the number with u + 3v ±5 mod 2 Theorem 25 [6] Sn T 24n + for n 0 and 2S n T 24n for n For any integer m positive or negative satisfying m mod 6 and m let m p e p e 2 2 p e r r be the prime factorisation where each p i mod 6 or p i is the negative of a prime 5 mod 6 Then we have Theorem 26 [6] 25 T m T p e T p e 2 2 T p er r where 0 if p mod 24 and e is odd if p 3 9 mod 24 and e is even 26 T p e e/2 if p 7 mod 24 and e is even e + if p mod 24 and T p 2 e e + if p mod 24 and T p 2 We are now ready to complete the proof of Theorem 3 First we write the prime factorisation 27 24n p a p a 2 2 p a r r q b q b 2 2 q b s s where each p j 5 mod 6 and q j mod 6 so that and 24n p a p 2 a2 p r a r q b q b 2 2 q b s s 28 a + a a r mod 2

8 8 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG From 25 we have 29 T 24n T p a T p 2 a 2 T p r ar T q b By 2 Theorem 2 6 and 24 we have T q b 2 2 T q b s s 20 N SC n S n T 24n 2 Part i of Theorem 3 now follows immediately from Theorem 26 and 29 We prove part ii We observe from 8 24 and 20 that T 24n is even and 2 sptn T 24n mod 2 2 Now suppose sptn is odd so that T 24n 0 From 28 we see that at least one of the a j is odd say a Since T 24n 0 we deduce that p 23 mod 24 and the factor T p a ±a + is even If j a j is odd and p j 23 mod 24 then the factor T p j a j would also be even and and 2 would imply that sptn is even which is a contradiction Therefore each a j is even for j Similarly each b j is even Hence each exponent in the factorisation 27 is even except a So 2 T p a ± 2 a + mod 2 a mod 4 and 24n p 4a+ m 2 where p 23 mod 24 is prime and m p Conversely if 24n p 4a+ m 2 where p 23 mod 24 is prime and m p then it easily follows that 2 T 24n is odd and sptn is odd This completes the proof of Theorem 3 24 Examples and Folsom and Ono s results We illustrate Part i of Theorem 3 with an example Below is a table of the 6 self-conjugate S-partitions of 5 weight π π π π π π Thus 6 N SC 5 ω π j j

9 as predicted by the Theorem since PARITY OF THE SPT-FUNCTION In [] Folsom and Ono incorrectly stated that sptn is odd if and only if 24n pm 2 where p 23 mod 24 is prime and m is an integer We give some examples illustrating the difference between their statement and ours We also make some comments on their proof Example 27 n 507 Then 24n 23 3 Calculation gives spt which is clearly even as predicted by our theorem In fact if p 23 mod 24 is prime and n 24 p3 m 2 + where m 6p then 24n p 3 m 2 and sptn 0 mod 24 This congruence is a special case of [3 Theorem 3i] Example 28 n 2688 Then 24n 23 5 and spt mod 2 a number with 574 decimal digits as predicted by our theorem Again using [3 Theorem 3i] we have spt a+ + mod 8 We clarify Folsom and Ono s proof We let Lz Sz be defined as in equations and 4 of [] We proceed as in Section 4 of [] to obtain q 2n 2n+24m+ + q 2n 52n+24m+5 Lz n m 0 q 2n+2n+24m + q 2n+52n+24m 5 mod 2 + n m 0 From [ Lemma 4] we have q S24z sptnq 24n L24z mod 2 n so that sptn mod 2 d <d 2 d d 2 24n

10 0 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG and 22 sptn d24n mod 2 2 where dm is the number of positive divisors of m The spt-parity result Theorem 3ii follows in a straightforward manner 25 Proof of Theorem 5 We begin with some preliminary facts q n 2 + q n + 4 q n 23 2 q 2n 2 24 fq + 4Ψ q q; q 2 ϑ 4 0 q where fq is the third order mock theta function q n2 fq q; q 2 n ϑ 4 z q is the theta-function and 25 fq ϑ 4 z q q; q n n exp2πinz q n2 + 4 by [9 p36] n q n3n+/2 + q n by [9 p24] We restate Theorem 4 from [5] 26 sptnq n nq n q; q q + n q; q It is well known that 27 ϑ 4 0 q m0 m+ q 2m+ + q 2m+ See for example [3 Eqn 333 p462] By 25 we have 28 q; q n q n3n+/2 + q n 4q; q q; q fq n q n3n+/2 + q n q n 2 Ψ q + 4 q; q2 ϑ 4 0 q by 24 4q; q Ψ q + ϑ4 0 q 2 4q; q

11 PARITY OF THE SPT-FUNCTION Ψ q + q; q by 27 Therefore by 24 and 26 we have 29 sptnq n q; q q; q q; q Consequently as desired Ψ q + q; q q; q 4 q; q 2 nq n q + n q; q m0 by [4 p23222] m+ q 2m+ + q 2m+ n q n3n+/2 mod 4 + q n nq n q + Ψ q + n q; q q 4n+ q 4n+ Ψ q mod 4 m0 q 8n+2 q 2 8n+2 m+ q 2m+ by 27 + q 2m+ m0 q 4n q + 2 4n q 4m+ + q + 4m+ q 8n+2 Ψ q q8n+2 m q 4n+2 q 4n+2 q 4m + q 4m q 8n 2 q + 2 8n 2 sptn n ψn mod 4 3 Other Results q 4n+2 Ψ q q 4n+2 In this section we give some results that we discovered along the way in our quest to prove Theorem 2 and the following Theorem 3 [8] 3 N S m n 0 for all m n

12 2 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG For example before considering the result 3 for general m one might first consider the special case m 0 In Theorem 34 we express the generating function of N S 0 n in terms of a series involving tails of infinite products The theorem also contains some natural variations We first need to extend a result from [9] Proposition 32 Prop2 p403 [9] Suppose that f α z α nz n is analytic for z < If α is a complex number such that i α α n < + and then ii lim nα α n 0 n + lim z The extension we need is d dz zf αz α α n Lemma 33 Suppose f α z α nz n f β z β nz n and f αβ z α nβ n z n are analytic for z < And suppose that i ii hold for the each of the three sequences α n β n α n β n with corresponding limits α β and αβ Then Proof β n α n α lim z β n α n α α d dz z αf βz f αβ z αβ αβ n + α n β n αβ β β n The result follows easily from Proposition 32 αβ α n β n Theorem 34 We have 32 q q 2 2n q N S 0 nq n n 33 nq n2 q q 2 n q n q 2 n 34 n q n2 q q 2 ; q 2 2n q n q; q 2 n

13 35 PARITY OF THE SPT-FUNCTION 3 q n q n q Proof From 5 we have N S m nz m q n m q q + n2 qn q n q n q n q n q n+ ; q zq n ; q z q n ; q q q n q n zq n ; q z q n ; q q n q n k0 zq n k q k m0 z q n m q m by [4 p9225] Picking out the coefficient of z 0 we have N S 0 nq n q n q 2nk q q n q 2 k0 k q n2k+ q q 2 k0 k q n q + by [4 p9225] q 2 k0 k q 2k+ ; q q q 2 2k q k k0 which gives 32 To prove 33 we apply Lemma 33 with α n q; q n β n q; q 2 n α q; q β q; q 2 36 n q nn+/2 z; q n q; q n lim τ 0 2ϕ τ q z; q τ lim τ 0 z; q q; q t; q 2ϕ z n z; q q; q q; q 2 n 0 0 τ; q z q by [4 p24iii]

14 4 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG and we have the identity z n 39 q; q 2 n q; q z; q Thus we have q q 2 n q n q; q lim z lim z d z dz d lim z dz z; q q; q d dz z z n q; q 2 n z; q lim z n q nn+/2 z; q n q; q n d dz z n q nn+/2 z; q n q; q n n q nn+/2 z; q n q; q n n q nn+/2 q n because d 30 lim zf z F z dz when F z is analytic at z On the other hand nq n2 d z n q n2 q; q 2 n dz q; q 2 n z d τ dz lim τ 0 2 ϕ qτ ; q zτ 2 q z d dz lim qτ; q zτ; q z τ 2ϕ ; q qτ τ 0 q; q zτ 2 ; q zτ d + n z; q n q nn+/2 dz q; q q; q q; q n n q nn+/2 q; q q n z n q; q n d lim z dz zq; q by 39 and [4 p9225] z z by [4 p24iii2] again by 30 Thus both sides of equation 33 are equal to the same thing and therefore equal to each other

15 PARITY OF THE SPT-FUNCTION 5 Equation 34 is contained in Theorem 2 and is equation 4 The proof is given in Section 2 Finally we turn to 35 The identity 3 q n q n q is well known See for example [7 p463] q n q n m m0 q mn q nn+m + q n2 + qn q n mn which completes the proof of 35 q mn + m nm+ n m q mn q n q n q mn m0 q nn+m + m q mm+n Some Remarks on Theorem 34 As mentioned before we originally wanted to obtain identities for N S m n in order to approach the result 3 The first identity we obtained was 32 The series on the left side of 33 is a natual tweak To our suprise this series seemed to also have nonnnegative coefficients and the identity 33 was discovered emprically A quick search in Neil Sloane s On-line Encyclopedia of Integer Sequences [6] reveals that 32 q q 2 n q n nq n2 q 2 n q + n2 m n mmm nq n where Mm n is the number of partitions of n with crank m See sequence A5995 [8] It is clear that the left sides of equations 32 amd 34 are congruent mod 2 The right hand side of 34 was found empirically The coefficients of this series appear to grow very slowly and many of the coefficients are zero Such q-series are quite rare These properties led us to quickly identify this series with the special mock theta function σ q which was studied previously by the first author Dyson and Hickerson [6] We note that these coefficients also appear in Sloane s On-line Encylopedia See sequence A [7] It was only later we discovered the connection with self-conjugate S-partitions The FFW-Function The initial study of the spt-function [5] was inspired by a result of Fokkink Fokkink and Wang [0] Recall that D denotes the set of partitions

16 6 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG into distinct parts Define 33 FFWn : π D π n Fokkink Fokkink and Wang [0] proved that #π sπ 34 FFWn dn the number of positive divisors of n In [5] a q-series proof of this result was given using the identity 35 FFWnq n n q nn+/2 q n q n We extend the FFW-function and obtain similar expressions for the spt-function and spt-crank generating functions We define 36 FFWz n : π D π n so that Theorem #π + z + + z sπ FFW n FFWn FFWz nq n n q nn+/2 zq n q n q z zq z k q k q q k k0 Proof Given a partition into n distinct parts and smallest part k we may subtract k from the smallest part k + from the next smallest part k + n from the largest part to obtain an unrestricted partition into at most n parts This process can be reversed and we see that q nn /2 q nk q n is the generating function for partitions into n distinct parts with smallest part k Thus FFWz nq n

17 since Now PARITY OF THE SPT-FUNCTION 7 q n + + zq 2n z + + z k q kn + n q nn /2 n q nn+/2 zq n q n k z k x k z n q nn+/2 zq n q n z z x zx x n q nn+/2 z n q n zq n q zq arguing as on [5 p34] Lastly we show that 320 FFWz nq n z k q k+ ; q We see that the coefficient of z k q n in RHS320 is #π + z + + z sπ π D π n k+ sπ k0 q n which is also to coefficient of z k q n in LHS320 We note that right side of 320 coincides with the right side of 39 This completes the proof of Corollary FFW n k0 #π π D π n sπ odd k q k q k q j { 0 if n j 2 j if n j 2 j q j2 Proof Equations follow from setting z in Theorem 35 and using Gauss s result [4 p23] that q + 2 n q n2 q

18 8 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG Remark 37 The result 32 is due to Alladi [ Thm2] Equation 322 appears to be new Alladi [2] has found an extension of 32 that is a combinatorial interpretation of a partial theta-function identity [2 ] that appears in Ramanujan s Lost Notebook [5 p38] Theorem Sz q zq 324 sptnq n q 325 N SC nq n q n q nn+/2 q n z q n n n q nn+/2 q n q n n n q nn+/2 q n + q n z n z Proof In [4 p24iii2] we replace z by q and let a z b z and c 0 to obtain z n z n 326 q n z q n q nn+/2 z z n q n q z q n q n From 5 we have q n q n+ ; q Sz q zq n ; q z q n ; q q z n z n q n q z z q n z z n q nn+/2 z z n + q z z z q n q n z q From we have Hence Sz q z z zq q z q + n q nn+/2 z z q n q n n q nn+/2 z z n z q n q n z n z n q nn+/2 q n z q n n q nn+/2 z z q n q n

19 PARITY OF THE SPT-FUNCTION 9 which is 323 Equation 324 follows from 6 by letting z in 323 Before proving 325 we need to correct a result in [9] By Theorem 2 in [9] with a q b 0 and c q we have 327 q; q q; q n q nn+/2 q; q q; q n q n q nn+/2 + n n q; q q; q n q n 2 q 2n2 +3n+ q; q q; q 2n+ q 2n+ + q; q 2q q; q q 2 lim τ 0 3 ϕ 2 2q q 2 ; q 2 n q n + + q; q n q n2 2 + q; q 2 n q; q τ q q τ q; q 2 τ 2 q 3 q 3 q 3 + q n q n by q; q q n q n by [4 p24iii0] q n q n q n q n by [4 p29ex6] with x q 2 and y q We have corrected the proof of Case 6 in [9 pp ] From 327 we have 328 Now q; q q nn+/2 q; q n q n 2 n q n2 q; q 2 n 329 n z n q nn+/2 q; q n + q n τ 2 lim τ 0 2 ϕ q; q zτ q 2 lim τ; q qz; q z τ 2ϕ q; q τ τ 0 q; q zτ; q qz q; q q n q n by [4 p24iii2]

20 20 GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG qz; q + qz; q 2 q; q q; q z; q n q nn+/2 q; q n qz; q n After dividing both sides of 329 by q applying d and letting z we find dz that 330 q n n q nn+/2 q n + q n 2 q n q n2 q; q 2 n q n q + n 2 q by 328 The result 325 follows from 7 and 330 q nn+/2 q n q n Acknowledgements We would like to thank Ken Ono and Rob Rhoades for their comments and suggestions References K Alladi A partial theta identity of Ramanujan and its number-theoretic interpretation Ramanujan J URL: 2 K Alladi A combinatorial study and comparison of partial theta identities of Andrews and Ramanujan Ramanujan J URL: 3 G E Andrews Applications of basic hypergeometric functions SIAM Rev URL: 4 G E Andrews The Theory of Partitions Encycl Math Appl Vol 2 Addison-Wesley Reading Mass 976 Reissued: Cambridge Univ Press Cambridge G E Andrews The number of smallest parts in the partitions of n J Reine Angew Math URL: 6 G E Andrews F J Dyson and D Hickerson Partitions and indefinite quadratic forms Invent Math URL: 7 G E Andrews and P Freitas Extension of Abel s lemma with q-series implications Ramanujan J GE Andrews F G Garvan and J L Liang Combinatorial interpretations of congruences for the spt-function preprint URL: fgarvan/papers/spt-crankpdf 9 G E Andrews J Jiménez-Urroz and K Ono q-series identities and values of certain L- functions Duke Math J URL:

21 PARITY OF THE SPT-FUNCTION 2 0 R Fokkink W Fokkink and Z B Wang A relation between partitions and the number of divisors Amer Math Monthly URL: A Folsom and K Ono The spt-function of Andrews Proc Natl Acad Sci USA URL: ono/publications-cv/pdfs/pdf 2 F G Garvan New combinatorial interpretations of Ramanujan s partition congruences mod 5 7 and Trans Amer Math Soc URL: 3 F G Garvan Congruences for Andrews spt-function modulo and extension of Atkin s Hecke-type partition congruences preprint URL: fgarvan/papers/spt3pdf 4 G Gasper and M Rahman Basic Hypergeometric Series Encycl Math Appl Cambridge Univ Press Cambridge S Ramanujan The lost notebook and other unpublished papers Springer-Verlag Berlin N J A Sloane The On-Line Encyclopedia of Integer Sequences published electronically 200 URL: 7 N J A Sloane Sequence A in The On-Line Encyclopedia of Integer Sequences published electronically 200 URL: 8 N J A Sloane Sequence A5995 in The On-Line Encyclopedia of Integer Sequences published electronically 200 URL: 9 G N Watson The final problem: an account of the mock theta functions J Lond Math Soc II URL: Department of Mathematics The Pennsylvania State University University Park PA address: andrews@mathpsuedu Department of Mathematics University of Florida Gainesville FL address: fgarvan@ufledu Department of Mathematics University of Central Florida Orlando FL address: jiel@knightsucfedu