CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit and Yee introduced partition functions associated with the Ramanujan/Watson mock theta functions ωq) and νq). In this paper, we study arithmetic properties of the partition functions. Based on one of the results of Andrews, Dixit and Yee, mod 2 congruences are obtained. In addition, infinite families of mod 4 and mod 8 congruences are presented. Lastly, an elementary proof of the first explicit examples of congruences for ωq) given by Waldherr is presented.. Introduction In his last letter to Hardy in 920, Ramanujan introduced the notion of a mock theta function along with a number of examples of order, 5, and 7. Since then, mock theta functions have been the subject of intense study. Recently, the first and fourth authors with A. Dixit found a new partition function p ω n) that is associated with the third order mock theta function ωq) []: where ωq) = q 2n2 +n) q; q 2 ) 2, n+ a; q) n := a) aq) aq n ). One of the results in [] yields a mod 2 congruence of the coefficients of ωq), which led us to a further search for congruences. S. Garthwaite and D. Penniston [4] showed that the coefficients of ωq) satisfy infinitely many congruences of a similar type to Ramanujan s partition congruences, and M. Waldherr [5] found the first explicit examples of congruences, suggested by some computations done by J. Lovejoy. We define p ω n) by p ω n)q n = q n q n )q n+ ; q) n q 2n+2 ; q 2 ), where a; q) := lim n a; q) n. From its generating function definition, we see that p ω n) counts the number of partitions of n in which each odd part is less than twice the smallest part. In [], it is shown that p ω n)q n = qωq). The main result of this paper is: Theorem.. For nonnegative integers n and k, p w 2 2k+ n + ) 22k + p w 2 2k+ n + 7 ) 22k + Keywords: partition congruences, mock theta functions 2 200 AMS Classification Numbers: Primary, P8, P8 0 mod 4), 0 mod 8),
2 GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE p w 2 2k+4 n + 8 ) 22k + 0 mod 4). In [], another partition function p ν n) is defined. Namely, p ν n)q n = q n q n+ ; q) n q 2n+2 ; q). It is shown that p ν n)q n = ν q), where νq) is a third order mock theta function, q nn+) νq) = q; q 2. ) n+ This mock theta function νq) is related to ωq) as follows [, p. 62, Equation 26.88)]: where ν q) = qωq 2 ) + q 2 ; q 2 ) ψq 2 ), ) ψq) = q nn+)/2. By ), we can derive congruences of p ν n) from p ω n), which will be given in Section 4. This paper is organized as follows. In Section 2, we present mod 2 congruences of p ω n) and p ν n). In Section, we prove Theorem.. In Section 4, we prove congruences of p ν n). As noted earlier, Waldherr [5] provided the first explicit congruences for ωq): p ω 40n + 28) 0 mod 5), p ω 40n + 6) 0 mod 5). In Section 5, we provide an elementary proof of the above congruences. and 2. Mod 2 congruences We recall the following results from []: q n q n ; q) n+ q 2n+2 ; q 2 = ) j q 6j2 +4j+ + q 4j+2 ), 2) ) j=0 q n q n+ ; q) n q 2n+2 ; q 2 ) = ) j q jj+2) + q 2j+ ). ) Let p ω,o n) and p ω,e n) be the number of partitions of n counted by p ω n) into an odd number of parts and an even number of parts, respectively. Then it follows from 2) that { ) j, if n = 6j 2 + 4j + or 6j 2 + 8j + for some j 0, p ω,o n) p ω,e n) = 0, otherwise. This yields the following obvious result with the recongnition that 6j 2 + 8j + = 6j + ) 2 4j + ) +. Theorem 2.. We have p ω n) j=0 { mod 2), if n = 6j 2 + 4j + for some j Z, 0 mod 2), otherwise.
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) Corollary 2.2. Let p 5 be prime, and let r be such that 0 r p, and r )p + )/2 is a quadratic nonresidue mod p. Then, for all n 0, p ω pn + r) 0 mod 2). Proof. Assume that there exists an integer j such that pn + r = 6j 2 + 4j +. Then pn + r) = 2j + ) 2. This yields r 2j + ) 2 mod p) or ) p + r ) j + ) 2 mod p). 2 But, we have chosen r such that r ) ) p+ 2 is a quadratic nonresidue mod p, so it cannot be congruent to a square. Similarly, ) yields the following mod 2 result. Theorem 2.. We have p ν n) Corollary 2.4. For all n 0 and r = 2,, { mod 2), if n = j 2 + 2j for some j Z, 0 mod 2), otherwise. p ν 4n + r) 0 mod 2). Proof. If j is even, then j 2 + 2j 0 mod 4). If j is odd, then j 2 + 2j mod 4). So 4n + 2 and 4n + can never be represented as j 2 + 2j. Corollary 2.5. Let p 5 be prime, and let r be such that 0 r p, and r + ) is a quadratic nonresidue mod p. Then, for all n 0, p ν pn + r) 0 mod 2). Proof. For some j, let pn + r = j 2 + 2j, which is equivalent to pn + r) + = j + ) 2, and this gives r + j + ) 2 mod p). But, we have chosen r such that r + is a quadratic nonresidue mod p, so it cannot be congruent to a square. We start with a formula from page 6 of [6]: where fq) is a mock theta function, and φq) =. Proof of Theorem. fq 8 ) + 2qωq) + 2q ω q 4 ) = φq)φq2 ) 2 q 4 ; q 4 ) 2 =: F q), 4) ψq) = n= q n2 = q; q 2 ) 2 q 2 ; q 2 ), q n+ 2 ) = q 2 ; q 2 ) q; q 2 ).
4 GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE First, note that q 2 ; q 2 ) 5 φq) = q; q) 2 q 4 ; q 4 ) 2, 5) ψq) = q2 ; q 2 ) 2 q; q). 6) Also, by Entry 25 i), ii), and Entry 25 v), vi) in [2, p.40], respectively, we have Lemma.. The 4-dissection of F q) is where F 0 q) = φq) q; q) 2 Proof. By 7) and 8), φq) = φq 4 ) + 2qψq 8 ), 7) φq) 2 = φq 2 ) 2 + 4qψq 4 ) 2. 8) F q) = F 0 q 4 ) + qf q 4 ) + q 2 F 2 q 4 ) + q F q 4 ),, F q) = 2φq)2 ψq 2 ) q; q) 2, F 2 q) = 4φq)ψq2 ) 2 q; q) 2 F q) = φq)φq2 ) 2 q 4 ; q 4 ) 2 φq 4 ) + 2qψq 8 ) ) φq 4 ) 2 + 4q 2 ψq 8 ) 2) = from which the result follows. q 4 ; q 4 ) 2 = φq4 ) + 2qψq 8 )φq 4 ) 2 + 4q 2 φq 4 )ψq 8 ) 2 + 8q ψq 8 ) q 4 ; q 4 ) 2,, F q) = 8ψq2 ) q; q) 2. The following lemmas can easily be proved by the binomial theorem and induction so we state without proof. Lemma.2. For any positive integer n, Lemma.. For any prime p, + x) 2n + x 2 ) 2n mod 2 n ). + x) p + x p ) mod p). We first prove the case k = 0 in Theorem.4 in a separate theorem. Theorem.4. For any nonnegative integer n, p ω 8n + 4) 0 mod 4), p ω 8n + 6) 0 mod 8), p ω 6n + ) 0 mod 4). Proof. Examining the left hand side of 4), we see that the only terms producing q n for n 0,, 7 mod 8) come from 2qωq). Consequently, for the first and third congruences in the theorem, if we can prove that in F q) the terms for q 8n+4 and q 6n+, respectively, have coefficients divisible by 8, we will be done. Also, for the second congruence, it will be sufficient to show that in F q), the terms for q 8n+6 have coefficients divisible by 6. For the first congruence, it suffices to prove that the odd powers in F 0 q) vanish mod 8, which implies that the coefficient of q 8n+4 in F q) is divisible by 8. By Lemma. and 5), F 0 q) = φq) q; q) 2
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) 5 q 2 ; q 2 ) 5 = q; q) 8 q 4 ; q 4 ) 6 q 2 ; q 2 ) 5 q 2 ; q 2 ) 4 q 4 ; q 4 ) 6 = q2 ; q 2 ) q 4 ; q 4 ) 6, mod 8) where the congruence follows from Lemma.2. Thus, mod 8, F 0 q) is an even function, and returning to the observations at the beginning of this proof, we see that p w 8n + 4) is divisible by 4. We now prove the second congruence, which is equivalent to the statement that the odd powers in F 2 q) vanish mod 6, which implies that the coefficient of q 8n+6 in F q) is divisible by 6. By Lemma., 5), and 6), 4 F 2q) = φq)ψq2 ) 2 q; q) 2 = q2 ; q 2 ) q 4 ; q 4 ) 2 q; q) 4 q2 ; q 2 ) q 4 ; q 4 ) 2 q 2 ; q 2 ) 2 = q 2 ; q 2 ) q 4 ; q 4 ) 2. mod 4) By the analysis at the beginning of this proof, we see that p w 8n + 6) is divisible by 8. Lastly, for p w 6n + ), by looking at the dissection of F q) in Lemma., the terms q 6n+ in F q) must come from qf q 4 ). Next, since 6n + = 44n + ) +, if we can prove that the powers of the form q 4n+ in 2 F q) vanish mod 4, we will be done. By Lemma., 5), and 6), 2 F q) = φq)2 ψq 2 ) q; q) 2 q 2 ; q 2 ) 9 = q; q) 6 q 4 ; q 4 ) 2 q 2 ; q 2 ) 9 q; q) 2 q 2 ; q 2 ) 2 q 4 ; q 4 ) 2 mod 4) Now, by Lemma.2, so for some even function Xq 2 ). Thus, q 2 ; q 2 ) 7 = q; q) 2 q 4 ; q 4 ) 2 = φq)q 2 ; q 2 ) 2. q 2 ; q 2 ) 2 q 4 ; q 4 ) mod 2), q 2 ; q 2 ) 2 = q 4 ; q 4 ) + 2q 2 Xq 2 ) 2 F q) φq 4 ) + 2qψq 8 ) ) q 4 ; q 4 ) + 2q 2 Xq 2 ) ) mod 4) φq 4 )q 4 ; q 4 ) + 2qψq 8 )q 4 ; q 4 ) + 2q 2 φq 4 )Xq 2 ), giving the required result. For the first congruence above, 7) is used. We now prove Theorem. by induction on k. For k = 0, the congruences are given in Theorem.4. To prove the congruence for any positive integer k, we consider the following sequence g k defined by g k = 4g k, 9)
6 GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE which has the solution g k = 2 2k g 0 22k. In particular, for any nonnegative integer n, 2 2k+ n + 22k +, if g 0 = 8n + 4, g k = 2 2k+ n + 7 22k +, if g 0 = 8n + 6, 2 2k+4 n + 8 22k +, if g 0 = 6n +. From 9), we see that g k mod 4) for k > 0. Thus, on the left hand side of 4), the coefficient of q g k comes from 2qωq) + 2q ω q 4 ). By the definition of p w n), we have 2qωq) + 2q ω q 4 ) = 2qωq) 2q q 4 ω q 4 ) ) ) = 2 p w n)q n p w n) ) n q 4n, and the coefficient of q g k in 2qωq) + 2q ω q 4 ) is 2 p ω g k ) ) g k p ω g k ) ). We first consider the case when g 0 = 8n + 4 or 6n +. By the induction hypothesis, we know that p ω g k ) 0 mod 4). Therefore, to show that p ω g k ) 0 mod 4), it suffices to show that the coefficient of q g k in F q) in 4) vanishes mod 8, which is indeed true as seen in F q) in Lemma. since g k mod 4). For the case when g 0 = 8n + 6, we know that p ω g k ) 0 mod 8) by the induction hypothesis. Thus, we have to show that the coefficient of q g k in F q) in 4) vanishes mod 6. Again, since g k = 4g k = 4g k ) +, it suffices to show that the coefficient of q g k in F q)/8 vanishes mod 2. By Lemma. and 6), 8 F q) = ψq2 ) q; q) 2 = q 4 ; q 4 ) 6 q; q) 2 q 2 ; q 2 ) q4 ; q 4 ) 6 q 2 ; q 2 ) 4 mod 2) q 4 ; q 4 ) 4 mod 2). Since g 0 2 and g j mod 4 for j > 0, g k 0 mod 4. Thus the required result follows. 4. Congruences for p ν n) The formula ) is the key to the results in this section. Theorem 4.. For any nonnegative integer n, Proof. Note that q 2 ; q 2 ) ψq 2 ) = q4 ; q 4 ) q 2 ; q 2 ) 2 p ν 8n + 6) 0 mod 4). q4 ; q 4 ) q 8 ; q 8 ) 4 q 2 ; q 2 ) 2 q 6 ; q 6 ) 2 mod 4) = q 8 ; q 8 ) φq 2 )φq 4 ) = q 8 ; q 8 ) φq 8 ) + 2q 2 ψq 6 ) ) φq 6 ) + 2q 4 ψq 2 ) ) q 8 ; q 8 ) φq 8 )φq 6 ) + 2q 2 φq 6 )ψq 6 ) + 2q 4 φq 8 )ψq 2 ) ) mod 4), in which the terms for q 8n+6 vanish. This completes the proof. Again, by ), we see that p ν 2n ) = p ω n). Thus the following congruences immediately follow from Theorem.. 0)
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) 7 Theorem 4.2. For any nonnegative integers n and k, p v 2 2k+4 n + ) 22k+ 0 mod 4), p v 2 2k+4 n + 7 ) 22k+ 0 mod 4), p v 2 2k+5 n + 8 ) 22k+ 0 mod 4). 5. Waldherr s congruences In this section, we provide an elementary proof of the first explicit examples of congruences for p ω n). Theorem 5. Waldherr [5]). p ω 40n + 28) 0 mod 5), p ω 40n + 6) 0 mod 5). Proof. First note that the terms for q 40n+28 and q 40n+6 in F q) in 4) come from 2qωq) only. Also, since 40n + 28 = 40n + 7) and 40n + 6 = 40n + 9), it suffices to show that in F 0 q) the terms for q 0n+7 and q 0n+9 vanish mod 5. By 5), 6), and 8), we have which is equivalent to By Lemma., q 2 ; q 2 ) 0 q; q) 4 q 4 ; q 4 ) 4 = q; q) 4 = q 4 ; q 4 ) 0 q 2 ; q 2 ) 4 q 8 ; q 8 ) 4 q 4 ; q 4 ) 4 q 2 ; q 2 ) 4 q 8 ; q 8 ) 4 + 4q q8 ; q 8 ) 4 q 4 ; q 4 ) 2, + 4q q4 ; q 4 ) 2 q 8 ; q 8 ) 4 q 2 ; q 2 ) 0, F 0 q) = φq) q 2 ; q 2 ) 5 q; q) 2 = q; q) 8 q 4 ; q 4 ) 6. Since 0n + 7 and 0n + 9 are odd, we consider odd powers of q only in F 0 q), which appear in 8q q4 ; q 4 ) 6 q 2 ; q 2 ) 24 q 2 ; q 2 ) 5 q 4 ; q 4 ) 6 = 8q q4 ; q 4 ) 0 q 2 ; q 2 ) q 2 ; q 2 ) 0 8q q20 ; q 20 ) 2 q 0 ; q 0 ) 2 n= ) n q nn ) mod 5), where the congruence follows from Lemma. and Euler s pentagonal number theorem. Now we can easily check that nn ) 0, 2, 4 mod 0, from which it follows that the coefficients of q 0n+7 and q 0n+9 are divisible by 5. This completes the proof. Acknowledgements The fourth author was partially supported by a grant #28090) from the Simons Foundation. References [] G. E. Andrews, A. Dixit, and A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions ωq), νq) and φq),. [2] B. C. Berndt, Ramanujan s Notebooks, Part III, Springer-Verlag, New York, 99. [] N. J. Fine, Basic hypergeometric series and applications, Amer. Math. Soc., Providence, 988. [4] S. A. Garthwaite and D. Penniston, p-adic properties of Maass forms arising from theta series, Math. Res. Lett., 5 2008), 459 470. [5] M. Waldherr, On certain explicit congruences for mock theta functions, Proc. Amer. Math. Soc., 9 20), 865 879. [6] G. N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 96), 55 80.
8 GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Department of Mathematics, The Pennsylvania State University, University Park, PA 6802, USA E-mail address: gea@psu.edu E-mail address: dxp299@psu.edu E-mail address: sellersj@psu.edu E-mail address: auy2@psu.edu