Asymptotic behavior of odd-even partitions

Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's"Lost"Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions.


Introduction and Statement of results
Andrews [1] considered a certain family of functions and noticed a mysterious phenomenon. More precisely, Andrews looked into q-series identities involving hypergeometric functions, for example in particular ( [1] and [2, Page 19 and Page 104]) (1 + q n ) , While the others can be nicely written in terms of infinite product (so that it turns out that they are modular forms up to q powers), Andrews did not find any such shape of identities for (1.1). Moreover, Zagier [12, Table 1] figured out that (1.1) is not modular. Nonetheless, Andrews [1] provided a combinatorial interpretation for this function, namely odd-even partitions.
Recall that a partition of positive integer n is a nonincreasing positive integer sequence whose sum is n. Define a partition function OE(n) by the number of partitions of n in which the parts alternate in parity starting with the smallest part odd. In other 1 words, OE(n) counts the number of odd-even partitions of n. For instance, there are no odd-even partitions of 2 and the odd-even partitions of 3 are 3 and 2+1, and thus OE(2) = 0 and OE(3) = 2. Then the generating function for the odd-even partitions is given which is exactly identical to (1.1). Here the q-Pochhammer symbol is defined as (a) n := (a; q) n := n j=1 (1 − aq j−1 ) for n ∈ N 0 ∪ {∞}. In this paper, we investigate the asymptotic behavior of OE(n). In order to study the asymptotic behavior of the coefficients of a series, one can either use the Circle Method [5,9,11] or apply Ingham's Tauberian Theorem [6]. Since O(q) has a pole at every root of unity and it is not easy to find the bounds for O(q) at every root of unity, it is difficult to use the Circle Method in our case. Moreover, as OE(n) is not monotonically increasing, we cannot directly apply Ingham's Tauberian Theorem to our case either (see Section 2 for more details). Thus, we need to slightly modify our function so that we can apply Ingham's Tauberian Theorem.
We also investigate the asymptotics of odd-even overpartitions, studied by Lovejoy [8]. Recall that an overpartition of positive interger n is a partition of n in which the first occurrence (equivalently, the final occurrence) of a number may be overlined. An odd-even overpartition is an overpartition with the smallest part odd and such that the difference between successive parts is odd if the smaller is nonoverlined and even otherwise. For example, there are no odd-even overpartitions of 2, the odd-even overpartitions of 3 are 3, 3, 2 + 1, and 2 + 1, and the odd-even partitions of 4 are 3 + 1 and 3 + 1. Notice that if all parts are non-overlined, then we have the odd-even partitions. We denote OE(n) by the number of odd-even overpartitions of n and define OE(0) := 1. The generating function is given in [8] is one of Ramanujan's third order mock theta functions. These functions appeared in Ramanujan's deathbed letter to Hardy and are now known as the holomorphic parts of weight 1/2 harmonic Maass forms (see [13]). We remark that the generating function for the odd-even overpartitions is a mixed mock modular form, i.e., the product of a modular form and a mock theta function. From this fact, we can apply Wright's Circle Method [11] to obtain the asymptotic formula for OE(n). e π √ n 3 as n → ∞.
The paper is organized as follows. In Section 2, we study some basic properties of odd-even partitions and introduce an auxiliary Theorem which play important roles to prove Theorem 1.1. The proof is given in Section 3. We conclude the paper with the proof of Theorem 1.2 in Section 4.

Acknowledgement
These results are part of the author's PhD thesis, written under the direction of Kathrin Bringmann. The author thanks her for suggesting this problem and valuable advice, Don Zagier and Byungchan Kim for insightful comments and for providing the numerical results to the main theorems, and Jeremy Lovejoy for affording the idea to consider odd-even overpartitions which expanded the scope of this paper. The author also thanks Steffen Löbrich and Michael Woodbury for their support and fruitful conversation regarding this topic. n relevant partitions of n OE(n)

Preliminaries
From these values, we see that OE(n) is not monotonically increasing. Nevertheless, OE(n) ≤ OE(n + 2) holds for every n due to the fact that we can always make an odd-even partition of n + 2 from the one of n by adding 2 to the largest part. Thus, OE(n) is monotonically increasing for even (odd resp.) n. This suggests that the appropriate approach to understand the asymptotic behavior of OE(n) is to split the power series of OE(n) into two parts, one with even n and the other with odd n, as follows: Here, for convenience we define OE(0) := 1. We further split the q hypergeometric series in (1.2) accordingly by considering the parity of powers of q for each summand.

2.2.
Ingham's Tauberian Theorem. From the asymptotic behavior of a power series, Ingham's Tauberian Theorem [6] gives an asymptotic formula for its coefficients.
Theorem (Ingham [6]). Let f (q) = n≥0 a(n)q n be a power series with weakly increasing nonnegative coefficients and radius of convergence equal to 1. If there are constants 3. Proof of Theorem 1.1

3.1.
Asymptotics for the generating functions. In this section, we estimate the functions O e (q) and O o (q). Throughout the section we set q = e −ε . In order to get the asymptotic formulas for these functions, we exploit the second proof of [12,Proposition 5]. The idea of the proof is based on the asymptotics of the individual terms in the series. We first study the asymptotic behavior of the summand and then sum up the asymptotics. We denote the mth term in the series (1. 2) by The sequence (f m ) m∈N is unimodal, meaning that f m increases until f m reaches a maximum value and then decreases. More precisely, for 0 < |q| < 1 the ratio goes to ∞ as m → 0, decreases as m grows, and tends to 0 as m → ∞.
To determine when f m takes the maximum value, we check when the ratio (3.1) becomes 1. This ratio is equal to 1 exactly for q 2m the unique root of the equation Q In other words, f m approaches the maximum value when q 2m is close to Q and m near Log(Q)/(2 Log(q)). We further note that Thus, the main contribution occurs when the terms are of the form q 2m = Qq −2ν (or q m = Q 1 2 q −ν ) with ν ∈ ν 0 + Z satisfying ν = o(m) and ν 0 denotes the fractional part of Log(Q)/(2 Log(q)). In this setting, we evaluate the size of f m . For this, we use the asymptotic expansion from Zagier [12,Page 53]. Here the dilogarithm function Li 2 (z) is defined for |z| < 1 by as ε → 0.
Remark. In fact, Zagier obtained the asymptotic expansion with arbitrary many main terms. Since we only use the first few main terms in this paper, we do not need to consider the complete expansion.
We set q → q 2 , A → 1/2, and B → 1/4 in Lemma 3.1. Thus, R becomes Q and we have, recalling that Q Furthermore, we use the special value of the dilogarithm function from [12, Section I.1]
Proof: Using (3.5), we can also rewrite S j in terms of ϕ(ν) as To estimate S j , we begin by rewriting the sum in ν on the right-hand side of (3.7) as where α := 2 + ν 0 + j and the Jacobi Theta function is given for z ∈ C and τ ∈ H by ϑ (z; τ ) := n∈ 1 2 +Z e πin 2 τ +2πin(z+ 1 2 ) .
The modular inversion formula for the Jacobi theta function [13, Proposition 1.3 (7)] implies that for a, b ∈ C with Re(a) > 0 Plugging in a → 8 √ 5 and b → √ 5(2α − 1) and simplifying the summation yields that (3.9) The last equality comes directly from the fact that as ε → 0 + e √ 5(2α−1) 2 ε 32 From (3.7), (3.8), and (3.9), we obtain for any j ∈ {0, 1, 2, 3} as ε → 0 + . Recalling (3.6), we have the desired result.  ). We first deal with the even case. Setting a(n) = OE(2n) and replacing q by q 2 in Theorem 2.2 determines the constants We reamrk that since OE(n) does not satisfy weakly increasing property with n = 0, we only consider when n ≥ 1. Thus, we have By letting n → n/2, we obtain the desired asymptotic formula for OE(n) with even n, namely For odd n, we rewrite the series as Since by Theorem 3.2 .
Finally from (3.10) and (3.11) we get the desired asymptotic formula for OE(n), for every n, as n → ∞.

Proof of Theorem 1.2
We follow the same method of the proof of Theorem 4 in [4]. The strategy is to estimate the generating function near and away from a dominant pole, and then apply Wright's Circle Method. Although the method of proof is not new, because we are dealing with a different function, the result does not follow directly from the statement of Theorem 4 in [4], and thus we include its proof here. However, it is basically the same proof.

Asymptotics of O(q).
Using the Watson's identity for Ramanujan's third order mock theta function f (q) [10] f (q) = 2 (q) ∞ n∈Z From this expression we can see that O(q) has a dominant pole at q = 1.
Remark. One can find M > Therefore, we find that Combining (4.2) and (4.4) gives the proof of the part (i).
where C = {|q| = e − π 2 √ 3n }. In fact, the integral I 1 contributes the main term as the integral I 2 is an error term.
In order to evaluate I 1 , we introduce a function P s (u), defined by Wright [11], for fixed M > 0 and u ∈ R + P s (u) := 1 2πiˆ1 This functions is rewritten in terms of the I-Bessel function up to an error term. where I ℓ denotes the usual the I-Bessel function of order ℓ.
Using Theorem 4.1 (i), we write the integral I 1 as By making the change of variables v = 1 − i4 √ 3nx, we arrive at where we use the asymptotic formula for the I-Bessel function [3, 4.12.7] Now we turn to the integral I 2 . From the Corollarly 4.2, we have for My < |x| ≤ 1/2 (1−ǫ) , which together with (4.7) completes the proof.