Representations of mock theta functions Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

Representations of mock theta functions

Dandan Chen^a, Liuquan Wang^b,∗

aSchoolofMathematicalSciences,EastChinaNormalUniversity,Shanghai 200241,People’sRepublicofChina

bSchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,Hubei, People’sRepublicofChina

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received17December2018 Receivedinrevisedform14October 2019

Accepted3February2020 Availableonlinexxxx

CommunicatedbyGeorgeAndrews DedicatedtoProfessorBruceBerndt ontheoccasionofhis80thbirthday MSC:

05A30 11B65 33D15 11E25 11F11 11F27 11P84

Keywords:

Mockthetafunctions Hecke-typeseries Appell-Lerchseries

3φ2Summationformulas

MotivatedbytheworksofLiu,weprovideauniﬁedapproach to ﬁnd Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters a and b.

Specializing the choices of (a,b), we not only give various known and new representations for the mock theta func- tionsoforders2,3,5,6 and8,butalso presentmany other interesting identities. We ﬁnd that some mock theta functions of diﬀerent orders are related to each other, in the sense that their representations can be deduced from the same(a,b)-parameterizedidentity.Furthermore,weintroduce the concept of false Appell-Lerch series. We then express theAppell-Lerchseries,falseAppell-LerchseriesandHecke- typeseriesinthispaperusingthebuildingblocksm(x,q,z) andf_a,b,c(x,y,q) introducedbyHickersonandMortenson,as wellasm(x,q,z) andf_a,b,c(x,y,q) introducedinthispaper.

We also show the equivalences of our new representations forseveralmock thetafunctionsandtheknownrepresenta- tions.

* Correspondingauthor.

E-mailaddresses:ddchen@stu.ecnu.edu.cn(D. Chen),wanglq@whu.edu.cn,mathlqwang@163.com (L. Wang).

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Contents

1. Introduction . . . . 2

2. Preliminaries . . . . 8

2.1. Usefulbasichypergeometricseriesidentities . . . . 9

2.2. BuildingblocksforAppell-Lerchseries,falseAppell-LerchseriesandHecke-typeseries 12 3. Mockthetafunctionsoforder2 . . . 18

3.1. RepresentationsforA⁽²⁾(q) . . . 20

3.2. RepresentationsforB⁽²⁾(q) . . . 23

3.3. Representationsforμ⁽²⁾(q) . . . 25

4. Mockthetafunctionsoforder3 . . . 27

4.1. Representationsforf⁽³⁾(q) . . . 28

4.2. Representationsforφ⁽³⁾(q) . . . 28

4.3. Representationsforψ⁽³⁾(q) . . . 30

4.4. Representationsforω⁽³⁾(q) . . . 35

4.5. Representationsforν⁽³⁾(q) . . . 35

4.6. Representationsforρ⁽³⁾(q) . . . 36

5.1. Representationsforf₀⁽⁵⁾(q) . . . 39

5.2. Representationsforφ⁽⁵⁾₀ (q) andψ⁽⁵⁾₀ (q) . . . 40

5.3. RepresentationsforF₀⁽⁵⁾(q) . . . 43

5.4. Representationsforf₁⁽⁵⁾(q) . . . 46

5.5. Representationsforφ⁽⁵⁾₁ (q) andψ⁽⁵⁾₁ (q) . . . 47

5.6. RepresentationsforF₁⁽⁵⁾(q) . . . 48

6.1. Representationsforφ⁽⁶⁾(q) . . . 51

6.2. Representationsforψ⁽⁶⁾(q) . . . 53

6.3. Representationsforρ⁽⁶⁾(q),σ⁽⁶⁾(q) andλ⁽⁶⁾(q) . . . 54

6.4. Representationsforμ⁽⁶⁾(q) . . . 55

6.5. Representationsforγ⁽⁶⁾(q) . . . 58

6.6. Representationsforφ⁽⁶⁾₋ (q) . . . 59

6.7. Representationsforψ₋⁽⁶⁾(q) . . . 60

7.1. RepresentationsofS⁽⁸⁾₀ (q) andS₁⁽⁸⁾(q) . . . 62

7.2. RepresentationsofT₀⁽⁸⁾(q) . . . 62

7.3. RepresentationsforT₁⁽⁸⁾(q),U₀⁽⁸⁾(q),U₁⁽⁸⁾(q) . . . 63

7.4. RepresentationsforV₀⁽⁸⁾(q) . . . 63

7.5. RepresentationsforV₁⁽⁸⁾(q) . . . 65

8. Concludingremarks . . . 67

Acknowledgments . . . 70

References . . . 70

1. Introduction

In his last letter to Hardydated on January12, 1920, Ramanujan gavea list of 17 functionswhichhecalled“mockthetafunctions”.Hedeﬁnedeachfunctionasaq-series in Eulerian form and separatedthem into four classes:oneclass of thirdorder, two of

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ﬁfthorder,andoneofseventhorder.Ramanujanalsostatedidentitiessatisﬁedbymock theta functions of the same order. In his lost notebook [39], identities for mock theta functionsof thesixthand tenthorderswererecorded.Sincethen, mockthetafunctions haveattracted theattentionofmanymathematicians.

It wassometime before researchersunderstoodthe modularityof mocktheta functions. From the Eulerian forms of mock theta functions, it is difficult to observe any significant transformation properties. Therefore, finding alternative representations for mockthetafunctionsbecomesthefirsttaskforstudyingtheirmodularbehaviors.With thecontributionofmanyworks,Watson[46,47],Andrews[4],AndrewsandHickerson[8], Berndtand Chan[10],Choi[14–17], Garvan[21],Gordonand McIntosh[24],Hickerson [28], andZwegers [50], toname afew,we nowknow thatmockthetafunctionsusually admitatleastoneofthetwokindsofrepresentations:Appell-LerchseriesorHecke-type series. A complete list of Appell-Lerch series representations for classical mock theta functionscanbefound in[29,Section5].

Appell-Lerchseriesareseriesoftheform ∞

n=−∞

(−1)ⁿq^n(n+1)/2bⁿ

1−aqⁿ . (1.1)

Here and throughout this paper, we assume that |q| <1. After multiplying the series (1.1) by thefactora^/2 andviewing itas functioninthe variablesa,b and q, itis also usually refereed as a level Appell function. This kind of series was ﬁrst studied by Appell[9] andLerch[31].

AseriesisofHecke-typeifithasthefollowingform:

(m,n)∈D

(−1)^H(m,n)qQ(m,n)+L(m,n),

whereH andLare linearforms,Qisaquadratic form,and Dis somesubsetofZ×Z such thatQ(m,n) ≥0 for every (m,n) ∈ D. Historically Q(m,n) was assumed to be indefinite(see[3] forexample).HereweallowQ(m,n) tobedefiniteaswell.Thefollowing classicalidentity of Jacobi expresses aninfinite product as aHecke-type series [3, Eq.

(3.15)]:

(q;q)³_∞= ∞ n=−∞

m≥|n|

(−1)^mq^(m²^+m)/2.

Hereandlater weusethestandardq-seriesnotation:

(x;q)₀:= 1, (x;q)_n:=

n−1

k=0

(1−xq^k), (x)_∞= (x;q)_∞:=

∞ k=0

(1−xq^k).

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Motivated by Jacobi’s identity, Hecke [26] systematically investigated theta series related toindeﬁnitequadraticforms.Forinstance,Hecke[26, p.425] foundthat

∞ n=−∞

|m|≤n/2

(−1)^m+nq⁽ⁿ²^−3m²)/2+(n+m)/2= ∞ n=1

(1−qⁿ)²,

whichisoriginallyduetoRogers[40,p.323].KacandPeterson[30] illustratedwaysfor proving Hecke-typeidentitiesusingaﬃneLiealgebra.

Appell-LerchseriesandHecke-typeseriesplayedimportantrolesinq-series.Itserves asbridgesbetweenmockthetafunctionsandthetheoryofmodularforms.Forexample, forthethirdordermockthetafunction¹

f⁽³⁾(q) :=

∞ n=0

qⁿ²

(−q;q)²_n, (1.2)

Watson [46] foundthefollowingAppell-Lerchseriesrepresentation f⁽³⁾(q) = 2

(q;q)_∞ ∞ n=−∞

(−1)ⁿq³²ⁿ²⁺¹²ⁿ

1 +qⁿ . (1.3)

Fortheﬁfthorder mockthetafunction f₀⁽⁵⁾(q) :=

∞ n=0

qⁿ² (−q;q)n

, (1.4)

Andrews [4] showedthatithasaHecke-typeseriesexpression as:

f₀⁽⁵⁾(q) = 1 (q;q)_∞

∞ n=0

|j|≤n

(−1)^jq⁵²ⁿ²⁺¹²^n−j²(1−q⁴ⁿ⁺²). (1.5)

ByusingtheAppell-LerchseriesorHecke-typeseriesexpressionsofmockthetafunctions suchas(1.3) and(1.5),Zwegers[49] successfullyﬁtmockthetafunctionsintothetheory of modular forms. For more detailed introduction to the developments of mock theta functions, we refer the reader to the survey of Gordon and McIntosh [25], the paper of Hickerson and Mortenson [29], the recent books of Andrews and Berndt [6], and Bringmann etal.[11] aswellas thereferenceslistedthere.

There are several ways for establishing Appell-Lerch series or Hecke-type series expressions for mock theta functions. To deduce Appell-Lerch series representations of third order mocktheta functions, Watson used atransformation formulaconnecting a

1 Throughout this paper, toavoid confusion,we use a superscript (n) to indicatethata mock theta functionisofordern.

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terminated 8φ7 series to a terminated 4φ3 series. Andrews [4] used Bailey chain the- orytoproduce Hecke-typeseriesexpressionsfortheﬁfthandseventhordermocktheta functions. Liu [32] derived some q-series expansion formulas and gave new proofs to (1.5). Ina series of works, Liu[33,34] established some transformation formulas for q- hypergeometricseriesandtherebyprovedmanyinterestingHecke-typeidentitiessuchas (see[34,Proposition1.11])

∞ n=0

(q;q²)nqⁿ

(q²;q²)_n = (q;q²)_∞ (q²;q²)_∞

∞ n=0

n j=−n

(−1)^n+jqⁿ²^+n−j². (1.6) AmonganumberofnicetransformationformulasofLiu,thefollowingonehasshown itspowerinestablishingHecke-typeidentities(see[33,Theorem1.7] or[34,p.2089]).

Theorem1.1. Formax{|uab/q|,|ua|,|ub|,|c|,|d|}<1,wehave² (uq, uab/q;q)_∞

(ua, ub;q)_∞ ³φ₂

q/a, q/b, v c, d ;q,uab

q

= ∞ n=0

(1−uq²ⁿ)(u, q/a, q/b;q)n

(1−u)(q, ua, ub;q)_n (−uab)ⁿq⁽ⁿ²⁻^3n)/2×3φ₂

q⁻ⁿ, uqⁿ, v c, d ;q, q

. (1.7) Utilizing Theorem 1.1, several new Hecke-type identitieshave been found byWang andYee[44,45].Inparticular,theyprovedthat[45, Theorem1.1]

∞ n=1

qⁿ(q;q²)n

(−q;q²)n(1 +q²ⁿ) = ∞ n=1

|j|≤n

(−1)^jqⁿ²^+j²−^∞

n=1

(−1)ⁿq²ⁿ². (1.8) Note thatthis isaHecke-type identity associated with adeﬁnite thetaseries, which is quiterareintheliterature.Furthermore,Wang[43] usedTheorem1.1togivenewproofs forﬁvefalsethetafunctionidentitiesofRamanujan.Inarecentwork,ChanandLiu[12]

usedTheorem1.1 toestablishthreeHecke-typeidentitiessuchas ∞

n=1

(q;q)_n (−q;q)n

(−1)ⁿqⁿ⁽ⁿ⁻^1)/2= ∞ n=1

n j=−n+1

(1−qⁿ)²(−1)^n+j+1q²ⁿ²⁻ⁿ⁻^j². (1.9) Inthis paper, we continue to employTheorem 1.1 to illustrate a systematicway to establish various representations for q-series in the Eulerian form. We ﬁrst generalize Theorem1.1to thefollowingform.

Theorem 1.2. Suppose max{|αabz/q|,|αa|,|αb|,|αc₁|,· · ·,|αc_m|} < 1 and m is a non- negativeinteger.Wehave

2 Fortheconvergence,weonlyneedtoassumethat|uab/q|<1.Werequiremax{|ua|,|ub|,|c|,|d|}<1 sothatthedenominatorofeachtermappearingintheidentitydoesnotvanish.

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(αq, αab/q;q)_∞

(αa, αb;q)_∞ ^m+2φm+1

q/a, q/b, αb1,· · ·, αbm

0, αc₁,· · ·, αc_m ;q, αabz/q

= ∞ n=0

(1−αq²ⁿ)(α, q/a, q/b;q)n(−αab/q)ⁿq^n(n−1)/2 (1−α)(q, αa, αb;q)_n

×m+2φ_m+1

q⁻ⁿ, αqⁿ, αb1,· · ·, αbm

0, αc₁,· · ·, αc_m ;q, zq

. (1.10)

Theorems 1.1 and 1.2provide an elegant way forﬁnding alternativerepresentations for basic hypergeometric series. Indeed, these theorems allow us to write a series in Eulerian formas asuminvolvingtruncatedm+2φm+1series. Insuitablesituations,this truncatedsummaybefurthersimpliﬁedandinturnweobtainverynicerepresentations oftheoriginalseries.SuchnicerepresentationsinvolveAppell-LerchseriesorHecke-type series. For example, using Theorem 1.1 we establish the following (a,b)-parameterized identity (seeTheorem5.12):formax{|ab|,|aq²|,|bq²|}<1,

(q², ab;q²)_∞ (aq², bq²;q²)_∞³φ2

q²/a, q²/b, q² 0, q³ ;q², ab

= (1−q) ∞ n=0

(1 +q²ⁿ⁺¹)(q²/a, q²/b;q²)_n (aq², bq²;q²)n

(−ab)ⁿq³ⁿ²⁺²ⁿ n j=−n

q^−2j²^−j. (1.11)

Itturnsoutthatbychoosingsuitablevaluesfor(a,b) inthisidentity,wegetHecke-type seriesrepresentationsforfourmockthetafunctionsoforders5,6and8.Namely,weﬁnd that

F₁⁽⁵⁾(q) :=

∞ n=0

q^2n(n+1) (q;q²)n+1

= 1

(q²;q²)_∞ ∞ n=0

2n j=0

(−1)ⁿq⁵ⁿ²⁺⁴ⁿ⁻(^j+1₂ )(1 +q²ⁿ⁺¹), (1.12) ψ₋⁽⁶⁾(q) :=

∞ n=1

qⁿ(−q;q)2n−2

(q;q²)_n =q(−q;q)_∞ (q;q)_∞

∞ n=0

(−1)ⁿq³ⁿ³⁺³ⁿ n j=−n

q^−2j²^−j, (1.13)

T₁⁽⁸⁾(q) :=

∞ n=0

qⁿ⁽ⁿ⁺¹⁾(−q²;q²)n

(−q;q²)_n+1

=(−q²;q²)_∞ (q²;q²)_∞

∞ n=0

q⁴ⁿ²⁺³ⁿ(1−q²ⁿ⁺¹) n j=−n

(−1)^jq^−2j²^−j, (1.14)

V₁⁽⁸⁾(q) :=

∞ n=0

q⁽ⁿ⁺¹⁾²(−q;q²)n

(q;q²)n+1

=q(−q;q²)_∞ (q²;q²)_∞

∞ n=0

(−1)ⁿq⁴ⁿ²⁺⁴ⁿ n j=−n

q^−2j²^−j. (1.15)

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Furthermore,theparameterizedidentity(1.11) alsogeneratesnewHecke-typeidentities.Forinstance,ifwetake(a,b)→(0,1) in (1.11),wegetthefollowingidentitywhich seemsto benew:

∞ n=0

(−1)ⁿ (q²;q²)_n (q;q²)n+1

qⁿ⁽ⁿ⁺¹⁾= ∞ n=0

n j=−n

(1 +q²ⁿ⁺¹)q⁴ⁿ²⁺³ⁿ⁻^2j²⁻^j. (1.16) By establishing diﬀerent (a,b)-parameterized identities, we are able to provide dif- ferentrepresentationsfor thesamemocktheta function.Anexample isthatusing two parameterizedidentities other than(1.11),we ﬁnd two Appell-Lerch series representa- tionsforV₁⁽⁸⁾(q).Namely,

V₁⁽⁸⁾(q) = (−q⁴;q⁴)_∞ (q⁴;q⁴)_∞

∞ n=−∞

(−1)ⁿq⁽²ⁿ⁺¹⁾²

1−q⁴ⁿ⁺¹ (1.17)

=q(−q;q)_∞ (q;q)_∞

∞ n=−∞

(−1)ⁿqⁿ²⁺²ⁿ

1 +q⁴ⁿ⁺² . (1.18)

Theformula(1.15) wasfoundbySrivastava[41] andCui,GuandHao [18] usingBailey pairs,and(1.17) isduetoGordonandMcIntosh[24].Therepresentation(1.18) appears tobe new,andwewillshowthatitisequivalentto(1.17) (seeSection7.5).

Wewillpresent24parameterizedidentitieslike(1.11).Threeofthemwerediscovered byLiu [33]. Bychoosing suitablevaluesfor theparameters, we provide newproofs for most of the known Appell-Lerch series or Hecke-type series representations for mock theta functions of orders 2, 3, 5, 6 and 8. Meanwhile, we will also show many new Hecke-typeidentitiesassociatedtodeﬁnite orindeﬁnitequadraticforms.

Inorder to write Appell-Lerchseries or Hecke-type series representations inastan- dardwayand thushavingclearer understandingoftheirmodularproperties,Hickerson and Mortenson [29] introduced two functions m(x,q,z) and f_a,b,c(x,y,q) (see (2.19) and (2.20) fordeﬁnition). They serve as building blocksfor Appell-Lerch series and a largefamilyofHecke-type series,respectively.Moreover,their modularpropertieshave beenwell studiedbyZwegers [49].Hickerson andMortenson[29] also developedanap- proachtoexpressaHecke-typeseriesintermsofAppell-Lerchseries.Inparticular,they [29, Section 5] gave representations for all classical mock theta functions in terms of m(x,q,z).

In this paper, we will follow [29] and write the Appell-Lerch series and Hecke-type seriesweobtainedintermsofthesebuildingblocks.Alongthisprocess,weﬁndthatthere arecertainserieswhoseshape issimilar toAppell-Lerchseriesbutcannotbeexpressed intermsofm(x,q,z).Forexample,in(3.15) weestablishthefollowing identity

∞ n=0

(−1)ⁿqⁿ⁽ⁿ⁺¹⁾(q²;q²)n

(q;q²)²_n+1 = ∞ n=0

q²ⁿ²⁺²ⁿ 1−q²ⁿ⁺¹ − ⁻

1 n=−∞

q²ⁿ²⁺²ⁿ

1−q²ⁿ⁺¹. (1.19)

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This doesnot satisfythe deﬁnitionof Appell-Lerch series in(1.1). Similarly,there are someHecke-typeserieswhichseemsnotexpressiblebyfa,b,c(x,y,q).Anexampleis(1.8) (see(2.46)).Therefore,weintroducetwonewfunctionsm(x,q,z) andf_a,b,c(x,y,q) (see (2.37) and(2.38)).Sincethesummandsinm(x,q,z) andm(x,q,z) diﬀerbysigns,wecall m(x,q,z) afalseAppell-Lerchseries.Wewillexpressalmost alltheAppell-Lerchseries, falseAppell-LerchseriesandHecke-typeseriesinthis paperusingthesebuildingblocks m(x,q,z), fa,b,c(x,y,q), m(x,q,z) and f_a,b,c(x,y,q). In particular, we use the method in[29] toconvertarepresentation intermsoffa,b,c(x,y,q) toarepresentationinterms of m(x,q,z).

Besides seeing the modularity of a series clearly, there is another advantage for ex- pressingAppell-LerchseriesandHecke-typeseriesusingm(x,q,z) orf_a,b,c(x,y,q).That is, wecanuse propertiesof these buildingblocksto transformbetweendiﬀerent forms.

By doing this, we can see if diﬀerent series representations are equivalent or not. For example, after writing (1.17) and (1.18) in terms of m(x,q,z) and using propertiesof m(x,q,z) establishedin[29],weﬁndthattheyareinfactequivalent (seeSection7.5).

The paper is organized as follows. In Section 2, we ﬁrst recall some formulas from the theory of basichypergeometricseries. We also discuss somelimiting casesof Wat- son’s q-analogofWhipple’stheorem.Theformulas listedinSection2.1 willbe used in evaluatingcertainterminated3φ2series,whicharefundamentalforestablishingparame- terizedidentities.TheninSection2.2wegivethedeﬁnitionsandusefulpropertiesofthe building blocksof Appell-Lerch series, false Appell-Lerch series and Hecke-type series.

As examples, we will rewritethe identities(1.8) and (1.9) usingthese building blocks.

InSection3,weshall ﬁrstproveTheorems1.1 and1.2. Thenweapply Theorem1.1 in Sections 3-7,where wediscuss mock thetafunctions of orders 2,3,5, 6and 8,respectively. For eachmocktheta function,we correspondinglyestablishsomeparameterized identities.Each oftheseidentitiesgivesus arepresentationforthemockthetafunction and producesnewinterestingidentities.

2. Preliminaries

In thissection, we ﬁrstcollectsomeuseful identitiesonbasichypergeometricseries.

ThenweintroduceseveralbuildingblocksforexpressingAppell-LerchseriesandHecke- typeseriesand someformulasforsimplifyingsuchexpressions.

Throughoutthispaper,wedenoteζn:=e^2πi/n.Forconvenience,weadoptthefollow- ingcompact notation:

(a1, a2,· · ·, am;q)n= (a1;q)n(a2;q)n· · ·(am;q)n, (a₁, a₂,· · ·, a_m;q)_∞= (a₁;q)_∞(a₂;q)_∞· · ·(a_m;q)_∞. WerecalltheJacobi’stripleproductidentity:

j(x;q) := (x)_∞(q/x)_∞(q)_∞= ∞ n=−∞

(−1)ⁿq(ⁿ2)xⁿ.

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Againforconvenience,wedenote

j(x₁, x₂, . . . , x_n;q) :=j(x₁;q)j(x₂;q)· · ·j(x_n;q). (2.1) Assomespecialcases,weletaandmbe rationalnumberswithmpositive anddeﬁne

Ja,m:=j(q^a;q^m), Ja,m:=j(−q^a;q^m) andJm:=Jm,3m= (q^m;q^m)_∞. Wewillusethefollowingidentitieswithoutmention(see [29,Section2]).

J0,1= 2J1,4= 2J₂²

J₁, J1,2= J₂⁵

J₁²J₄², J1,2= J₁²

J₂, J1,3= J2J₃² J₁J₆, J1,4= J₁J₄

J2

, J1,6= J₁J₆² J2J3

, J1,6= J₂²J₃J₁₂ J1J4J6

.

We also recall the classical partial fraction expansion for the reciprocal of Jacobi’s thetaproduct(see[39,p.1] or[42, p.136]):

∞ n=−∞

(−1)ⁿq(ⁿ⁺¹₂ )

1−qⁿz = J₁³

j(z;q). (2.2)

Herezisnotanintegralpowerofq.Thisformulawillbeusedseveraltimesforsimplifying expressions.

2.1. Usefulbasic hypergeometric seriesidentities

Inthissubsection,wecollectsomeusefulidentities,whichwillbeimportantindeduc- ingAppell-LerchseriesorHecke-typeseriesrepresentationsfromq-seriesoftheEulerian form.

Thebasichypergeometricseriesrφs isdeﬁnedas[23,Eq. (1.2.22)]

rφ_s

a₁,· · ·, a_r b1, . . . , bs

;q, z

= ∞ n=0

(a₁,· · ·, a_r;q)_n (q, b1,· · ·, bs;q)n

((−1)ⁿqⁿ⁽ⁿ⁻^1)/2)^1+s⁻^rzⁿ. From[34, Eq.(3.14)],we ﬁnd

3φ2

q⁻ⁿ, αqⁿ, β c, d ;q, q

= (−c)ⁿq^n(n−1)/2(qα/c;q)n

(c;q)_n ³φ2

q⁻ⁿ, αqⁿ, d/β

d, qα/c ;q, qβ/c

. (2.3) Wealsoneedthefollowingformula[23,p.28] whichrelatesa3φ2series toa2φ1 series:

3φ₂

q⁻ⁿ, b, bzq⁻ⁿ/c 0, bq¹⁻ⁿ/c ;q, q

= (c;q)_n (c/b;q)n2φ₁

q⁻ⁿ, b, c ;q, z

. (2.4)

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Watson’sq-analogofWhipple’stheorem(see,forexample,[23,Eq.(2.5.1)];[32,The- orem5])canbestated inthefollowingform.

Lemma 2.1(Watson).Letnbe anonnegativeinteger.Wehave (αq, αab/q;q)_n

(αa, αb;q)n 4φ3

q⁻ⁿ, q/a, q/b, αcd/q αc, αd, q²/αabqⁿ ;q, q

(2.5)

=8φ7

q⁻ⁿ, q√ a,−q√

a, α, q/a, q/b, q/c, q/d

√α,−√

α, αa, αb, αc, αd, αqⁿ⁺¹ ;q, α²abcdqⁿ⁻²

.

Intherestofthissubsection,wediscusssomeconsequencesofLemma2.1,whichwill be usedfrequently.

From [34,Proposition2.2] weﬁnd

3φ2

q⁻ⁿ, αqⁿ⁺¹, αcd/q αc, αd ;q, q

=(−α)ⁿq^n(n+1)/2 (q;q)n

(qα;q)n

n j=0

(−1)^j(1−αq^2j)(α, q/c, q/d;q)j

(1−α)(q, αc, αd;q)j

(cd/q)^jq^−j(j+1)/2. (2.6) Lemma 2.2.Forany nonnegative integern,we have

3φ₂

q⁻ⁿ, αqⁿ⁺¹, q/c αd, q²/c ;q, d

=(q/c)ⁿ (αc, q;q)_n (q²/c, αq;q)n

n j=0

(−1)^j(1−αq^2j)(α, q/c, q/d;q)_j (1−α)(q, αc, αd;q)j

(cd/q)^jq⁻^j(j+1)/2. (2.7)

Proof. Replacingαbyαq,β byαcd/q,cbyαcanddbyαdin(2.3),wehave

3φ2

q⁻ⁿ, αqⁿ⁺¹, αcd/q αc, αd ;q, q

=(−αc)ⁿqⁿ⁽ⁿ⁻^1)/2(q²/c;q)_n (αc;q)n 3φ₂

q⁻ⁿ, αqⁿ⁺¹, q/c αd, q²/c ;q, d

. (2.8)

Combining thiswith(2.6),wegetthedesiredidentity.

Lettingα→q⁻¹inLemma2.2,wegetthefollowingresult.

Lemma 2.3.Forany nonnegative integern,we have

3φ2

q⁻ⁿ, qⁿ, q/c d/q, q²/c ;q, d

= (q/c)ⁿ(1−qⁿ)(c/q;q)_n (q²/c;q)n

(2.9)

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×

(c+d)q+cd(q−2)−q³ (c−q)(d−q)(1−q) +

n j=2

(−1)^j(1−q^2j−1) (1−q^j)(1−q^j⁻¹)

(q/c, q/d;q)j

(c/q, d/q;q)_j(cd)^jq⁻^j(j+3)/2

.

Lettingd→ ∞in(2.6),weﬁndthefollowingtransformationformula, whichappears asProposition2.3in[34].

Corollary2.4. Foranynonnegative integer n,wehave

2φ1

q⁻ⁿ, αqⁿ⁺¹ αc ;q, c

=(−α)ⁿq^n(n+1)/2 (q;q)n

(qα;q)n

n j=0

(1−αq^2j)(α, q/c;q)j

(1−α)(q, αc;q)j

(c/α)^jq^−j(j+1). (2.10)

Lemma2.5. Forany nonnegativeinteger n,wehave

3φ2

q⁻ⁿ, αqⁿ⁺¹, q 0, q²/c ;q, q

= (αc, q;q)n

(q²/c, αq;q)_n(α/c)ⁿqⁿ²⁺²ⁿ n j=0

(1−α)(q, αc;q)_j c^jα^−jq^−j²^−j. (2.11) Proof. Replacing (b,c) by(αb,αc) in (2.4),wehave

3φ2

q⁻ⁿ, αb, bzq⁻ⁿ/c 0, bq¹⁻ⁿ/c ;q, q

= (αc;q)_n (c/b;q)n2φ1

q⁻ⁿ, αb αc ;q, z

. (2.12)

Now,setting(b,z)= (qⁿ⁺¹,c) in (2.12),weobtain

3φ₂

q⁻ⁿ, αqⁿ⁺¹, q 0, q²/c ;q, q

= (αc;q)_n (cq⁻ⁿ⁻¹;q)n2φ₁

q⁻ⁿ, αqⁿ⁺¹ αc ;q, c

. (2.13) Substituting(2.10) into (2.13),wecompletetheproofof Lemma2.5.

Takingα→q⁻¹ inLemma2.5,we obtain Lemma2.6. Forany nonnegativeinteger n,wehave

3φ₂

q⁻ⁿ, qⁿ, q 0, q²/c ;q, q

= (1−qⁿ)(c/q;q)_n (q²/c;q)n

c⁻ⁿqⁿ²⁺ⁿ

×

q−2c+cq (1−q)(c−q)+

n j=2

1−q^2j⁻¹ (1−q^j−1)(1−q^j)

(q/c;q)_j (c/q;q)j

c^jq^−j²

. (2.14)

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From [34,Proposition2.5] weﬁnd³ (−1)ⁿ(αq;q)n

(q;q)_n q^n(n+2)/2₂φ₁

q⁻ⁿ, αqⁿ⁺¹ αc ;q,1

= n j=0

(1−αq^2j)(α, q/c;q)_j (1−α)(q, αc;q)j

q^j²⁻^j(αc)^j. (2.15)

Lemma 2.7.Forany nonnegative integern,we have

3φ2

q⁻ⁿ, αqⁿ⁺¹, q/c 0, q²/c ;q, q

= (αc, q;q)n

(q²/c, αq;q)_n(q/c)ⁿ n j=0

(1−α)(q, αc;q)_j (αc)^jq^j²^−j. (2.16) Proof. Taking(b,c,z)= (αqⁿ⁺¹,αc,1) in(2.4),weobtain

3φ2

q⁻ⁿ, αqⁿ⁺¹, q/c 0, q²/c ;q, q

= (αc;q)n

(cq⁻ⁿ⁻¹;q)n2φ1

q⁻ⁿ, αqⁿ⁺¹ αc ;q,1

. (2.17) Togetherwith (2.15),wecomplete theproof ofLemma2.7.

Wewill alsoneedtheq-Pfaﬀ–Saalschützsummationformula[23,p.40,Eq. (2.2.1)]:

3φ2

q⁻ⁿ, aqⁿ, aq/bc aq/b, aq/c ;q, q

= (b, c;q)n

(aq/b, aq/c;q)_n aq

bc n

. (2.18)

2.2. BuildingblocksforAppell-Lerchseries,falseAppell-Lerch seriesandHecke-type series

Following HickersonandMortenson[29], wedeﬁne m(x, q, z) := 1

j(z;q) ∞ r=−∞

(−1)^rq(^r₂)z^r

1−q^r−1xz, (2.19)

where x,z∈C^∗:=C\{0}withneitherz norxzanintegralpowerofq. Let f_a,b,c(x, y, q) :=

sg(r)=sg(s)

sg(r)(−1)^r+sx^ry^sq^a(^r₂)+brs+c(^s₂). (2.20)

Here x,y∈C^∗ and sg(r):= 1 for r≥0 andsg(r):=−1 forr <0.

3 Thereisatypoin[34] thatq^n(n+1)/2shouldbereplacedby(−1)ⁿq^n(n+1)/2.

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Intermsoftheabovebuildingblocks,identities(1.3) and(1.5) canbe rewritten(see [29,Eq.(5.4)])as

f⁽³⁾(q) = 2m(−q, q³, q) + 2m(−q, q³, q²) = 4m(−q, q³, q) +J_3,6²

J₁ (2.21) and[29,Eq. (8.12)]

f₀⁽⁵⁾(q) = 1 J1

f_3,7,3(q², q², q) +q³f_3,7,3(q⁷, q⁷, q) = 1 J1

f_3,7,3(q^5/8,−q^5/8,−q^1/4).

(2.22) HickersonandMortensonwerealsoabletoconvert(2.22) intoanexpressioninterms of m(x,q,z) (see [29, Corollary 1.12]). Since the modular properties of m(x,q,z) and f_a,b,c(x,y,q) have been well studiedin [49], it will be easier for us to understand the modularpropertiesofanAppell-Lerchseries orHecke-typeseriesafterexpressing them usingthese buildingblocks.

In this paper, we will also express the Appell-Lerch series or Hecke-type series we encounteredintermsofthesebuildingblocks.Insomecases,theexpressionsinm(x,q,z) orfa,b,c(x,y,q) maybefurthersimpliﬁed.Wecollectthefollowingpropertiesofm(x,q,z) and f_a,b,c(x,y,q), which will be helpful for simplifyingthe ﬁnal expressions. Following [29],theterm“generic”willbeused tomeanthattheparametersdonotcausepolesin theAppell-Lerchsumsor inthequotientsofthetafunctions.

Lemma2.8. (Cf.[29,Proposition 3.1].) Forgenericx,z∈C^∗

m(x, q, z) =m(x, q, qz), (2.23)

m(x, q, z) =x⁻¹m(x⁻¹, q, z⁻¹), (2.24)

m(qx, q, z) = 1−xm(x, q, z). (2.25)

Lemma2.9. (Cf.[29,Theorem3.3].) Forgenericx,z0,z1∈C^∗

m(x, q, z₁)−m(x, q, z₀) = z₀J₁³j(z₁/z₀;q)j(xz₀z₁;q)

j(z0;q)j(z1;q)j(xz0;q)j(xz1;q). (2.26) Lemma2.10.(Cf. [29,Theorem3.5].) Forgenericx,z,z∈C^∗

m(x, q, z) =

n−1 r=0

q⁻(^r+12 )(−x)^rm(−q(ⁿ2)−nr(−x)ⁿ, qⁿ², z)+

zJ_n³ j(xz;q)j(z;qⁿ²)

n−1

r=0

q(^r₂)(−xz)^rj(−q(ⁿ₂)+r(−x)ⁿzz;qⁿ)j(q^nrzⁿ/z;qⁿ²) j(−q(ⁿ₂)(−x)ⁿz, q^rz;qⁿ)

. (2.27)

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As acorollaryofthis powerfulformula,wehavethefollowing result.

Lemma 2.11.(Cf.[29,Corollary3.7].) Forgenericx,z∈C^∗

m(x, q, z) =m(−qx², q⁴, z⁴)−x qm(−x²

q , q⁴, z⁴)− J₂J₄j(−xz²;q)j(−xz³;q) xj(xz;q)j(z⁴;q⁴)j(−qx²z⁴;q²).

(2.28) Thefollowing lemmacontainsusefulformulas forsimplifyingexpressionsintermsof f_a,b,c(x,y,q).

Lemma 2.12.(Cf.[29,Propositions6.1-6.2,Corollary 6.4].)Forx,y∈C^∗ fa,b,c(x, y, q) =fa,b,c(−x²q^a,−y²q^c, q⁴)−xfa,b,c(−x²q^3a,−y²q^c+2b, q⁴)

−yfa,b,c(−x²q^a+2b,−y²q^3c, q⁴) +xyq^bfa,b,c(−x²q^3a+2b,−y²q^3c+2b, q⁴), (2.29) fa,b,c(x, y, q) =−q^a+b+c

xy fa,b,c(q^2a+b/x, q^2c+b/y, q), (2.30) f_a,b,c(x, y, q) =−yf_a,b,c(q^bx, q^cy, q) +j(x;qâ), (2.31) f_a,b,c(x, y, q) =−xf_a,b,c(qâx, q^by, q) +j(y;q^c). (2.32) In[29, Eq.(4.6)] itwasdefinedforxbeingneither0noranintegralpowerofqthat

h(x, q) := 1 j(q;q²)

∞ n=−∞

(−1)ⁿqⁿ⁽ⁿ⁺¹⁾

1−qⁿx . (2.33)

It wasalsoprovedthatforgenericx∈C^∗ [29,Proposition4.4]

h(x, q) =−x⁻¹m(x⁻²q, q², x). (2.34) Meanwhile, Hickerson and Mortenson [29, Eq. (4.8)] also deﬁned for x² being neither zeronoranintegralpowerofq²that

k(x, q) := 1 xj(−q;q⁴)

∞ n=−∞

q^n(2n+1)

1−q²ⁿx². (2.35)

It wasprovedthat[29,Eq.(4.10)]

xk(x, q) =m(−x², q, x⁻²) + J₁⁴

2J₂²j(x²;q). (2.36) Wewill use(2.34) and(2.36) later.

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Whilem(x,q,z) andfa,b,c(x,y,q) canbeusedtoexpressAppell-Lerchseriesandabig familyofHecke-typeseriesweencountered,therearestillsomeserieswhichareunlikely toberepresentedbythem.Fortheseserieswehavetointroducetwonewbuildingblocks.

Wedeﬁneforx,z∈C^∗ withxznotbeinganintegralpowerofqthat m(x, q, z) =

∞ r=−∞

(−1)^rsg(r)q(^r₂)z^r

1−q^r−1xz . (2.37)

Thiscloselyresembles(2.19) exceptthatwedonothaveafactorofinﬁniteproductand wehaveintroducedsg(r) in thesummand.

Inawaysimilar to(2.20),wedeﬁne f_a,b,c(x, y, q) :=

sg(r)=sg(s)

(−1)^r+sx^ry^sq^a(^r₂)+brs+c(^s₂). (2.38) The diﬀerence between this function and fa,b,c(x,y,q) is that we have removed sg(r) fromthesummand.

WehavesomeformulassimilartoLemma2.12forf_a,b,c(x,y,q).

Lemma2.13.Wehave

f_a,b,c(x, y, q) =f_a,b,c(−x²q^a,−y²q^c, q⁴)−xf_a,b,c(−x²q^3a,−y²q^c+2b, q⁴)

−yf_a,b,c(−x²q^a+2b,−y²q^3c, q⁴) +xyq^bf_a,b,c(−x²q^3a+2b,−y²q^3c+2b, q⁴), (2.39) f_a,b,c(x, y, q) =−q^a+b+c

xy f_a,b,c(q^2a+b/x, q^2c+b/y, q). (2.40) Proof. Decomposethedeﬁnition(2.38) dependingontheparityofrands,weget(2.39).

Replacing(r,s) by(−r−1,−s−1) weget(2.40).

Atthismoment,themodularpropertiesofm(x,q,z) andf_a,b,c(x,y,q) areunclearand deserveinvestigationinthefuture.Inparticular, m(x,q,z) doesnotmeetthedeﬁnition of Appell-Lerch series given in (1.1). However, since the summands in m(x,q,z) and m(x,q,z) onlydiﬀer by signs, which is similar to the case of false theta functions, we wouldliketocall m(x,q,z) asafalse Appell-Lerchseries.

Specialcases ofthefunctionsm(x,q,z) and f_a,b,c(x,y,q) existintheliterature. For example,WangandYee [45,Eqs.(5.1),(5.5)] establishedthefollowingidentity

∞ n=1

(−1)ⁿ⁻¹qⁿ(q²;q²)_n−1 (−q²;q²)n

= ∞ n=1

q^n(n+1)/2 1 +q²ⁿ −

−1 n=−∞

q^n(n+1)/2

1 +q²ⁿ . (2.41) Upondecomposingthesumsontherightsideaccordingtotheparityofn,wecanwrite therightsideas