Asymptotics of Bailey-type mock theta functions

We compute asymptotic estimates for the Fourier coefficients of two mixed mock modular forms, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the circle method due to Wright and a modified Tauberian theorem. We encounter cancellation in our estimates for one of the mock theta functions due to the auxiliary function $\theta_{n,p}$ arising from the splitting of Hickerson and Mortenson. We deal with this by using higher order asymptotic expansions for the Jacobi theta functions.


History.
We recall the defintion of a classical mock theta function. Let q be a complex variable with |q| < 1. A classical mock theta function M (q) is a function for which near each root of unity ξ there exists a weakly holomorphic modular form, F ξ , and a rational number a ξ , such that near ξ We then eliminate the possibility of having holomorphic theta functions from the definition by declaring that no F ξ satisfies the above condition for all roots of unity. A nice list of the classical mock theta functions exists in the appendix of [7] and Section 4 of [14]. Large families of new examples of modular type functions that satisfy Eq. (1) were discovered after S. Zwegers wrote his thesis [20] on mock theta functions in 2002, whereby the classical mock theta functions were found to be linked to harmonic Maass forms. As a result, functions that are finite sums of normalized Appell sums can be viewed as mock theta functions. This result brings the theory of mock theta functions and combinatorial generating functions closer together. For example, let ζ := e 2πiz , then the famous partition rank generating function, R(z; τ ) := ∞ n=0 m∈Z N (m, n)ζ m q n := n≥0 q n 2 (ζq, ζ −1 q; q) n , can be written as a sum of normalized Appell sums and is thus a mock theta function when z ∈ H (Lemma 3.1 in [10]). Understanding how the coefficients of mock theta functions grow is important, especially when a combinatorial interpretation is available. For example (see Theorem 1.2 in [10]), where p(n) is the partition function, β := √ nlog(n) π √ 6 , and f (n) ∼ g(n) denotes that the ratio of f (n) and g(n) goes to 1 as n → ∞.

Bailey pairs and Mock theta functions.
The inspiration for this work comes from the fact that we want to find similar asymptotic estimates for mock theta functions that come from Bailey pairs. Let α n (q) =: α n and β n (q) =: β n be two sequences of q-series. The tuple (α n , β n ) is referred to as a Bailey pair with respect to a ∈ C (assuming a causes no poles in what follows) if β n = n k=0 α k (q) n−k (aq) n+k .
The fact that Bailey pairs and mock theta functions are related is not immediately obvious, and it wasn't until Andrews showed that Eq. (2) can be iterated to obtain an infinite family of Bailey pairs that a true connection was found [1,2]. This is the content of Bailey's lemma [1,2,3,4]. Bailey's lemma leads to families of sums, known as higher level Appell sums, which are not necessarily mock theta functions, but mixed mock theta function [7,18]. Occasionally, certain pairs lead to normal Appell sums via Bailey's lemma, and we call the resulting functions Bailey-type mock theta functions.
The study of Bailey-type mock theta functions became more interesting with a key result by Hickerson and Mortenson [14], which gave an explicit decomposition of indefinite theta functions in terms of Appell sums and theta functions. This result was used by many authors in works such as [13,17,18] to write families of Bailey-type mock theta functions in terms of classical mock theta functions. For example, Lovejoy and Osburn in [18] derived a Bailey-type mock theta function, R 1 (q), and used the decomposition of [14] to find the formula where φ is the 10th order classical mock theta function given by and M 1 (q) is a weakly holomorphic modular form. Understanding how the coefficients of certain Baileytype mock theta functions grow is an interesting question, which was proposed by Lovejoy and Osburn in [18], and which we will begin to answer in this work. To the best of our knowledge, no works have investigated the growth of Bailey-type mock theta functions in depth. Doing so here for two example functions, we hope to lay the groundwork for future and more advanced studies of the asymptotic properties of Bailey-type mock theta functions. Let a(n) denote the coefficients of R and b(n) the coefficients of R 1 , which are two Bailey-type mock theta functions defined in Definition 3 (the a(n) and b(n) are explicitly defined in Examples 1 and 2) . We will show the following. Theorem 1. The following estimates hold as n → ∞: 24n .
The following To obtain asymptotic estimates like the ones we give in our main Theorem 1, it is often useful to use a modified circle method due to Wright [19], which allows one to look at a finite number of poles. Wright's technique has been used by several authors in recent years [6,8,10,12] to deal with combinatorial generating functions like R(z; τ ), for example. The common theme here and in the works [6,8,10,12] is that the functions are generically mixed mock modular forms, which are more suited for the adapted circle method of Wright. This work is organized as follows: In Section 2, we define the main objects of this work. In Section 3, we provide estimates near τ = 0 of the Jacobi theta function and normalized Appell sum. In Sections 4 and 5, we employ the Wright circle method to prove the first part of our main theorem, and in Section 6 we use results from [5,15] to prove the second part of our theorem. Finally, we offer some remarks on our results and thoughts on future work regarding this topic in Section 7.

Acknowledegments
This work was supervised by Kathrin Bringmann, and we would like to thank her for her contributions. We would like to give special thanks to Caner Nazaroglu for giving insight into many of the calculations in this work and Jeremy Lovejoy for his helpful comments and suggestions regarding many identities. We finally want to thank the anonymous reviewer, Chris Jennings-Schaffer, Alexandru Ciolan, and Markus Schwagenscheidt for their helpful suggestions and edits.

Preliminaries and basic definitions
The basic objects that appear in this work, and some of their properties, are collected in this section. We begin by recalling the definitions of the normalized Appell sum and the Jacobi theta function: where z 1 , z 2 ∈ C, ζ j := e 2πizj , q := e 2πiτ , τ ∈ H, and ϑ is the Jacobi theta function (or ϑ-function, for short) given by Furthermore, we have the Jacobi product representation for the ϑ-function: where ζ := e 2πiz . Many of the important functions discussed here were originally defined in [14,18]. In those works, the authors used a slightly different notation for the ϑ-function (denoted by j) and the Appell sum (denoted by m). One can go between the two via the formulas ϑ(z 2 ; τ ) = −iq 1/8 ζ −1/2 2 j(ζ 2 , q), m(ζ 1 ; q; ζ 2 ) = iq 1/8 ζ −1/2 1 µ(z 1 + z 2 , z 2 ; τ ).
We will use the following identities frequently.
The first definition comes from the work of [14], and uses the standard combinatorial notation for the Jacobi triple product x is a non-zero complex number. When x is an integral or half integral power of q, we will always write j in terms of a ϑ-function as discussed in Section 2, via the transformations Definition 4 (see Section 2, [18] and Theorem 1.3, [14]). Let x and y be complex numbers so that they do not cause poles in the quotients that follow. Then for positive integers n, p, r := r * + (n−1) 2 and s := s * + (n−1) 2 , with {a} denoting the fractional part of the number a, define the function θ n,p (x, y, q) by, θ n,p (x, y, q) := We then have the following theorem.
is a mock theta function and satisfies the formula Example 6 (The function R 3,3 ). The Fourier expansion for R 3,3 (q) takes the shape − 6 q 27 + 5 q 28 − 6 q 29 + 6 q 30 − 5 q 31 + 6 q 32 − 6 q 33 + 7 q 34 − 9 q 35 + 9 q 36 − 9 q 37 + 9 q 38 − 9 q 39 + 11 q 40 − 10 q 41 + 12 q 42 One can see the alternating sign changes and our main theorem shows that this behavior holds in the limit n → ∞ We can explicitly get the function R 3,3 (q) in a form that is suitable for applying the circle method: where we defined Q(r, s) := r(r − 1) 2 + s(s + 1) 2 + 5s + 6r + 5rs, above. The study of T (τ ) will be the main focus for the rest of the chapter.
With this information, we can show the following formula holds with τ → τ

Preliminary estimates for modular theta functions and Appell sums near τ = 0
We collect all of the necessary estimates for the accessory objects that appear in this work near the point τ = 0. We have two subcategories of estimates that we need to deal with: the classical estimates that only need one error term, and the higher order estimates that keep many error terms in the asymptotic expansion.

Classical estimates.
We begin with the ϑ-functions near the origin.
Proof. We begin with Eq. (8). Using the Jacobi product formula and Proposition 2.8 , we have as τ → 0 Thus, where the second to last step follows from the fact that 1 − e 2πiα is O(1). Similarly for Eq. (9), Finally, the estimate for the η-function follows directly from the transformation law in Proposition 2.
We also need similar estimates for the Appell function near τ = 0. Equation (4) gives Before moving forward, we show that the integral h(2τ ; 12τ ) can be bounded by a standard Gaussian integral.
Proof. The proof follows from the transformation law for h given in Proposition 2.4: which implies that Since 0 ≤ α < 1 2 , the term in the parentheses is bounded above by a constant. Therefore, Where we used the fact that y > 0 and that R e − yw 2 |τ | 2 dw = π y |τ |. This leads to the claimed estimate as τ → 0.

Higher order estimates.
The main terms in the estimates of the previous section will not be sufficient in proving the growth of the a(n), thus we need the following.
Lemma 10. Let α be as in Lemma 8. Then as τ → 0 within a cone, where, Proof. Let w := e 2πiα . The proof of Eq. (12) follows directly by applying the technique in the proof of Lemma 8 and observing that On the other hand, for Eq. (13), we consider the associated Jacobi product Plugging this into the calculation in the proof of Lemma 8 gives the result.

The a(n)
The function T (τ ) defined in Example 6 can be simplified greatly.
4.1. The pole at τ = 1 2 . As was claimed in Section 3.2, we require higher order asymptotic expansions to accurately determine the growth of the a(n). We break the study near τ = 1 2 into two parts: T (τ ) and the Appell function, where Lemma 10 will prove useful for the study of T (τ ). Now we can prove the following.
Remark 14. The growth of R 3 near τ = 1 2 is determined by the estimate in Theorem 13.

Growth on the minor arcs
We now want to show that the growth at the other cusps is negligible to that given in Theorem 13. Thus, our target is to beat the bound exponentially  Remark 16. Notice that the right hand side of Eq. (18) does not depend on a. Furthermore, the bounds above also hold for the functions ϑ(aτ ; bτ ), ϑ 1 2 ; bτ and ϑ 1 2 + aτ ; bτ which can by seen by replacing Log(•) with − Log(•) in the proof below.
Proof. The proof uses the same ideas as in [8,10,12] to prove their bounds away from the dominant pole. We recall the Taylor expansion for Log(1 − z) = − n≥1 z n n . This implies, The trick now, as described by many works such as [8,10,12], is to extract the first term in the sum, and add an extra term, which will be the first term in the expansion for Taking the absolute value of this equation and using the fact that 1 − |q| b ≤ |1 − q b |, we have the upper bound The Log term can be estimated by the asymptotic formulas derived for the ϑ-functions in Lemma 8. Namely, as n → ∞ where C a,b is a constant. Now we bound the fractions. Recall that we are away from the root of unity, q = 1 corresponding to l = 0, by the amount |u| > M v. Following the procedure on page 10 of [8], we can bound 1 |1−q b | by using the fact that cosine is a decreasing function near 0. Namely, as n → ∞ This implies that For the other fraction, we have as n → ∞ Combining Eqs. (21), (22), and (23), we have which proves the first part of Eq. (18). The second part of Eq. (18) follows by noticing that as before.
The following result shows that we can bound the above sum by a classical single variable theta function, Θ(τ ). A similar result was also mentioned by the authors of [8], but was not carried out explicitly.
Proposition 17. Let Θ(τ ) := n∈Z q n 2 . Then, Proof. Splitting the sum in Eq. (24) into negative and positive index, and then recombining, we find Since |q| < 1, we have that 1 − |q| ≤ |1 + q|. Combined with the fact that |q| m is a decreasing function in m, we have that, which proves the claim.
Remark 18. Recall that Θ(τ ) is a holomorphic modular form for the group Γ 0 (4). Γ 0 (4) has three inequivalent cusps represented by 0, 1 2 , and ∞. Recall that in Section 4.1.2 we computed the estimate near 0 and 1 2 , which gives Due to Proposition 17 and the corresponding remark, we can see we only need to check the growth of Θ(τ ) near ∞. Since Θ(τ ) is modular, it's growth at ∞ is at most O(1). Since under the transformation τ → τ + 1 2 the Jacobi theta remains unchanged, that is ϑ 1 2 ; 24τ → ϑ 1 2 ; 24τ , we can use Lemmas 15 and 17 to obtain the following.
Theorem 19. Let M > 0 such that 0 < yM < x − 1 2 . Then there is a β > 0 such that as n → ∞, The estimates in Lemma 15 alone are not sufficient to bound T (τ ) away from the dominant pole since they do not provide accurate information about the decay of the ϑ-functions near generic cusps p h . However, we can use Lemma 15 in combination with a generalization of Lemma 12 to rule out contributions from cusps not equal to 1 2 . Our goal is to prove the following. Theorem 20. Let M > 0 and let v : holds uniformly as n → ∞.
We note additionally that T (τ ) decays rapidly near 0.
To deal with the other cusps, we recall the fact that ϑ(z; τ ) is a Jacobi form of weight and index 1 2 . Thus, the following properties hold [7,11]: Thus, lim w→0 A(σ) = p h . Regardless of whether h is even or odd, using Eq. (25) yields If h is even, d must be odd. Therefore, Eq. (26) implies that Eq. (28) reduces to, which upon using the Jacobi triple product and taking the limit w → 0 gives Subbing in w = τ − p h proves the first claim. The second case when h is odd has two separate situations to contend with, depending on whether d is odd or even. Assume first that d is odd. Then h 2 and d 2 are both half integers. Therefore using Remark 22 as before, we have where • is the floor function. Using the Jacobi product again and taking the limit as w → 0, we have The last step follows since h odd implies h 2 = h−1 2 . If d is even, the only thing that changes in Eq. (29) is that ϑ σ+1 2 ; σ becomes ϑ σ 2 ; σ , which both yield the same estimate up to a constant factor as w → 0 by examining the triple product representations.
With the previous lemma, we have the following useful corollary.
Corollary 28. Let τ := u + iv. Suppose |u| is uniformly bounded by v for all n. Then we have the lower bound as n → ∞ . The machinery of the the proof of Lemma 27 with p :=γ and w := 24w now applies. Ifh is odd, we are done. Ifh is even, we find that as n → ∞ in a cone where the last step follows from the fact that Im(w) = Im(τ ) = v.

Proof of Lemma 20.
As stated at the beginning of this section, we claimed that there are only a finite number of cusps we need to check. The following proposition gives us a rough bound for this number, but it more importantly tells us that all of the cusps that could cause a large pole have h even. As a reminder, we have already checked explicitly that T (τ ) decays exponentially near 0 in Lemma 21, which means we do not have to investigate this case in the proof below.
Proposition 29. Let δ := √ 192. With this choice, the only cusps that could lead to T (τ ) having larger growth than that at 1 2 are cusps p h with h ≤ 24 even. Proof. We use the notation from the proof of Theorem 13, where we saw that we could write T (τ ) as Referring back to Remark 16 we can use Lemma 15 to bound the combination D(τ ) := ϑ 3 (96τ ; 288τ )S(τ ).
Recalling that each one of the four terms in S(τ ) is of the form , and that the bounds in Lemma 15 only depend on the factor in the second slot, we find that If h is odd, by Lemma 27 and Cor. 28, we have that Using Lemma 15 on D gives

The b(n)
We now turn our attention to R 1 (q) =: n≥0 b(n)q n . To begin, we note that the b(n) form a weakly increasing sequence, which is apparent from the definition of ν(q).
Lemma 31. Let b(n) denote the n th Fourier coefficient of the function ν(−q). Then the sequence {b(n)} ∞ n=0 is weakly increasing and no b(n) < 0. The following Tauberian Theorem allows us to capture the growth of the b(n) by only computing an estimate for the growth of R 1 in an angular region around τ = 0. The original theorem is due to Ingham [15], but we state it in a more modern form taking into account some additional technicalities regarding the growth of functions in angular regions around the origin.
Theorem 32 (See Theorem 1.1 of [5] with α = 0). Let c(n) denote the coefficients of a power series C(q) := ∞ n=0 c(n)q n with radius of convergence equal to 1. Define z := x + iy ∈ C. If the c(n) are non-negative, are weakly increasing, and we have as t → 0 + that Remark 33. We will show the bound in Theorem 32 for R as τ → 0 with τ ∈ H, which is sufficient to show the bound for general z since we can define an even extension of R The last line follows by splitting the sum into n < 0 and n > 0, and then swapping n → −n in the sum over n < 0. We then have, We can use the triple product formula to deal with the ϑ-function to find as τ → 0 that Therefore, we can state the following.