Asymptotics and sign patterns for coefficients in expansions of Habiro elements

We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich–Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier’s result on asymptotics for the Fishburn numbers.


Introduction
The expression first occurred in a talk entitled "Analytic continuation of Feynman integrals" by Kontsevich as part of the Seminar on Algebra, Geometry and Physics at MPIM Bonn on October 14, 1997.
Here and throughout, (a) n = (a; q) n := is the standard q-hypergeometric notation.Note that (1.1) does not converge on any open subset of C, but is well-defined when q is a root of unity (where it is finite) and when q is replaced by 1 − q.Moreover, F (q) is an element of the Habiro ring [13] H := lim ← − n Z[q]/ (q) n .
Motivated by Kontsevich's lecture and Stoimenow's work on regular linearized chord diagrams [16], Zagier determined the asymptotic behavior for the Fishburn numbers ξ(n) which are the coefficients in the formal power series expansion Namely, as n → ∞ [19, Theorem 4] where C 0 = 12 12 , C 1 = C 0 3 8 − 17π 2 144 + π 4 432 and all C i are effectively computable constants.A key step in proving (1.2) is the "strange identity" where " = " means that the two sides agree to all orders at every root of unity (for further details, see [19,Sections 2 and 5]) and 12 * is the quadratic character of conductor 12.The idea is to first express ξ(n) in terms of the Taylor series coefficients of F (e −t ), then employ (1.3) to, ultimately, obtain estimates for these coefficients.These estimates, in turn, lead to (1.2). "Identities" such as (1.3) are not only important in proving asymptotics, but also play a crucial role in obtaining congruences for ξ(n) modulo prime powers [2] and quantum modularity for F (q) [20].For developments in these latter two directions, see [1,2,5,[8][9][10][11][12]17].The positivity of the Fishburn numbers ξ(n) is a consequence of any of its numerous combinatorial interpretations [18, A022493].The purpose of this paper is to prove asymptotics and study sign patterns for coefficients in expansions of elements in H which satisfy a general type of strange identity.Before stating our main result, we introduce some notation.
Let f : Z → C be a function of period M ≥ 2. For integers a ≥ 0 and b > 0, consider the partial theta series where ν ∈ {0, 1}.Suppose there exists We write and define (1.5) Assume there exists a smallest positive integer k ν such that G (ν) ≥ 0 be the smallest integer such that for n ≥ N Moreover, there exists an integer 0 has the same sign as , then for all non-negative integers ℓ, ξ f (ℓ) has the same sign as The paper is organized as follows.In Section 2, we prove Theorem 1.1.In Section 3, we give some applications, including asymptotics and positivity statements for the generalized Fishburn numbers ξ t (n) which arise from the Kontsevich-Zagier series F t (q) associated to the colored Jones polynomial for the family of torus knots T (3, 2 t ), t ≥ 2 [8].This extends (1.2) and gives an alternative proof of the positivity of the Fishburn numbers ξ(n) (see Corollary 3.1 and Remark 3.2).In Section 4, we comment on asymptotics for other expansions of F f (q) and then conclude with conjectures concerning the positivity of coefficients for these expansions in three situations: F t (q), the Kontsevich-Zagier series associated to the colored Jones polynomial for the family of torus knots T (2, 2m + 1), m ≥ 1 and a "Habiro-type" q-series with origins in Ramanujan's lost notebook [3].

Proof of Theorem 1.1
Proof of Theorem 1.1.We follow the strategy of [19].To find the asymptotics of ξ f (n), we first consider the expansions and define the L-function The Mellin transform of where Γ(s) is the usual Gamma function.Applying Mellin inversion to (2.3), we obtain . It is well-known that L(s, f ) can be analytically continued to the whole complex plane except for a possible simple pole at s = 1 with residue By a standard complex analytic computation, we have as t → 0 + .By (1.4) and (2.2), we compare coefficients to obtain Also, it is clear from (1.4) and (2.2) that R f,M = 0. Via [15, Eqn. ( 24)] or [6, Chapter 12], we have for any ε > 01 .Using (2.4) and (2.6), it follows (2.7) From (2.1), we have From (2.7) and (2.8), we deduce An application of Stirling's formula Combining (2.7), (2.9) and (2.11) yields where and all the remaining constants α i,f,ν are effectively computable.Now, we recall where S n,m denotes the Stirling numbers of the first kind.From (2.1) and (2.13), we interchange sums and thus Next, from S n,n = 1 and the recursion S n+1,m = S n,m−1 + nS n,m we have 12), it follows and by (2.10) Finally, we combine (2.14)-(2.17) to obtain The result (1.8) now follows from (2.12).Now for fixed f , we have from (2.4) and (2.5) (2.18) Next, (1.5) implies and this yields 1 where f (k ν ) have the same sign for all n ≥ N f,ν .Next, we note using (2.4) and [2, Lemma 3.2] that where for k ≥ 0, B k (x) denotes the kth Bernoulli polynomial.Hence (2.20) implies that if C n,f , or equivalently (1.9), has the same sign as (−1) ν G (ν) ) and (2.14) imply that ξ f (ℓ) has the same sign as (−1) ν G (ν) f (k ν ) for all ℓ ≥ 0.

Examples
In this section, we illustrate Theorem 1.1 with three examples.
3.1.Kontsevich-Zagier series for torus knots T (3, 2 t ).For t ≥ 2, consider the Kontsevich-Zagier series associated to the family of torus knots T (3, 2 t ) where , if t is even, is the q-binomial coefficient. 2The expression F t (q) matches the N th colored Jones polynomial for T (3, 2 t ) at a root of unity q = e 2πi N , converges in a similar manner as F (q) and is an element 2 For t = 1, one may define the sum over the j ℓ to be 1 in (3.1) to recover (1.1).
of H (see [8] for further details).The generalized Fishburn numbers ξ t (n) are defined by An application of Theorem 1.1 is the following.Note that (1.2) follows after taking t = 1 and simplifying (for brevity, we only state the leading term).
Moreover, let N t ≥ 0 be the smallest integer such that for all 0 ≤ n < N t , then ξ t (ℓ) > 0 for all ℓ ≥ 0 and t ≥ 1.

An example with
and write The k = 1 case of (3.  82k+1) .
Moreover, ξ G k (n) > 0 for all n ≥ 0 and k ≥ 1. where Observe that (3.20) is not a strange identity but an actual identity valid for |q| < 1 and every odd order root of unity q (see [2, Example 3.2]).Although G k (q) ∈ H, we note that when q = e −t with t → 0 + , the expression (q; q 2 ) n k in the right-hand side of (3.18) will have an asymptotic expansion starting with t n k .Hence, the expansions (2.1) and (2.2) for this q-series as t → 0 + are still valid and we can apply Theorem 1.1.First, we have that χ k (n) is an odd function with period M = 4k + 2. Next, we claim that k 0 = 1.To see this, observe that for any ℓ ≥ 1, (1.5) and (3.21) yield and thus which is non-zero for all k ≥ 1.By Theorem
Using Theorem 1.1, we may deduce asymptotics for the coefficients of F f 1 1 + q and F f 1 − q 1 + q .
Namely, if we write This follows upon first noting and so then applying Theorem 1.1.Similarly, if then one can check Asymptotics for the coefficients of F t (q), X m (q) and G k (q) with q replaced by 1 1+q or 1−q 1+q now follow readily from (4.1), (4.2) and Corollaries 3.1, 3.3 and 3.6.Thus, all but finitely where ζ(s) is the Riemann zeta function.Observe that N (max) f,ν exists since ζ(2n + ν + 1) − 1 → 0 as n → ∞ and M f,ν is independent of n.Finally, let B k (x) denote the kth Bernoulli polynomial for an integer k ≥ 0. Our main result is now the following.Theorem 1.1.Assume (1.4) is true.Then as n → ∞, we have