aa r X i v : . [ m a t h . N T ] J a n ASYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS
Taylor Garnowski [email protected]¨at zu K¨oln
Abstract.
We compute asymptotic estimates for the Fourier coefficients of two mock theta functions,which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the circle methoddue to Wright and a modified Tauberian theorem. We encounter cancellation in our estimates for oneof the mock theta functions due to the auxiliary function θ n,p arising from the splitting of Hickersonand Mortenson. We deal with this by using higher order asymptotic expansions for the Jacobi thetafunctions. Keywords.
Mock theta functions, Bailey pairs, Wright circle method. Introduction
History.
The classical study of mock theta functions originated with Ramanujan in the early 20thcentury and continues to the present day. There are varying versions of definitions for what a classicalmock theta function is (see for example, [6, 12]), but they generally all encompass the following: Let q be a complex variable with | q | <
1. A classical mock theta function M ( q ) is a function for which neareach root of unity ξ there exists a weakly holomorphic modular form, F ξ , such that near ξM ( q ) − F ξ ( q ) = O (1) . (1)We then eliminate the possibility of having holomorphic theta functions from the definition by declaringthat no F ξ satisfies the above condition for all roots of unity. A nice list of the classical mock thetafunctions exists in the appendix of [6] and Section 4 of [14]. Large families of new examples of modulartype functions that satisfy Equation (1) were discovered after S. Zwegers wrote his thesis [20] on mocktheta functions in 2002, whereby the classical mock theta functions were found to be linked to harmonicMaass forms. From [20], we can define a mock theta function to be the holomorphic part of a weight harmonic Maass form (see [6] for more on harmonic Maass forms), and thus, functions that are finitesums of normalized Appell sums can be viewed as mock theta functions. This result brings the theory ofmock theta functions and combinatorial generating functions closer together. For example, let ζ := e πiz ,then the famous partition rank generating function, R ( z ; τ ) := ∞ X n =0 X m ∈ Z N ( m, n ) ζ m q n can be written as a sum of normalized Appell sums and is thus a mock theta function when z ∈ H (Lemma3.1 in [9]). Understanding how the coefficients of mock theta functions grow is important, especially whena combinatorial interpretation is available. For example (see Theorem 1.2 in [9]), N ( m, n ) ∼ β (cid:18) βm (cid:19) p ( n ) = β √ n sech (cid:18) βm (cid:19) e π √ n , where p ( n ) is the partition function, β := √ n log( n ) π √ , 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 thatwe 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 Baileypair with respect to a if β n = n X k =0 α k ( q ) n − k ( aq ) n + k , (2)where ( a ; q ) n := Q n − j =0 (1 − aq j ), ( a ; q ) ∞ := Q ∞ j =0 (1 − aq j ), and ( q ; q ) n =: ( q ) n are the usual q -Pochammersymbols. The fact that Bailey pairs and mock theta functions are related is not immediately obvious, andit wasn’t until Andrews showed that Equation (2) can be iterated to obtain an infinite family of Baileypairs that a true connection was found [1, 2]. This is the content of Bailey’s lemma [1, 2, 3, 4]. Bailey’slemma leads to families of sums, known as higher level Appell sums , which are not necessarily mock thetafunctions, but a more general object called a mixed mock theta function [6, 18]. Occasionally, certainpairs lead to normal Appell sums via Bailey’s lemma , and we call the resulting functions
Bailey-typemock theta functions .The study of Bailey-type mock theta functions became more interesting with a key result from Hick-erson and Mortenson [14], which gave an explicit decomposition of indefinite theta functions in termsof 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 . Forexample, Lovejoy and Osburn in [18] derive a Bailey-type mock theta function, R (4)1 ( q ), and used thedecomposition of [14] to find the formula R (4)1 ( q ) = − φ ( q ) + M ( q ) , (3)where φ is the 10th order classical mock theta function given by φ ( q ) := ∞ X n =0 q n ( n +1)2 ( q ; q ) n +1 and M ( q ) is a weakly holomorphic modular form. Understanding how the coefficients of certain Bailey-type mock theta functions grow is an interesting question, which was proposed by Lovejoy and Osburnin [18], and which we will begin to answer in this work. To the best of our knowledge, no workshave investigated the growth of Bailey-type mock theta functions in depth. Doing so here for twoexample functions, we hope to lay the groundwork for future and more advanced studies of the asymptoticproperties of Bailey-type mock theta functions. Let a ( n ) denote the coefficients of R (3)3 and b ( n ) thecoefficients of R (3)1 , which are two Bailey-type mock theta functions defined in Definition 2.3. We willshow the following. Theorem 1.1.
The following asymptotic expansions hold as n → ∞ : a ( n ) ∼ ( − n √ √ n e π √ n ,b ( n ) ∼ (cid:0) π (cid:1) sin (cid:0) π (cid:1) + 1 ! e π √ n √ n . The following table shows the ratio between the estimated values in Theorem 1.1 and the actual valuesfor some values of n . n a ( n ) b ( n )100 0 . . . . . . SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 3
To obtain asymptotic estimates, like the ones we give in our main Theorem 1.1, it is often useful touse 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 [7, 9, 11] to deal with combinatorialgenerating functions like R ( z ; τ ), for example. While the functions we deal with in this work have noknown combinatorial interpretations, we are able to adapt the circle method of Wright to give us anestimate for one of the functions we investigate.This work is organized as follows: In Section 2, we define the main objects of this work. In Section3, we provide estimates near τ = 0 of the Jacobi theta function and normalized Appell sum. In Section4, we employ the Wright circle method to prove the first part of our main theorem, and in Section 5 weuse results from [5, 15] to prove the second part of our theorem. Finally, we offer some remarks on ourresults and thoughts on future work regarding this topic in Section 6. Acknowledegments
This work is part of an ongoing PhD thesis advised by Kathrin Bringmann, and we would like to thankher for her contributions. We would like to give special thanks to Caner Nazaroglu for giving insightinto many of the calculations in this work and Jeremy Lovejoy for his helpful comments and suggestionsregarding many identities. We finally want to thank Chris Jennings-Schaffer, Alexandru Ciolan, andMarkus Schwagenscheidt for their helpful suggestions and edits.2.
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 that the definitions of the normalized Appell sum and the Jacobi theta function: µ ( z , z ; τ ) := ζ ϑ ( z ; τ ) X n ∈ Z ( − n q n n ζ n − q n ζ , (4)where z , z ∈ C , ζ j := e πiz j , q := e πiτ , τ ∈ H , and ϑ is the Jacobi theta function (or ϑ function, forshort) given by ϑ ( z ; τ ) := X m ∈ + Z ( − m q m ζ m . (5)Furthermore, we have the Jacobi product representation for the ϑ function: ϑ ( z ; τ ) = − iq ζ − ( ζ ; q ) ∞ ( q ; q ) ∞ ( qζ − ; q ) ∞ , (6)where ζ := e πiz . Many of the important functions discussed here were originally defined in [18] and [14].In those works, the authors use a slightly different notation for the ϑ function (denoted by j ) and theAppell sum (denoted by m ). One can go between the two via the formulas ϑ ( z ; τ ) = − iq / ξ − / j ( ξ , q ) ,m ( ξ ; q ; ξ ) = iq / ξ − / µ ( z + z , z ; τ ) . We will use the following identities frequently in this work.
Proposition 2.2 ([20]) . The normalized Appell sum and Jacobi ϑ function satisfy :(1) µ ( z , z ; τ + 1) = e − πi µ ( z , z ; τ ) , (2) µ ( z + 1 , z ; τ ) = µ ( z , z + 1; τ ) = − µ ( z , z ; τ ) , ASYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS (3) µ ( z τ , z τ ; − τ ) = −√− iτ e − πi ( z − z τ µ ( z , z ; τ ) + √− iτ i e − πi ( z − z τ h ( z − z ; τ ) , where h ( z ; τ ) isthe Mordell integral given by h ( z ; τ ) := Z R e πiτx e − πzx cosh( πx ) dx, (4) h (cid:0) zτ ; − τ (cid:1) = √− iτ e − πiz τ h ( z ; τ ) , (5) ϑ ( z + τ ; τ ) = − e − πiτ − πiz ϑ ( z, τ ) , (6) ϑ ( z ; τ + 1) = e πi ϑ ( z ; τ ) , (7) ϑ (cid:0) zτ ; − τ (cid:1) = − i √− iτ e πiz τ ϑ ( z ; τ ) , and (8) η ( τ ) = √− iτ η (cid:0) − τ (cid:1) and η ( τ + 1) = e πi η ( τ ) , where η is Dedekind’s eta function. For k ≥ Definition 2.3 ([18]) . Let k ≥ and define B k ( n k , n k − , ..., n ; q ) := ( − n ( − q ) n k − q (cid:18) n k − + 12 (cid:19) Q k − j =2 q j − n k − j (cid:16) − q j − ; q j − (cid:17) n k − j Q kj =1 (cid:0) q j − ; q j − (cid:1) n k − j +1 − n k − j , with n := 0 . Then we define R ( k )1 ( q ) := X n k ≥ n k − ≥ ... ≥ n ≥ q ( nk +12 ) B k ( n k , ..., n ; q ) ,R ( k )3 ( q ) := X n k ≥ n k − ≥ ... ≥ n ≥ ( − n k q n k +2 n k ( q ; q ) n k ( − q ; q ) n k B k ( n k , ..., n ; q ) . The authors of [18] showed that(7) R (3)1 ( q ) = ν ( − q ) , where ν ( q ) := P n ≥ q n n ( − q ; q ) n +1 is a classical third order mock theta function.The first definition comes from the work of [14], and uses the standard combinatorial notation for theJacobi triple product j ( x, q ) := ( x ) ∞ ( q/x ) ∞ ( q ) ∞ , where x is a non-zero complex number. When x an integral or half integral power of q , we will alwayswrite j in terms of a ϑ function, as discussed in Section 2, via the transformations ϑ ( aτ ; bτ ) = − iq b q − a j ( q a , q b ) ,ϑ (cid:18) aτ + 12 ; bτ (cid:19) = − q b q − a j ( − q a , q b ) . Definition 2.4 (see Section 2, [18] and Theorem 1.3, [14]) . Let x and y be complex numbers so that theydo not cause poles in the quotients that follow. Then for positive integers n, p , r := r ∗ + { ( n − } and s := s ∗ + { ( n − } , with { a } denoting the fractional part of the number a , define the function θ n,p ( x, y, q ) SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 5 by, θ n,p ( x, y, q ) := j (cid:16) q p (2 n + p ) , q p (2 n + p ) (cid:17) j (cid:0) − , q np (2 n + p ) (cid:1) ( p − X r ∗ =0 p − X s ∗ =0 q n ( r − ( n − ) +( n + p )( r − ( n − )( s + ( n +1)2 )+ n ( s + ( n +1)22 ) · ( − x ) r − n − ( − y ) s + ( n +1)2 · j (cid:16) − q pn ( s − r ) x n y n , q np (cid:17) j (cid:16) q p (2 n + p )( r + s )+ p ( n + p ) x p y p , q p (2 n + p ) (cid:17) j (cid:16) q pr (2 n + p )+ p ( n + p )2 ( − y ) n + p ( − x ) n , q p (2 n + p ) (cid:17) j (cid:16) q ps (2 n + p )+ p ( n + p )2 ( − y ) n + p ( − x ) n , q p (2 n + p ) (cid:17) ) , where (cid:0) bc (cid:1) is the standard binomial coefficient. Recall that m ( ξ ; q ; ξ ) = iq / ξ − / µ ( z + z , z ; τ ) . We then have the following theorem:
Theorem 2.5 ([18]) . For k ≥ the function R ( k )3 ( q ) is a mock theta function and satisfies the formula R ( k )3 ( q ) = 2 q − k − (2 k − +1) m (cid:16) q k − , q k − +2 k , − (cid:17) − q − θ , (cid:16) q k − +1 , − q k − +1 , q (cid:17) ϑ (cid:0) ; τ (cid:1) = 2 iq − k − µ (cid:18) k − τ + 12 ,
12 ; (cid:0) k − + 2 k (cid:1) τ (cid:19) − q − θ , (cid:16) q k − +1 , − q k − +1 , q (cid:17) ϑ (cid:0) ; τ (cid:1) . Example 2.6 (The function R (3)3 ) . R (3)3 ( q ) = 2 iq − µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) − q − θ , (cid:0) q , − q , q (cid:1) ϑ (cid:0) ; τ (cid:1) = 2 iq − µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) − q ϑ (cid:0) ; τ (cid:1) j ( q , q ) j ( − , q ) X r,s =0 ( − r q r ( r − + s ( s +1)2 +5 r ( s +1)+3( r + s ) j (cid:0) q s − r ) , q (cid:1) j (cid:0) q r + s )+44 , q (cid:1) j ( − q r +22 , q ) j ( − q s +22 , q )= 2 iq − µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) + 2 iq ϑ (cid:0) ; τ (cid:1) ϑ (96 τ ; 288 τ ) ϑ (cid:0) ; 24 τ (cid:1) X r,s =0 n ( − r q Q ( r,s ) · ϑ (4( s − r ) τ ; 16 τ ) ϑ ( { r + s ) + 44 } τ ; 96 τ ) ϑ (cid:0) + { r + 22 } τ ; 96 τ (cid:1) ϑ (cid:0) + { s + 22 } τ ; 96 τ (cid:1) o := 2 iq − µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) + T ( τ ) , where Q ( r, s ) := r ( r − s ( s + 1)2 + 5 s + 6 r + 5 rs. Example 2.7 (The function R (3)1 ) . Recall that R (3)1 ( q ) = ν ( − q ) . ASYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS
With this information, we can show the following formula holds with τ τ + (see A.2, pg. 355 of [6]) R (3)1 ( q ) = − iq − µ (5 τ, τ ; 12 τ ) + e − πi q − η ( τ + ) η (3 τ + ) η (12 τ ) η (2 τ ) η (6 τ ) . Preliminary estimates for modular theta functions and Appell sums near τ = 0We collect all of the necessary estimates for the accessory objects that appear in this work near thepoint τ = 0. We have two subcategories of estimates that we need to deal with: the classical estimates thatonly need one error term, and the higher order estimates that keep many error terms in the asymptoticexpansion.3.1. Classical estimates.Lemma 3.8.
Let α ∈ (0 , , let q := e πiτ , q := e − πiτ , and k > be a rational number. As τ → , ϑ ( ατ ; τ ) = − i sin( πα ) q − α q √− iτ (1 + O ( q )) , (8) ϑ (cid:18) k + ατ ; τ (cid:19) = − q − α e πiα (1 − k ) √− iτ q k − k + (cid:16) O ( q k ) (cid:17) , (9) η ( τ ) = q √− iτ (1 + O ( q )) . (10). Proof.
We begin with Equation (8). Using the transformation law and the Jacobi product formula forthe ϑ function in property 3 of Proposition 2.2 , we have as τ → ϑ ( ατ ; τ ) = iq − α √− iτ ϑ (cid:18) α ; − τ (cid:19) = e − πiα q − α q √− iτ ( e πiα ; q ) ∞ ( q e − πiα ; q ) ∞ ( q ; q ) ∞ . Thus, ϑ ( ατ ; τ ) = e − πiα q − α q √− iτ (cid:16) − e πiα + O ( q ) (cid:17)(cid:16) O ( q ) (cid:17)(cid:16) O ( q ) (cid:17) = e − πiα q − α q √− iτ (1 − e πiα + O ( q )) = e − πiα q − α (1 − e πiα ) q √− iτ (1 + O ( q ))= − i sin( πα ) q − α q √− iτ (1 + O ( q )) , where the second to last step follows from the fact that 1 − e πiα is O (1). Similarly for Equation (9), ϑ (cid:18) k + ατ ; τ (cid:19) = ie − πi ( k ατk + α τ ) τ √− iτ ϑ (cid:18) kτ + α ; − τ (cid:19) = ie − πiαk q − α q k √− iτ (cid:16) − iq e − πi ( α + kτ ) (cid:17) ( e πi ( α + kτ ) ; q ) ∞ ( q e − πi ( α + kτ ) ; q ) ∞ ( q ; q ) ∞ ∼ q k + + k e − πiα e − πi αk √− iτ (cid:16) − e πiα q − k (cid:17) = − q − α e πiα (1 − k ) √− iτ q k − k + (cid:16) O ( q k ) (cid:17) . Finally, the estimate for the η function follows directly from the transformation law in Proposition 2.2. (cid:3) SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 7
We also need similar estimates for the Appell function near τ = 0. Equation (4) gives µ (5 τ, τ ; 12 τ ) = q ϑ (3 τ ; 12 τ ) X m ∈ Z ( − m q m ( m +1) q m − q m q . Proposition 2.2.3 implies that µ (5 τ, τ ; 12 τ ) = − q µ (cid:0) , ; − τ (cid:1) √− iτ + h (2 τ ; 12 τ )2 i . (11)Before moving forward, we show that the integral h (2 τ ; 12 τ ) can be bounded by a standard Gaussianintegral. Lemma 3.9.
Let α < . Then as τ → , h ( ατ ; τ ) ≪ | τ | − . Proof.
The proof follows from the transformation law for h given in Proposition 2.2.4: h ( ατ ; τ ) = q α √− iτ h (cid:18) α ; − τ (cid:19) = q α √− iτ Z ∞−∞ e − πix τ e − παx cosh( πx ) dx = q α √− iτ Z ∞ e − πix τ e − παx cosh( πx ) − Z ∞ e − πix τ e παx cosh( πx ) ! dx, which implies that | h ( ατ ; τ ) | ≤ (cid:12)(cid:12)(cid:12) q α √− iτ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Z ∞ e − πix τ (cid:16) e (2 α − πx e − πx + e − (2 α +1) πx e − πx (cid:17) dx (cid:12)(cid:12)(cid:12) ≪ p | τ | , where the last line follows from the fact that α < and we bounded the integral above by a Gaussianintegral. This leads to the claimed estimate as τ → (cid:3) Higher order estimates.
The main terms in the estimates of the previous section will not besufficient in proving the growth of the a ( n ), thus we need the following. Lemma 3.10.
Let α be as in Corollary 3.8. Then as τ → , (12) ϑ ( ατ ; τ ) = − i sin( πα ) q √− iτ (cid:0) − a q + a q + O ( q ) (cid:1) , (13) ϑ (cid:18)
12 + ατ ; τ (cid:19) = − √− iτ (cid:16) − i cos(2 πα ) q + 2 i cos(4 πα ) q + O ( q ) (cid:17) , where, a := 1 + 2cosh(2 πiα ) ,a := 1 + 2cosh(2 πiα ) + 2cosh(4 πiα ) . Proof.
The proof of Equation (12) follows directly by applying the technique in the proof of Corollary3.8.1 and observing that(1 − ax )(1 − ax )(1 − ax )(1 − a − x )(1 − a − x )(1 − a − x )(1 − x )(1 − x )(1 − x )= 1 − x − a − x − ax + a − x + x + ax + a x + a − x + O ( x ) . ASYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS
Similarly, Equation (13) follows from the generic expansion ∞ Y j =0 (cid:16) − ax − x j (cid:17) ∞ Y j =0 (cid:16) − a − x x j (cid:17) ∞ Y j =0 (cid:0) − xx j (cid:1) = 1 + a − ax − − ( a + a − ) x + O ( x ) . (cid:3) The a ( n )The function T ( τ ) can be simplified greatly. Proposition 4.11. T ( τ ) = 4 iq ϑ (cid:0) ; τ (cid:1) ϑ (96 τ ; 288 τ ) ϑ (cid:0) ; 24 τ (cid:1) ( q ϑ (4 τ ; 16 τ ) ϑ (68 τ ; 96 τ ) ϑ (cid:0) + 46 τ ; 96 τ (cid:1) ϑ (cid:0) + 22 τ ; 96 τ (cid:1) − q − ϑ (12 τ ; 16 τ ) ϑ (20 τ ; 96 τ ) ϑ (cid:0) + 94 τ ; 96 τ (cid:1) ϑ (cid:0) + 22 τ ; 96 τ (cid:1) + q − ϑ (4 τ ; 16 τ ) ϑ (20 τ ; 96 τ ) ϑ (cid:0) + 46 τ ; 96 τ (cid:1) ϑ (cid:0) + 70 τ ; 96 τ (cid:1) − q − ϑ (4 τ ; 16 τ ) ϑ (68 τ ; 96 τ ) ϑ (cid:0) + 94 τ ; 96 τ (cid:1) ϑ (cid:0) + 70 τ ; 96 τ (cid:1) ) . (14)We will prove this through a series of lemmas. To do so, let S ( τ ) := X r,s =0 n ( − r q Q ( r,s ) · ϑ (4( s − r ) τ ; 16 τ ) ϑ ( { r + s ) + 44 } τ ; 96 τ ) ϑ (cid:0) + { r + 22 } τ ; 96 τ (cid:1) ϑ (cid:0) + { s + 22 } τ ; 96 τ (cid:1) o := X r,s =0 ς ( r, s ; τ ) . Lemma 4.12. ς ( r, r ; τ ) = 0 .Proof. This follows directly from the fact that ϑ (0; 16 τ ) = 0. (cid:3) Lemma 4.13. ς ( s, r ; τ ) = − ( − r + s ς ( r, s ; τ ) .Proof. Notice that Q ( r, s ) = Q ( s, r ). Using the fact that ϑ ( − s − r ) τ ; 16 τ ) = − ϑ (4( s − r ) τ ; 16 τ )completes the proof. (cid:3) Proposition 4.14. S ( q ) = 2 ( ς (1 , τ ) + ς (2 , τ ) + ς (3 , τ ) + ς (3 , τ )) .Proof. Follows directly from Lemmas 4.12-4.13. (cid:3)
Using this simplification in Proposition 4.11, we will investigate the asymptotic growth of the coeffi-cients of R in the next section.4.1. Essential singularity at τ = . As was claimed in Section 3.2, we require higher order asymptoticexpansions to accurately determine the growth of the a ( n ). We break the study near τ = into twoparts: T ( τ ) and the Appell function, where Lemma 3.10 will prove useful for the study of T ( τ ).4.1.1. T ( τ ) near τ = . The first result involves the function ϑ ( ; τ ), appearing in the denominator of T ( τ ). First recall the eta multiplier, given by (See Theorem 5.8.1 of [8]) ε ( M ) := (cid:16) d | c | (cid:17) e πi ( ( a + d − c − bd ( c − ) if c is odd , (cid:16) c | d | (cid:17) e πi ( ( a − d ) c − bd ( c − d − ) if c is even , where M := (cid:18) a bc d (cid:19) and (cid:0) •• (cid:1) is the Jacobi symbol.Now we can prove the following: SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 9
Lemma 4.15.
Define w := τ − . As w → we have ϑ (cid:18)
12 ; τ (cid:19) = 2 e πi e − πi w √− w (1 + O ( e − πi w )) . Proof.
Let z := − kw − k and define the matrices A := (cid:18) (cid:19) ,B := (cid:18) (cid:19) . Then, since we have the well known formula ϑ (cid:0) ; τ (cid:1) = 2 η (2 τ ) η ( τ ) , ϑ (cid:18)
12 ; Az (cid:19) = 2 η (2 Az ) η ( Az ) = 2 η ( B (2 z )) η ( Az )= 2 ǫ ( B ) (2 z + 1) η (2 z ) ǫ ( A ) η ( z ) = 2 ǫ ( B ) (2 z + 1) e πi z (cid:0) e πiz ; e πiz (cid:1) ∞ ǫ ( A ) e πiz ( e πiz ; e πiz ) ∞ = 2 ǫ ( B ) ( kz + 1) e πi z (cid:0) e πiz ; e πiz (cid:1) ∞ ǫ ( A ) ( e πiz ; e πiz ) ∞ = ϑ (cid:18)
12 ; τ (cid:19) = 2 e πi e − πi w √− w (1 + O ( e − πi w )) , which proves the claim. (cid:3) Theorem 4.16.
Let Q := e − πiτ − and write τ = x + iy . Define y := √ n and let M > such that | x − | < M y . Then as n → ∞ T ( τ ) = − √ q τ − e πi Q − (cid:16) O ( e − π √ n ) (cid:17) . Proof.
We focus our attention on the term S ( q ) defined by, S ( τ ) := q ϑ (4 τ ; 16 τ ) ϑ (68 τ ; 96 τ ) ϑ (cid:0) + 46 τ ; 96 τ (cid:1) ϑ (cid:0) + 22 τ ; 96 τ (cid:1) − q − ϑ (12 τ ; 16 τ ) ϑ (20 τ ; 96 τ ) ϑ (cid:0) + 94 τ ; 96 τ (cid:1) ϑ (cid:0) + 22 τ ; 96 τ (cid:1) + q − ϑ (4 τ ; 16 τ ) ϑ (20 τ ; 96 τ ) ϑ (cid:0) + 46 τ ; 96 τ (cid:1) ϑ (cid:0) + 70 τ ; 96 τ (cid:1) − q − ϑ (4 τ ; 16 τ ) ϑ (68 τ ; 96 τ ) ϑ (cid:0) + 94 τ ; 96 τ (cid:1) ϑ (cid:0) + 70 τ ; 96 τ (cid:1) . We refer to the two outer/inner ϑ quotients (with the signs and powers of q ) as S /S and S /S respectively. That is, S ( τ ) = S ( τ ) + S ( τ ) + S ( τ ) + S ( τ ). Notice that S (cid:18) τ + 12 (cid:19) + S (cid:18) τ + 12 (cid:19) = S ( τ ) + S ( τ ) ,S (cid:18) τ + 12 (cid:19) + S (cid:18) τ + 12 (cid:19) = − S ( τ ) − S ( τ ) . Thus, we can capture the behavior near the cusp by investigating the behavior near 0. We can applyLemma 3.10 to the S i ( τ ), and we find that as τ → S ( τ ) + S ( τ ) = − √ (cid:16) π (cid:17) sin (cid:18) π (cid:19) q · n ( a + a − c − c ) q − (cid:18) (cid:18) πi (cid:19)(cid:19) ( a + a − c − c ) q · + O ( q ) o , where, a := 2 i cos (cid:18) π (cid:19) , a := 2 i cos (cid:18) π (cid:19) c := 2 i cos (cid:18) π (cid:19) , c := 2 i cos (cid:18) π (cid:19) . Similarly as τ → S ( τ ) + S ( τ ) = − √ (cid:16) π (cid:17) sin (cid:18) π (cid:19) q · n ( h + h − l − l ) q − (cid:18) (cid:18) πi (cid:19)(cid:19) ( h + h − l − l ) q · + O ( q ) o , where, h := 2 i cos (cid:18) π (cid:19) , h := 2 i cos (cid:18) π (cid:19) l := 2 i cos (cid:18) π (cid:19) , l := 2 i cos (cid:18) π (cid:19) . We then observe that,sin (cid:16) π (cid:17) sin (cid:18) π (cid:19) n a + a − c − c o − sin (cid:16) π (cid:17) sin (cid:18) π (cid:19) n h + h − l − l o = 0 , and sin (cid:18) π (cid:19) (cid:18) (cid:18) πi (cid:19)(cid:19) n a + a − c − c o − sin (cid:18) π (cid:19) (cid:18) (cid:18) πi (cid:19)(cid:19) n h + h − l − l o = 2 √ i. Thus, S ( q → −
1) = S ( q → − S ( q → − S ( q →
1) + S ( q → √ (cid:16) π (cid:17) )(2 √ i ) Q · (cid:16) O ( Q ) (cid:17) = 8 √ iQ · (cid:16) O ( Q ) (cid:17) . (15)We now turn our attention to the outside ϑ quotient on T ( τ ). Sending τ → τ + and using the appropriate ϑ transformations, we have4 iq ϑ (cid:0) ; τ (cid:1) ϑ (96 τ ; 288 τ ) ϑ (cid:0) ; 24 τ (cid:1) = τ → iq e πi ϑ (cid:0) ; τ + (cid:1) ϑ (96 τ ; 288 τ ) ϑ (cid:0) ; 24 τ (cid:1) . SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 11
Applying Lemma 4.15 and Lemma 3.8 leads to the near estimate4 iq e πi ϑ (cid:0) ; τ + (cid:1) ϑ (96 τ ; 288 τ ) ϑ (cid:0) ; 24 τ (cid:1) ∼ i (cid:0) − i sin (cid:0) π (cid:1)(cid:1) Q · ( − i ( τ − )) q − τ − ) Q − e πi − r − i ( τ −
12 ) ! = − √ e πi q τ − Q · − . (16)Combining Equations (15) and (16), we obtain the full estimate for T ( τ ) near τ = : T (cid:18) τ → (cid:19) = − √ i q τ − e πi Q · − · (cid:16) O ( Q ) (cid:17) = √ q τ − e πi Q − (cid:16) O ( Q ) (cid:17) , which proves the claim. (cid:3) The Appell sum near τ = . In order to simplify our calculations, we introduce the notation ˙=to mean equal up to a constant multiple and ˙ ∼ to mean asymptotic up to a constant multiple . Since ourAppell sum is invariant under the transformation τ τ + , it suffices to look at the behavior near τ = 0. Thus, using the transformation law of Proposition 2.2.3, µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) = − q √− iτ µ (cid:18)
112 + 148 τ , τ ; − τ (cid:19) + h (2 τ ; 24 τ )2 i . (17)The Mordell integral of this type grows as a polynomial in | τ | as was shown in Lemma 3.9. Looking solelyat the remaining Appell sum gives µ (cid:18)
112 + 148 τ , τ ; − τ (cid:19) = e πi e πi τ ϑ (cid:0) τ ; − τ (cid:1) X n ∈ Z ( − n q n n q − n − e πi q − q n ˙= q − q · q (cid:16) − q − + O ( q ) (cid:17) − e − πi q − e − πi q + O ( q ) ! ˙= q − q q · q − (cid:16) O ( q ) (cid:17) (cid:16) O ( q ) (cid:17) ˙ ∼ τ → q . Plugging this back into Equation (17) and using the polynomial estimate for the Mordell integral, wefind as τ → µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) ˙ ∼ p | τ | . Thus, we have the following.
Theorem 4.17.
The growth of R (3)3 near τ = is determined by the estimate in Theorem 4.16. Growth away from τ = . We now want to show that the growth at the other cusps is negligibleto that given in Theorem 4.16. Thus, our target is to beat the bound exponentially Q − q τ − ≪ n e π √ n . This is because when we do the integration in Section 4.4, we can bound the integral (for generic bounds a ) Z a − a e − πiτ e π √ n − ε dx ≪ e π √ n − ε for some ε >
0. Thus, we can incorporate the estimates away from into an error term, and ignore themin our final estimate for the a ( n ).4.3. Bounding the Appell sum away from τ = .Lemma 4.18. Let | x | > M y , and let a and b be positive integers with a < b . Furthermore, let y := δ √ n with δ > and for M > define the term ε := − √ M + 1 > . Then as n → ∞ (18) 1 ϑ ( aτ ; bτ ) ≪ n e δ √ n πb (cid:16) π − ε (cid:17) , (19) 1 ϑ ( + aτ ; bτ ) ≪ n e δ √ n πb (cid:16) π − ε (cid:17) , and (20) 1 ϑ ( ; bτ ) ≪ n e δ √ nπb (cid:16) π − ε (cid:17) . Remark 4.19.
The bounds above also hold for the functions ϑ ( aτ ; bτ ) , ϑ (cid:0) ; bτ (cid:1) and ϑ (cid:0) + aτ ; bτ (cid:1) whichcan by seen by replacing log( • ) with − log( • ) in the proof below. We will use this fact later on in Section4.3.1.Proof. The proof uses the same ideas as in [7, 9, 11] to prove their bounds away from the dominant pole.We recall the Taylor expansion for log(1 − z ) = − P n ≥ z n n . This implies,log (cid:18) q a ; q b ) ∞ ( q b − a ; q b ) ∞ ( q b ; q b ) ∞ (cid:19) = X n ≥ q an + q ( b − a ) n + q bn n (1 − q bn ) . (21)The trick now, as described by many works such as [7, 9, 11], is to extract the first term in the sum, andadd an extra term, which will be the first term in the expansion forlog (cid:18) | q | a ; | q | b ) ∞ ( | q | b − a ; | q | b ) ∞ ( | q | b ; | q | b ) ∞ (cid:19) . Explicitly, we have X n ≥ q an + q ( b − a ) n + q bn n (1 − q bn ) = X n ≥ q an + q ( b − a ) n + q bn n (1 − q bn ) + q a + q b − a + q b (1 − q b )+( | q | a + | q | b − a + | q | b ) (cid:18) − | q | b − − | q | b (cid:19) . SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 13
Taking the absolute value of this equation and using the fact that 1 − | q | b ≤ | − q b | , we have the upperbound (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) q a ; q b ) ∞ ( q b − a ; q b ) ∞ ( q b ; q b ) ∞ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≥ q an + q ( b − a ) n + q bn n (1 − q bn ) + q a + q b − a + q b (1 − q b )+ ( | q | a + | q | b − a + | q | b ) (cid:18) − | q | b − − | q | b (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ≥ | q | an + | q | ( b − a ) n + | q | bn n (1 − | q | bn ) + | q | a + | q | b − a + | q | b | (1 − q b ) | + ( | q | a + | q | b − a + | q | b ) (cid:18) − | q | b − − | q | b (cid:19) = X n ≥ | q | an + | q | ( b − a ) n + | q | bn n (1 − | q | bn ) + ( | q | a + | q | b − a + | q | b ) (cid:18) | − q b | − − | q | b (cid:19) = log (cid:18) | q | a ; | q | b ) ∞ ( | q | b − a ; | q | b ) ∞ ( | q | b ; | q | b ) ∞ (cid:19) + ( | q | a + | q | b − a + | q | b ) (cid:18) | − q b | − − | q | b (cid:19) . The log term can estimated by the asymptotic formulas derived for the ϑ functions in Lemma 3.8. Namely,as n → ∞ log (cid:18) | q | a ; | q | b ) ∞ ( | q | b − a ; | q | b ) ∞ ( | q | b ; | q | b ) ∞ (cid:19) ≪ log q bδ √ n sin ( π ab ) + δπ √ n b ≪ C a,b + log( n − ) + δπ √ n b , (22)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 | x | > M y . Following the procedure on page 10 of [7], wecan bound | − q b | by using the fact that cosine is a decreasing function near 0. Namely, as n → ∞| − q b | = 1 − πbx ) e − πyb + e − πby ≥ − πbM y ) e − πyb + e − πby = 1 − (cid:0) − π b M y + O ( y ) (cid:1) (cid:0) − πyb + 2 π b y + O ( y ) (cid:1) + 1 − πby + 8 π b y + O ( y ) ≫ π b y (1 + M ) . This implies that 1 | − q b | ≪ πby √ M . (23)For the other fraction, we have as n → ∞ − | q | b ≪ πby . (24)Combining equations (22), (23), and (24), we havelog (cid:18) q a ; q b ) ∞ ( q b − a ; q b ) ∞ ( q b ; q b ) ∞ (cid:19) ≪ C a,b + log( n − ) + δπ √ n b + 32 πby √ M − πby , which proves Equation (18). Equation (19) follows by noticing thatlog (cid:18) q b ; q b ) ∞ ( − q b − a ; q b ) ∞ ( − q a ; q b ) ∞ (cid:19) = X n ≥ ( − n ( q a + q b − a ) + q b n (1 − q bn ) ≪ log (cid:18) | q | a ; | q | b ) ∞ ( | q | b − a ; | q | b ) ∞ ( | q | b ; | q | b ) ∞ (cid:19) + ( | q | a + | q | b − a + | q | b ) (cid:18) | − q b | − − | q | b (cid:19) , as before.For Equation (20), we have that ϑ (cid:18) , τ (cid:19) = − q ( − q ) ∞ ( q ; q ) ∞ = − q ( − q ; q ) ∞ ( q ; q ) ∞ , which implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log ϑ (cid:0) , τ (cid:1) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − log (cid:18) − q − / (cid:19) + X n ≥ ( − n q n n (1 − q n ) + X n ≥ q n n (1 − q n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ B + X n ≥ | q | n n (1 − | q | n ) ≪ log( P ( | q | ) , where B is a constant and P ( q ) = q η ( τ ) . Using Lemma 4 . P ( | q | b ) ≪ n e δ √ nπb (cid:18) π + √ M − (cid:19) . This completes the proof of Equation (20). (cid:3)
We look at the non-normalized Appell sum(25) A (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) := ϑ (cid:18)
12 ; 24 τ (cid:19) µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) = − q X n ∈ Z ( − n q n + n ) q n +2 . 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 [7], but was not carried out explicitly. Proposition 4.20.
Let Θ( τ ) := X n ∈ Z q n . Then, (cid:12)(cid:12)(cid:12)(cid:12) A (cid:18) τ + 12 ,
12 ; 24 τ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Θ( iy )1 − | q | . Proof.
Splitting the sum in Equation (25) into negative and positive index, and then recombining, wefind A (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) = q q + q X n> ( − n q n + n ) (cid:18)
11 + q n +2 + q − q n − (cid:19) . Since | q | <
1, we have that 1 − | q | ≤ | q | . Combined with the fact that | q | m is an increasing functionin m , we have that, (cid:12)(cid:12)(cid:12)(cid:12) A (cid:18) τ + 12 ,
12 ; 24 τ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − | q | + X n> | q | n + n ) (cid:18)
21 + | q | n − (cid:19) ≤ − | q | X n> | q | n ! = 11 − | q | Θ( iy ) , which proves the claim. (cid:3) SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 15
Remark 4.21.
Recall that Θ( τ ) is a holomorphic modular form for the group Γ (4) . Γ (4) has threeinequivalent cusps represented by , , and ∞ . Recall that in Section 4.1.2 we computed the estimate near 0 and , which gives µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) ≪ p | τ | . Due to Proposition 4.20 and the corresponding remark, we can see we only need to check the growth ofΘ( τ ) near ∞ . since Θ( τ ) is modular, it’s growth at ∞ is at most O (1). Since under the transformation τ τ + the Jacobi theta remains unchanged, that is ϑ (cid:0) ; 24 τ (cid:1) ϑ (cid:0) ; 24 τ (cid:1) , we can use Lemmas 4.18and 4.20 to obtain: Theorem 4.22.
Let
M > such that < yM < (cid:12)(cid:12) x − (cid:12)(cid:12) . Then there is a β > such that as n → ∞ , µ (cid:18) τ + 12 ,
12 ; 24 τ (cid:19) ≪ e π √ n (1 − β ) n . Bounding T ( τ ) away from τ = . The estimates in Lemma 4.18 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 ph . However, we can use Lemma 4.18 in combination with generalizations ofLemma 4.15 and Lemma 3.8 to narrow down the problematic set of cusps to 5 specific cusps. To do so,we prove a series of three lemmas that give us the growth of all of the ϑ functions near a cusp ph . Wefirst note that T ( τ ) decays rapidly near 0. Lemma 4.23. As τ → , T ( τ ) ≪ e − πi τ . Proof.
This follows easily from Lemma 3.8. (cid:3)
To deal with the other cusps, we recall the fact that ϑ ( z ; τ ) is a Jacobi form of weight and index .Thus, the following properties hold [6, 10]: Remark 4.24 (See Chapter 2, [6]) . Let A = (cid:18) a bc d (cid:19) ∈ SL ( Z ) and λ, k ∈ Z . Then ϑ (cid:18) zcτ + d ; aτ + dcτ + d (cid:19) = χ ( A )( cτ + d ) e πicz cτ + d ϑ ( z ; τ ) ,ϑ ( z + λτ + k ; τ ) = β ( λ ) e − πi ( λ τ +2 λz ) ϑ ( z ; τ ) , where χ and β are multipliers. With this, we prove a series of three lemmas beginning with a generalization of Lemma 4.15. Through-out, we will work with the variable σ := − h w − dh for w ∈ H and d ∈ Z . Notice thatlim w → σ = −∞ . Lemma 4.25.
Let ( p, h ) = 1 , with p, h both positive integers. Then as τ → ph for h even ϑ (cid:18)
12 ; τ (cid:19) ˙ ∼ (cid:16) − h (cid:16) τ − ph (cid:17)(cid:17) − e − πi h ( τ − ph ) , and if h is odd, ϑ (cid:18)
12 ; τ (cid:19) ˙ ∼ (cid:16) − h (cid:16) τ − ph (cid:17)(cid:17) − . Proof.
Let A := (cid:18) p bh d (cid:19) with b, d so that A ∈ SL ( Z ), which exist since ( p, h ) = 1. Define z := hσ + d with σ := − h w − dh for w ∈ H . Notice that A ( σ ) = ph + w ( pd − hb ) = ph + w. Thus, lim w → A ( σ ) = ph . Regardless of whether h is even or odd, we have using the first equation of Remark4.24, ϑ (cid:18)
12 ; A ( σ ) (cid:19) ˙= ( hσ + d ) e πi h σ + hd ϑ (cid:18) hσ + d σ (cid:19) . (26)If h is even, d must be odd. Therefore, the second Equation of 4.24 implies that Equation (26) reducesto, ϑ (cid:18)
12 ; A ( σ ) (cid:19) ˙= ϑ (cid:18)
12 ; σ (cid:19) , which upon using the Jacobi triple product and taking the limit w → ϑ (cid:18)
12 ; A ( σ ) (cid:19) ˙ ∼ (cid:18) − hw (cid:19) e − πi h w . Subbing in w = τ − ph 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 and d are both half integers. Therefore using Remark4.24 as before, we have ϑ (cid:18)
12 ; A ( σ ) (cid:19) ˙= ( hσ + d ) e πi h σ ϑ (cid:18)(cid:18) ⌊ h ⌋ + 12 (cid:19) σ + ⌊ d ⌋ + 12 ; σ (cid:19) ˙= ( hσ + d ) e − πiσ (cid:16) ⌊ h ⌋ + ⌊ h ⌋− h (cid:17) ϑ (cid:18) σ + 12 ; σ (cid:19) , where ⌊•⌋ is the floor function. Using the Jacobi product again and taking the limit as w →
0, we have ϑ (cid:18)
12 ; A ( σ ) (cid:19) ˙ ∼ ( − hw ) − e πih w (cid:16) − h + ⌊ h ⌋ + ⌊ h ⌋ + (cid:17) = ( − w ) − . (27)The last step follows since h odd implies ⌊ h ⌋ = h − . If d is even, the only thing that changes in Equation(27) is that ϑ (cid:0) σ +12 ; σ (cid:1) becomes ϑ (cid:0) σ ; σ (cid:1) , which both yield the same estimate up to a constant factor as w → (cid:3) As stated at the beginning of this section, we claimed that there are only a finite number of cusps weneed to check. The following proposition gives us a rough bound for this number, but it more importantlytells us that all of the cusps that could cause a large pole have p = 1 and h even. Proposition 4.26.
The only cusps that could lead to T ( τ ) having larger growth than that at are cusps h with h even and h ≤ .Proof. We use the notation from the proof of Theorem 4.16, where we saw that we could write T ( τ ) as T ( τ ) ˙= ϑ (96 τ ; 288 τ ) ϑ (cid:0) ; τ (cid:1) ϑ (cid:0) ; 24 τ (cid:1) S ( τ ) . Referring back to Remark 4.19 we can use Lemma 4.18 to bound the combination D := ϑ (96 τ ; 288 τ ) S ( τ ) . Recalling that each one of the four terms in S ( τ ) is of the form ϑ ( • ; 16 τ ) ϑ ( • ; 96 τ ) ϑ (cid:0) + • ; 96 τ (cid:1) ϑ (cid:0) + • ; 96 τ (cid:1) , SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 17 and that the bounds in Lemma 4.18 only depend on the factor in the second slot, we find that D ≪ n − e √ √ n ( · π + · π + · π ) (cid:16) π − ε (cid:17) = n − e √ √ n ( π ) (cid:16) π − ε (cid:17) = n − e √ π √ n ( − π ε ) = n − e π √ n ( − π ε ) , where ε = 1 − √ M . If we choose M > π q − π ≈ . D ≪ n − e √ n ( − ) . We define G := 1 ϑ (cid:0) ; τ (cid:1) ϑ (cid:0) ; 24 τ (cid:1) . If h is odd, by Lemma 4.25, the product GD beats the target bound. Namely, GD ≪ e √ n (1 − β ) , for β >
0. Thus, we only need to consider even h . In order to beat the bound e √ n , we refer to Lemma4.25 and realize that we need to find the largest even h that satisfies the inequality25 h > , which is h = 34. (cid:3) Proposition 4.26 gives us a first estimate of the number of cusps we need to check, as well as reducingour investigation to the cusps with p = 1. As we will see in the next two lemmas, this helps us significantly.We now generalize the estimate in the first equation of Lemma 3.8 for functions of the form ϑ ( a τ ; b τ ) . Lemma 4.27.
Let ( p, h ) = 1 , with p, h both positive integers, and let a and b be positive integers suchthat h does not divide b . Define α := a b and let { x } denote the fractional part of x . Then as τ → ph , ϑ ( a τ ; b τ ) ˙ ∼ (cid:16) − hb (cid:16) τ − ph (cid:17)(cid:17) − e πih b τ − ph ) ( ( ⌊ αp ⌋ +2 ⌊ αp ⌋{ αp } + { αp } ) − − α p ) . Remark 4.28.
The cases when h divides both b reduces to the situation in the first and second equationsof Lemma 3.8.Proof. Let A and σ be as in the proof of Lemma 4.25. Define z := α ( pσ + b ). Then by the first Equationin Remark 4.24, ϑ ( αA ( σ ); Aσ ) = ϑ (cid:18) α (cid:18) pσ + bhσ + d (cid:19) ; Aσ (cid:19) ˙= ( cσ + d ) e πihz hσ + d ϑ ( z ; σ ) . (28)Dealing separately with ϑ ( z ; σ ) using the second equation of Remark 4.24, we have ϑ ( z ; σ ) = ϑ ( αpσ + αb ; σ ) = e − πi ( ⌊ αp ⌋ σ +2 ⌊ αp ⌋ ( { αp } σ + { αb } ) ) ϑ ( { αp } σ + { αb } ; σ ) . Since { αp } < w → ϑ ( { αp } σ + { αb } ; σ ) ˙ ∼ e πiσ −{ αp } πiσ . Therefore as Im( w ) → ϑ ( z ; σ ) ˙ ∼ e − πi ( ⌊ αp ⌋ σ +2 ⌊ αp ⌋{ αp } σ ) e πiσ −{ αp } πiσ . (29)Subbing (29) back into (28) and computing that as Im( w ) → e πihz hσ + d ˙ ∼ e πihα p σ hσ + d , We have ϑ ( αA ( σ ); Aσ ) ˙ ∼ ( − hw ) − e πihα p σ hσ + d e − πi ( ⌊ αp ⌋ σ +2 ⌊ αp ⌋{ αp } σ ) e πiσ −{ αp } πiσ ˙ ∼ ( − hw ) − e πih w ( ⌊ αp ⌋ +2 ⌊ αp ⌋{ αp } + { αp } ) − πi h w − πiα p h w = ( − hw ) − e πih w ( ( ⌊ αp ⌋ +2 ⌊ αp ⌋{ αp } + { αp } ) − − α p )Sending w b w and then w τ − ph proves the claim. (cid:3) We finally have to address functions of the form ϑ (cid:0) + a τ ; b τ (cid:1) . Lemma 4.29.
Let p, h, a , b , and α as in Lemma 4.27 with the restriction that h does not divide b . If λ := h +2 αp then, ϑ (cid:18)
12 + a τ ; b τ (cid:19) ˙ ∼ (cid:16) − h (cid:16) τ − ph (cid:17)(cid:17) − e πih b τ − ph ) ( ⌊ λ ⌋ +2 ⌊ λ ⌋{ λ } + { λ }− λ − ) . Remark 4.30.
The cases when h divides b are addressed in the first and second equation of Corollary3.8.Proof. Define z := h +2 αp σ + d +2 αb . Then, ϑ (cid:18)
12 + αA ( σ ); A ( σ ) (cid:19) = ϑ (cid:18) zhσ + d ; A ( σ ) (cid:19) . Using Remark 4.24, we have ϑ (cid:18) zhσ + d ; A ( σ ) (cid:19) = ( hσ + d ) e πihz hσ + d ϑ (cid:18) ( h + 2 αp )2 σ + d + 2 αb σ (cid:19) = ( hσ + d ) e πihz hσ + d e − πi ⌊ h +2 αp ⌋ σ +2 ⌊ h +2 αp ⌋ n h +2 αp o σ + n d +2 αb o !! ϑ (cid:18)n h + 2 αp o σ + n d + 2 αb o ; σ (cid:19) . (30)Let λ := h +2 αp . Then, as w → ϑ (cid:18)n h + 2 αp o σ + n d + 2 αb o ; σ (cid:19) ˙ ∼ e πiσ − πi { λ } σ , and e πihz hσ + d ˙ ∼ e πihλ σ hσ + d . Thus, taking the limit w → ϑ (cid:18) zhσ + d ; A ( σ ) (cid:19) ˙ ∼ ( − hw ) − e πih w ( ⌊ λ ⌋ +2 ⌊ λ ⌋{ λ } + { λ }− λ − ) . Sending w → b w and then w → τ − ph proves the claim. (cid:3) We can now prove the main result of this section.
Theorem 4.31.
Let
M > . For M y < (cid:12)(cid:12) x − (cid:12)(cid:12) , there exists a β > such that R (3)3 ( q ) ≪ e π √ n (1 − β ) . Proof.
We begin by recalling that the polynomial f ( x ) := ( x − x ) + restricted to the interval [0 , x = and x = 0 with the values f (cid:18) (cid:19) = 0 , (31) f (0) = 18 . (32) SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 19
Based on the form T ( τ ) given in Equation (4.11), we need to investigate functions of the form D := ϑ (96 τ ; 288 τ ) ϑ ( a τ ; 16 τ ) ϑ ( a τ ; 96 τ ) ϑ (cid:0) + a τ ; 96 τ (cid:1) ϑ (cid:0) + a τ ; 96 τ (cid:1) , for even integers a i . Additionally, we know we need to check cusps h with h even, so defining w := τ − h , γ i := a i (mod h ), and Q := e − πiw we know from Lemma 4.25 that G := 1 ϑ (cid:0) ; τ (cid:1) ϑ (cid:0) ; 24 τ (cid:1) ≪ | w | Q − · q . (33)We have two sub-cases to investigate depending on whether h |
96 or not. Assume that h |
96. Then wehave using Lemmas 4.27, 4.29, and Corollary 3.8 (noticing that a = 4 , ϑ ( a τ ; 16 τ ) ˙ ∼ | w | − Q · h | , and h | a , | w | − Q ( ( γ h ) − γ h ) + · h |
16 and h a , | w | − Q h else , (34) ϑ ( a τ ; 96 τ ) ˙ ∼ | w | − Q · h if h | a ,w | − Q ( ( γ h ) − γ h ) + · else , (35) ϑ (cid:18)
12 + a i τ ; 96 τ (cid:19) ˙ ∼ ( h | a i ,w | − Q ( ( γih ) − γih ) + · else . (36)Using the inequalities in Equations (31) and (32), we have(34) ≪ , (35) ≪ , (36) ≫ Q · . Combining this with (33), we find T ( τ ) ≪ ϑ (96 τ ; 288 τ ) Q − · Q − · h . For h > √ ≈ . β > T ( τ ) ≪ | w | − Q ≪ e π √ n (1 − β ) , which is our target bound. This means we need to check h = 4 , h
96. Recalling some of the objects from Lemma 4.29, notice that the equation (with λ i := h +2 α i ) ⌊ λ i ⌋ + 2 ⌊ λ i ⌋{ λ i } + { λ i } − λ i −
14 = α + α − > , (37)since for us α i := a i < a i = 22 , , ,
94 corresponding to the shifted ϑ functions that appearin the denominator of T ( τ ). Using Lemmas 4.27 and 4.29 in a similar manner to Equations (34)-(36) andmaking note of Equation (37) we can bound D as D ≪ | w | . Combining this with the estimate for G , we find T ( τ ) ≪ | w | − Q − · h . Setting this estimate equal to our target of Q , we find that there is a β > T ( τ ) ≪ Q − β for all h > Thus, the completion of the proof now boils down to bounding the growth of T ( τ ) near and . For h = 4, this is quite easy, since all of the coefficients in the modular variable of D are divisible by 4.Furthermore, all of the a i in the elliptic variables in the denominator of D are congruent to 2 (mod 4)and the elliptic variable + a i τ reduces to a i τ . Therefore, using the first equation of Lemma 3.8 D ≪ | w | − Q . Furthermore, G ≪ | w | Q − . Thus, T ≪ | w | − .The cusp at h = 6 follows similarly since all of the a i are congruent to 2 (mod 6), and the ellipticvariables in the denominator of D reduce to + a i τ . Using Lemma 4.27 with a = 4 ,
12, we have ϑ ( a τ ; 16 τ ) ≪ | w | − Q · . For the rest of the functions in D , we use the first two equations of Lemma 3.8 to find near D ≪ | w | − Q ,G ≪ | w | Q − · . Combining G and D , we find that as τ → , T ( τ ) ≪ | w | − Q − ≪ | w | − Q − β , for some β >
0, which completes the proof. (cid:3)
Integration and proof of Theorem 1.1 for a ( n ) . We follow the approach of [7] by approximatingour integral with Bessel functions. We take the standard counter clockwise path around the origin γ := { e − πix : x ∈ ( − , ] } . By Cauchy’s theorem, we have a ( n ) = Z γ R (3)3 ( q ) q − n dx = I + I , where I := Z | x − | 12 ) q τ − e − πiτn dx + E , where E ≪ n . Dealing with the remaining integral, we use the substitution w := τ + , and then w = ivy (with y = √ n ) to obtain, e πi √ Z | x − | 12 ) q τ − e − πiτn dx = − ( − n √ i √ y I , (38) SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 21 where we define I := Z iM − iM e u ( v + v ) √ v dv and u := √ nπ . Lemma 7 of [7] gives the asymptotic expansion for I for such a u : I = i i √ / n / e π √ n + O e π √ n n / ! . Subbing back into Equation (38) and sending n → ∞ gives I ∼ a ( n ) ∼ ( − n √ √ n e π √ n . The b ( n )We now turn our attention to R (3)1 ( q ) =: P n ≥ b ( n ). To begin, we show that the b ( n ) form a weaklyincreasing sequence. Lemma 5.32. 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 ) < .Proof. The proof is a direct proof from the definition of ν ( − q ): ν ( − q ) := X n ≥ q n + n ( q ; q ) n +1 . Let f ( n ) := n + n . Then f ( n ) is clearly positive and increasing as a function of n . Additionally, wedefine the series 1( q ; q ) n +1 = n Y j =0 X l j ≥ q (2 j +1) l j := X h ≥ a ( h ; n ) q h := A ( n ) . The a ( h ; n ) count the number of partitions of h < n + 1 with strictly odd parts. Thus, the a ( h ; n ) forma weakly increasing sequence in h and n with no negative values for all n, h ≥ ν ( − q ) = X n ≥ q f ( n ) A ( n ) = X n,h ≥ a ( h, n ) q f ( n )+ h =: X m ≥ b ( m ) q m . Thus, b ( m ) = X n,h> ,f ( n )+ h = m a ( h, n ) . Since f ( n ) is increasing, we have r ( m + 1) = X n,h> ,f ( n )+ h = m +1 a ( h, n ) ≥ b ( m ) , which completes the proof. (cid:3) The following Tauberian Theorem allows us to capture the growth of the b ( n ) by only computing anestimate for the growth of R (3)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 regardingthe growth of functions in angular regions around the origin. Theorem 5.33 (See Theorem 1.1 of [5] with α = 0) . Let c ( n ) denote the coefficients of a power series C ( q ) := P ∞ n =0 c ( n ) q n with radius of convergence equal to . Define z := x + iy ∈ C . If the c ( n ) arenon-negative, are weakly increasing, and we have as t → + that C ( e − t ) ∼ λt α e At , and if for each M > such that | y | ≤ M | x | C ( e − z ) ≪ | z | α e A | z | with A > , then as n → ∞ c ( n ) ∼ λA α + √ πn α + e √ An . Remark 5.34. We will show the bound in Theorem 5.33 for R (3)1 as τ → with τ ∈ H , which is sufficientto show the bound for general z since we can define an even extension of R (3)1 into the lower half planeto get a function on all of C . Growth near τ = 0 . We focus on the Appell sum µ (cid:0) , ; − τ (cid:1) appearing in Equation (11): µ (cid:18) , 14 ; − τ (cid:19) = e πi ϑ (cid:0) ; − τ (cid:1) X n ∈ Z ( − n q n ( n +1)24 e nπi − e πi q n = e πi ϑ (cid:0) ; − τ (cid:1) ( − e πi + X n> ( − n q n ( n +1)24 e πin − e πi q n + ( − n q n ( n − e − nπi − e πi q − n ) . (39)The last line follows by splitting the sum into n < n > 0, and then swapping n 7→ − n in the sumover n < 0. We then have, X n> ( − n q n ( n +1)24 e πin − e πi q n + ( − n q n ( n − e − nπi − e πi q − n = X n> ( − n q n ( n +1)24 e πin − e πi q n − ( − n e − πi q n q n ( n − e − nπi − e − πi q n = X n> ( − n q n ( n +1)24 e πin − e πi q n − e − nπi e − πi − e − πi q n ! = X n> ( − n q n ( n +1)24 e πin X l ≥ e πil q nl − e − nπi e − πi X k ≥ e − πik q nk ! = O ( q ) . We sub this back into Equation (39) to obtain µ (cid:18) , 14 ; − τ (cid:19) = − i sin (cid:0) π (cid:1) ϑ (cid:0) ; − τ (cid:1) (cid:16) O ( q ) (cid:17) . (40)We can use the triple product formula to deal with the ϑ function. Namely, ϑ (cid:18) 14 ; − τ (cid:19) = − ie − πi q ( q ; q ) ∞ ( i ; q ) ∞ ( − iq ; q ) ∞ = − ie − πi (1 − i ) q ( q ; q ) ∞ ( iq ; q ) ∞ ( − iq ; q ) ∞ ∼ τ → − (cid:16) π (cid:17) q . (41) SYMPTOTIC EXPANSIONS FOR BAILEY-TYPE MOCK THETA FUNCTIONS 23 Subbing Equations (40) and (41) into (11) and using Lemma 3.9, we have as τ → µ (5 τ, τ ; 12 τ ) = − q µ (cid:0) , ; − τ (cid:1) √− iτ + h (2 τ ; 12 τ )2 i = − q − i √− iτ sin (cid:0) π (cid:1) ϑ (cid:0) ; − τ (cid:1) (cid:16) O ( q ) (cid:17) + O ( | τ | − ) ∼ q i √− iτ sin (cid:0) π (cid:1) (cid:16) − (cid:0) π (cid:1) q (cid:17) ∼ − q − i sin (cid:0) π (cid:1) sin (cid:0) π (cid:1) √− iτ , where we used in the last step that q → τ → 0. Therefore, we can state the following. Theorem 5.35. As τ → , we have the estimate µ (5 τ, τ ; 12 τ ) ∼ − e πi τ i sin (cid:0) π (cid:1) sin (cid:0) π (cid:1) √− iτ . We now show that the eta product η ( τ + ) η (3 τ + ) η (12 τ ) η (2 τ ) η (6 τ ) , has similar growth near τ = 0. To get the behavior near 0 of the eta function involving the shift, wecan proceed as we did in the proof of Lemma 4.15. Define the transformation A := (cid:18) (cid:19) , with w := − τ − and then send τ → 0. We have that(42) η ( Aτ ) = ǫ ( A )(2 τ + 1) η ( τ ) = e − πi (2 τ + 1) η ( τ ) . Corollary 3.8 and Equation (42) say that near 0, η ( Aw ) = e − πi (2 w + 1) η ( w ) = e − πi (2 w + 1) e πiw ( e πiw ; e πiw ) ∞ ∼ e − πi (cid:18) − τ (cid:19) e − πi τ = ie − πi √ τ q . (43)Therefore as τ → η (cid:18) τ + 12 (cid:19) ∼ η ( Aw ) ∼ ie − πi √ τ q , (45) η (cid:18) τ + 12 (cid:19) ∼ η ( A (3 w )) ∼ ie − πi √ τ q . The other eta products satisfy the estimates near zero directly from Lemma 3.8 using the substitutions τ τ , τ τ , and τ τ respectively, we have as τ → η ( τ + ) η (3 τ + ) η (12 τ ) η (2 τ ) η (6 τ ) ∼ i e − πi √− iτ q − . (46) Proof of the estimate for the b ( n ) .Theorem 5.36. Let r ( n ) denote the coefficients of ν ( − q ) . Then as n → ∞ , r ( n ) ∼ (cid:0) π (cid:1) sin (cid:0) π (cid:1) + 1 ! e π √ n √ n . Proof. Combining Theorem 5.35 and Equation (46), we have as τ → ν ( − q ) = R ( q ) ∼ i q − i sin (cid:0) π (cid:1) sin (cid:0) π (cid:1) √− iτ + 1 √− iτ q = e πi τ √− iτ (cid:16) (cid:0) π (cid:1) sin (cid:0) π (cid:1) + 1 (cid:17) , where q − → τ → 0. Making the substitution τ := it π , we have that as t → + that R ( e − t ) = (cid:0) π (cid:1) sin (cid:0) π (cid:1) + 1 ! r π e π t √ t . The bound for the complex variable, z , in Theorem 5.33 is trivially satisfied by combining the estimatesin Theorem 5.35 and Equation (46).Define A := π and λ := (cid:18) ( π ) sin ( π ) + 1 (cid:19) p π . By Theorem 5.33 with α = , we have that r ( n ) ∼ λ √ πn e √ An = (cid:0) π (cid:1) sin (cid:0) π (cid:1) + 1 ! r π √ π √ n e √ n ∼ (cid:0) π (cid:1) sin (cid:0) π (cid:1) + 1 ! e π √ n √ n , which shows the claim. (cid:3) Conclusions and future work This work studied the estimates for the Fourier coefficients a ( n ) and b ( n ) for the base cases of the R ( k )1 and R ( k )3 , respectively. Both of this families are mock theta families that were derived from Bailey chainsin [18]. We expect that generalizing to k > θ ,p function that we encountered with p = 4 in this work, generalize for p > Q ( r, s ) that appears in the exponent of q in the sum of θ ,p , which willallow for simpler expressions for the θ ,p like we found in this work. Albeit possible to do without, itwould be nice to find more elegant methods for dealing with the asymptotics for these families of Baileymock theta functions. Based on numerical checks of the Fourier coefficients, we expect that the R ( k )1 haveweakly increasing coefficients for k > Conjecture 1. The R ( k )1 have weakly increasing coefficients for all k ≥ . Proving this by purely combinatorial means seems difficult, but possible using the many representationsof R ( k )1 given by Lovejoy and Osburn in [18]. One such way may involve appealing to some generalized q -binomial theorems and formulae for Gauss sums, like those posed in [16]. 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