On recursions for coefficients of mock theta functions

We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoglu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions. Additionally, we discuss an application of this identity to other mock theta functions.

Recently, Imamoglu, Raum and Richter [15] proved some intriguing results concerning recursive formulas for the coefficients of the 3rd order mock theta functions (1.2) For a fixed n ∈ Z + and for any a, b ∈ Z, set N := 1 12 (3a − b − 2) andÑ := 1 12 (3a + b − 4). Then For a fixed n ∈ Z + and for any a, b ∈ Z, set N : For a fixed n ∈ Z + and for any a, b ∈ Z, set N := 1 6 (a − 3b − 2) andÑ := 1 6 (a + 3b − 2), and let , if n is even; if n is odd. (1.6) The identities (1.2)-(1.6) were proven in [15] by applying holomorphic projection to the tensor product of a vector-valued harmonic weak Maass form of weight 1/2 and vector-valued modular form of weight 3/2. In Remark 1, ii) of [15], it was stated that these identities "can sometimes also be furnished by Appell sums because these are typically expressible in terms of divisors. However, it is not clear whether Theorem 1 and 9 could be obtained using this idea".
Motivated by this remark, the main purpose of this paper is explain how (1.1) can be used to give a q-series proof of these identities. The idea is to compare the coefficients of q n in identities which express a modular form times either f (q) or ω(q) (Appell-Lerch sums) in terms of Lambert series (divisor sums). Specifically, (1.2) and (1.3) will basically follow from (2.8) and (3.11), (1.4) from (3.17), (1.5) and (1.6) from (3.24) and (2.9), respectively.
The paper is organized as follows. In Section 2, we discuss some preliminary q-series identities. In Section 3, we prove Theorems 1.1 and 1.2. In Section 4, we discuss another application of (1.1) to other mock theta functions.
To prove (1.4), we need the following result whose proof is analogous to that of Theorem 2.3 and thus is omitted.
3. Proof of Theorems 1.1 and 1.2 Proof of Theorem 1.1. We begin with the proof of (1.2). First, note that By applying (see [5, page 114, and extracting the coefficient of q n on both sides of (3.1), we obtain m∈Z 3m 2 +m≤2n We now examine the right-hand side of (1.2). Observe that both a and b have the same sign, and when a = 3b, N is not an integer. Hence, by splitting the sum according to the values of a and b in the range 1 ≤ b < 3a, 1 < 3a < b, 3a < b ≤ −1, and b < 3a < −1, we find that where in the third equality, we note that Splitting the sum on the right-hand side of (1.3) in a way similar to (3.4), we find that The generating function for S 1 (n) is given by Note to have 12|3a − b − 2 and 4|ab − 1, we must have a odd and b ≡ 1 (mod 6). Hence we replace a = 2k + 1 and b = 6l + 1 in the first sum on the right-hand side of (3.7). Similarly, we replace a = 2k + 1 and b = 6l + 5 in the second sum. This leads us to where in the last equality, we replaced l by −l − 1 in the second sum. Similarly, the generating function for S 2 (n) is Now, to prove (3.11), we first set l = 1 and j = −3 in Theorem 2.3 to obtain Replacing q, a, b and c by q 6 , q 5 , q 5 and −1/q 2 , respectively, in [8, Corollary 3.2], we obtain which together with (3.12) gives (3.14) Substituting (3.14) into (3.13), we obtain where the last step follows from the fact that As (3.16) By (2.11), (3.2) and a calculation similar to (3.8) for the generating function of the right-hand side of (3.16), we see that (1.4) follows extracting the coefficient of q n from both sides of (3.17) To prove (3.17), we set j = −1, l = 1 in Theorem 2.4 to get Replacing q, a, b and c by q 6 , q, q and −q 2 , respectively, in [8,Corollary 3.2], we obtain We note that (see [6, Eq.
where the last step follows from the fact that By (3.21), we find that, to prove (3.17), it suffices to show that which is easily checked to be true by replacing n by −n in the sum on the left-hand side. This completes the proof of (3.17) and thus (1.4). Now, by taking the sum and difference of (1.5) and (1.6), respectively, we obtain the two equivalent formulas or equivalently We now set l = 1, j = 0 and replace q by −q in Theorem 2.3 to obtain This together with [14,Eq. (5.8) which is equivalent to (3.25 3b. Next, we examine the right-hand side of (2.9). Note that 6R n q n , whereR n := 1 3 (σ( n 4 ) − 2σ( n 2 )), if n is even; Hence, identity (2.9) is equivalent to 3b. (3.28) Applying (3.10) while extracting the coefficient of q n from both sides of (3.28), we obtain m∈Z 3m 2 +2m+1≤n  It is worth noting that (1.1) can also be used to obtain other identities involving mock theta functions. For example, in [4], Andrews, Rhoades and Zwegers consider the automorphic properties of the q-series which is related to the generating function for the number of concave compositions of n [2]. In particular, to show that ν 2 (q) is a mock theta function, they require the following key identity (see Theorem 1.3 in [4] We now prove a generalization of (4.1).
Using Theorem 4.1, one can also show where B(q) := ∞ n=0 q n (−q; q 2 ) n (q; q 2 ) n+1 is a 2nd order mock theta function (see [13]). Extracting the coefficient of q n on both sides, we obtain the following corollary. Similar results exist, for example, for the mock theta functions ψ(q), ρ(q) and λ(q) of order 6 and V 0 (q) of order 8 as they can be written in terms of Appell-Lerch series (see Section 5 of [14]).