Recently, Bringmann and Kane established two new Bailey pairs and used them
to relate certain q-hypergeometric series to real quadratic fields. We show how
these pairs give rise to new mock theta functions in the form of
q-hypergeometric double sums. Additionally, we prove an identity between one of
these sums and two classical mock theta functions introduced by Gordon and
McIntosh.Comment: 12 pages, to appear in Pacific J. Mat

Andrews

Jeremy Lovejoy

Robert Osburn

Warnaar

Zagier

English

arXiv

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q-hypergeometric double sums as mock theta functions

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Pacific Journal of Mathematics

<i>q</i>-hypergeometric double sums as mock theta functions

Recently, Bringmann and Kane established two new Bailey pairs and used them to relate certain q-hypergeometric series to real quadratic fields. We show how these pairs give rise to new mock theta functions in the form of q-hypergeometric double sums. We also prove an identity between one of these sums and two classical mock theta functions introduced by Gordon and McIntosh

Lovejoy, Jeremy

Osburn, Robert

Irish Universities

Q-hypergeometric double sums as mock theta functions

Research Repository UCD

Title Q-hypergeometric double sums as mock theta functionsAuthors(s) Lovejoy, Jeremy, Osburn, RobertPublication date 2013Publication information Lovejoy, Jeremy, and Robert Osburn. “Q-Hypergeometric Double Sums as Mock Theta Functions” 264, no. 1 (2013).Publisher Mathematical Science PublishersItem record/more informationhttp://hdl.handle.net/10197/7954Publisher's version (DOI) 10.2140/pjm.2013.264.151Downloaded 2023-12-02T04:02:18ZThe UCD community has made this article openly available. Please share how this accessbenefits you. Your story matters! (@ucd_oa)© Some rights reserved. For more informationq-HYPERGEOMETRIC DOUBLE SUMS AS MOCK THETA FUNCTIONSJEREMY LOVEJOY AND ROBERT OSBURNIn memory of Basil GordonAbstract. Recently, Bringmann and Kane established two new Bailey pairs and used themto relate certain q-hypergeometric series to real quadratic fields. We show how these pairs giverise to new mock theta functions in the form of q-hypergeometric double sums. Additionally, weprove an identity between one of these sums and two classical mock theta functions introducedby Gordon and McIntosh.1. IntroductionA Bailey pair relative to a is a pair of sequences (αn, βn)n≥0 satisfyingβn =n∑k=0αk(q)n−k(aq)n+k. (1.1)Here we have used the standard q-hypergeometric notation,(a)n = (a; q)n =n∏k=1(1− aqk−1),valid for n ∈ N ∪ {∞}. The Bailey lemma states that if (αn, βn) is a Bailey pair relative to a,then so is (α′n, β′n), whereα′n =(b)n(c)n(aq/bc)n(aq/b)n(aq/c)nαn (1.2)andβ′n =n∑k=0(b)k(c)k(aq/bc)n−k(aq/bc)k(aq/b)n(aq/c)n(q)n−kβk. (1.3)Inserting (1.2) and (1.3) into (1.1) with n→∞ gives∑n≥0(b)n(c)n(aq/bc)nβn =(aq/b)∞(aq/c)∞(aq)∞(aq/bc)∞∑n≥0(b)n(c)n(aq/bc)n(aq/b)n(aq/c)nαn, (1.4)valid whenever both sides converge. For more on Bailey pairs, including historical perspectivesand recent advances, see Chapter 3 of [2], [3], or [13].Date: September 19, 2012.2010 Mathematics Subject Classification. Primary: 33D15; Secondary: 05A30, 11F03, 11F37.12 JEREMY LOVEJOY AND ROBERT OSBURNIn a recent study of multiplicative q-series, Bringmann and Kane [6] established two new andinteresting Bailey pairs. They showed that (an, bn) is a Bailey pair relative to 1, wherea2n = (1− q4n)q2n2−2nn−1∑j=−nq−2j2−2j , (1.5)a2n+1 = −(1− q4n+2)q2n2n∑j=−nq−2j2, (1.6)andbn =(−1)n(q; q2)n−1(q)2n−1χ(n 6= 0), (1.7)and (αn, βn) is a Bailey pair relative to q, whereα2n =11− qq2n2+2n n−1∑j=−nq−2j2−2j + q2n2n∑j=−nq−2j2 , (1.8)α2n+1 = −11− qq2n2+4n+2 n∑j=−nq−2j2+ q2n2+2nn∑j=−n−1q−2j2−2j , (1.9)andβn =(−1)n(q; q2)n(q)2n+1. (1.10)These closely resemble Bailey pairs related to 7th order mock theta functions [1], but surpris-ingly no q-series obtained by a direct substitution of either (1.5)–(1.7) or (1.8)–(1.10) in (1.4) isa genuine mock theta function. For example, it turns out that substituting (1.5)–(1.7) in (1.4)with b, c→∞ yields−q(−q)∞ω(q)where ω(q) is one of the third order mock theta functions. The presence of the infinite productmeans that this is not a mock theta function but a mixed mock modular form.Recall that mock theta functions are q-series which were introduced by Ramanujan in his lastletter to G. H. Hardy on January 12, 1920. Until 2002, it was not known how these functionsfit into the theory of modular forms. Thanks to work of Zwegers [15] and Bringmann andOno [8, 9], we now know that each of Ramanujan’s examples of mock theta functions is theholomorphic part of a weight 1/2 harmonic weak Maass form f(τ) (as usual, q := e2πiτ whereτ = x + iy ∈ H). Following Zagier [14], the holomorphic part of any weight k harmonic weakMaass form f is called a mock modular form of weight k. If k = 1/2 and the image of f underq-HYPERGEOMETRIC DOUBLE SUMS AS MOCK THETA FUNCTIONS 3the operator ξk := 2iyk ∂∂τ is a unary theta function, then the holomorphic part of f is called amock theta function. Specializations of the Appell-Lerch seriesm(x, q, z) :=1j(z, q)∑r∈Z(−1)rq(r2)zr1− qr−1xzare perhaps the most well-known and most important class of mock theta functions [14, 15].Here x, z ∈ C∗ := C \ {0} with neither z nor xz an integral power of q, andj(x, q) := (x)∞(q/x)∞(q)∞.For more on mock theta functions, their remarkable history and modern developments, see [12]and [14].The goal of this paper is to obtain genuine mock theta functions from the Bailey pairs ofBringmann and Kane by first moving a step along the Bailey chain. Applying (1.2) and (1.3) to(an, bn) with (b, c)→ (−1,∞) and to (αn, βn) with (b, c)→ (−q,∞), we obtain the Bailey pairsrecorded in the following two lemmas.Lemma 1.1. The pair (a′n, b′n) is a Bailey pair relative to 1, wherea′2n = 2(1− q2n)q4n2−nn−1∑j=−nq−2j2−2j ,a′2n+1 = −2(1− q2n+1)q4n2+3n+1n∑j=−nq−2j2,andb′n =1(−q)nn∑j=1(−1)j(q; q2)j−1(−1)jq(j+12 )(q)n−j(q)2j−1.Lemma 1.2. The pair (α′n, β′n) is a Bailey pair relative to q, whereα′2n =11− qq4n2+3n n−1∑j=−nq−2j2−2j + q4n2+nn∑j=−nq−2j2 ,α′2n+1 = −11− qq4n2+7n+3 n∑j=−nq−2j2+ q4n2+5n+1n∑j=−n−1q−2j2−2j ,andβ′n =1(−q)nn∑j=0(−q)j(q; q2)j(−1)jq(j+12 )(q)n−j(q)2j+1.4 JEREMY LOVEJOY AND ROBERT OSBURNWith our main result, we present four mock theta functions arising from the Bailey pairs inLemmas 1.1 and 1.2. Defineθn,p(x, y, q) :=1J0,np(2n+p)p−1∑r∗=0p−1∑s∗=0qn(r−(n−1)/22 )+(n+p)(r−(n−1)/2)(s+(n+1)/2)+n(s+(n+1)/22 )×(−x)r−(n−1)/2(−y)s+(n+1)/2J3p2(2n+p)j(−qnp(s−r)xn/yn, qnp2)j(qp(2n+p)(r+s)+p(n+p)xpyp, qp2(2n+p))j(qp(2n+p)r+p(n+p)/2(−y)n+p/(−x)n, qp2(2n+p))j(qp(2n+p)s+p(n+p)/2(−x)n+p/(−y)n, qp2(2n+p)),where r := r∗+ {(n− 1)/2} and s := s∗+ {(n− 1)/2} with 0 ≤ {α} < 1 denoting the fractionalpart of α. Also, Jm := Jm,3m with Ja,m := j(qa, qm), and Ja,m := j(−qa, qm).Theorem 1.3. The following are mock theta functions:W1(q) :=∑n≥j≥1(−1)j(q; q2)j−1(−1)jqn2+(j+12 )(−q)n(q)n−j(q)2j−1(1.11)= 4m(−q17, q48,−1)− 4q−5m(−q, q48,−1)− 2q2θ3,2(q5, q5, q)j(q, q3),W2(q) :=∑n≥j≥1(q; q2)n(−1)j(q; q2)j−1(−1)n+jq(j+12 )(−q)n(q)n−j(q)2j−1(1.12)= 4m(−q, q8,−1) + 2qθ1,2(−q2,−q2, q)j(−1, q),W3(q) :=∑n≥j≥1(q; q2)n(−1)j(q2; q4)j−1(−1)n+jqn2+j2+j(−q2; q2)n(q2; q2)n−j(q2; q2)2j−1(1.13)= 4m(−q, q12,−1) + 2q3θ1,1(−q7,−q7, q4)j(−q, q4),W4(q) :=∑n≥j≥0(−q)j(q; q2)j(−1)jqn2+n+(j+12 )(−q)n(q)n−j(q)2j+1(1.14)= −2q−4m(−q5, q48,−1)− 2q−2m(−q11, q48,−1) + θ3,2(q3, q3, q)j(q, q3).It should be noted that the series defining W2(q) does not converge. However, similar to thesixth order mock theta function µ(q) [5], the sequence of even partial sums and the sequence ofodd partial sums both converge. We define W2(q) as the average of these two values.To prove Theorem 1.3 we first use the Bailey machinery to express the Wi in terms of Hecke-type double sums fa,b,c(x, y, q), wherefa,b,c(x, y, q) :=∑sg(r)=sg(s)sg(r)(−1)r+sxrysqa(r2)+brs+c(s2). (1.15)q-HYPERGEOMETRIC DOUBLE SUMS AS MOCK THETA FUNCTIONS 5Here x, y ∈ C∗ and sg(r) := 1 for r ≥ 0 and sg(r) := −1 for r < 0. Then we apply recent resultsof Hickerson and Mortenson [11] to express the Hecke-type double sums as Appell-Lerch seriesm(x, q, z) (up to the addition of weakly holomorphic modular forms).We highlight one connection to classical mock theta functions. Namely, we express the multi-sum (1.12) in terms of the “eighth order” mock theta functions S1(q) and T1(q), defined by (see[10])S1(q) :=∑n≥0qn(n+2)(−q; q2)n(−q2; q2)nandT1(q) :=∑n≥0qn(n+1)(−q2; q2)n(−q; q2)n+1.Corollary 1.4. We have the identityW2(q) = 2qT1(q)− qS1(q).Similar identities involving mock theta functions and multisums were given by Andrews [4,Section 13], and more could be deduced from [7, Theorem 2.4].The paper proceeds as follows. Some background material on Hecke-type double sums andAppell-Lerch series is collected in Section 2, and Theorem 1.3 and Corollary 1.4 are establishedin Section 3.2. PreliminariesWe recall some relevant preliminaries. The most important is a result which allows us toconvert from the Hecke-type double sums (1.15) to Appell-Lerch series. Definega,b,c(x, y, q, z1, z0) :=a−1∑t=0(−y)tqc(t2)j(qbtx, qa)m(−qa(b+12 )−c(a+12 )−t(b2−ac) (−y)a(−x)b, qa(b2−ac), z0))+c−1∑t=0(−x)tqa(t2)j(qbty, qc)m(−qc(b+12 )−a(c+12 )−t(b2−ac) (−x)c(−y)b, qc(b2−ac), z1).(2.1)Following [11], we use the term “generic” to mean that the parameters do not cause poles inthe Appell-Lerch series or in the quotients of theta functions.Theorem 2.1. [11, Theorem 0.3] Let n and p be positive integers with (n, p) = 1. For genericx, y ∈ C∗fn,n+p,n(x, y, q) = gn,n+p,n(x, y, q,−1,−1) + θn,p(x, y, q).We shall also require certain facts about j(x, q), m(x, q, z) and fa,b,c(x, y, q). From the defi-nition of j(x, q), we have6 JEREMY LOVEJOY AND ROBERT OSBURNj(qnx, q) = (−1)nq−(n2)x−nj(x, q) (2.2)where n ∈ Z andj(x, q) = j(q/x, q) = −xj(x−1, q). (2.3)Next, some relevant properties of the sum m(x, q, z) are given in the following (see (2.2b) ofProposition 2.1 and Theorem 2.3 in [11]).Proposition 2.2. For generic x, z, z0 ∈ C∗.m(x, q, z) = x−1m(x−1, q, z−1) (2.4)andm(x, q, z) = m(x, q, z0) +z0J31 j(z/z0, q)j(xzz0, q)j(z0, q)j(z, q)j(xz0, q)j(xz, q). (2.5)Finally, two important transformation properties of fa,b,c(x, y, q) are given in the following(see Propositions 5.1 and 5.2 in [11]).Proposition 2.3. For x, y ∈ C∗,fa,b,c(x, y, q) =fa,b,c(−x2qa,−y2qc, q4)− xfa,b,c(−x2q3a,−y2qc+2b, q4)− yfa,b,c(−x2qa+2b,−y2q3c, q4) + xyqbfa,b,c(−x2q3a+2b,−y2q3c+2b, q4)(2.6)andfa,b,c(x, y, q) = −qa+b+cxyfa,b,c(q2a+b/x, q2c+b/y, q). (2.7)3. Proof of Theorem 1.3Proof of Theorem 1.3. As the proofs of (1.11)–(1.14) are similar, we give full details only for(1.11) and (1.14). Recall that the goal is to express each double sum q-series in terms of Appell-Lerch series. For (1.11), apply Lemma 1.1 and let b, c→∞ in (1.4) to obtainq-HYPERGEOMETRIC DOUBLE SUMS AS MOCK THETA FUNCTIONS 7W1(q) =∑n≥0qn2b′n(q)=1(q)∞∑n≥0qn2a′n(q)=1(q)∞(∑n≥0q4n2a′2n(q) +∑n≥0q4n2+4n+1a′2n+1(q))=2(q)∞(∑n≥0q8n2−nn−1∑j=−nq−2j2−2j −∑n≥0q8n2+nn−1∑j=−nq−2j2−2j−∑n≥0q8n2+7n+2n∑j=−nq−2j2 −∑n≥0q8n2+9n+3n∑j=−nq−2j2).After replacing n with −n in the second sum and n with −n − 1 in the fourth sum, we letn = (r + s + 1)/2, j = (r − s − 1)/2 in the first two sums and n = (r + s)/2, j = (r − s)/2 inthe latter two sums to findW1(q) =2q2(q)∞(( ∑r,s≥0r 6≡s (mod 2)−∑r,s<0r 6≡s (mod 2))q32r2+5rs+ 72r+ 32s2+ 72s−( ∑r,s≥0r≡s (mod 2)−∑r,s<0r≡s (mod 2))q32r2+5rs+ 72r+ 32s2+ 72s)=2q2(q)∞(2q5f3,5,3(−q23,−q19, q4)− f3,5,3(−q13,−q13, q4)− q15f3,5,3(−q29,−q29, q4))= − 2q2(q)∞f3,5,3(q5, q5, q).In the penultimate equality, we have replaced (r, s) first by (2r+ 1, 2s) and then by (2r, 2s+ 1)in the first line, and then replaced (r, s) first by (2r, 2s) and then by (2r+1, 2s+1) in the secondline, and then invoked (1.15). In the last step, we have used (2.6). By Theorem 2.1, (2.1), (2.2)and (2.3), we havef3,5,3(q5, q5, q) = −2q−2j(q, q3)m(−q17, q48,−1) + 2q−7j(q, q3)m(−q, q48,−1) + θ3,2(q5, q5, q)and soW1(q) = 4m(−q17, q48,−1)− 4q−5m(−q, q48,−1)−2q2θ3,2(q5, q5, q)j(q, q3).For (1.12), apply Lemma 1.1 and let b = −√q and c = √q in (1.4) to get8 JEREMY LOVEJOY AND ROBERT OSBURNW2(q) =∑n≥0(−1)n(q; q2)nb′n(q)=(q; q2)∞2(q2; q2)∞∑n≥0(−1)na′n(q)=(q; q2)∞2(q2; q2)∞(∑n≥0a′2n(q)−∑n≥0a′2n+1(q))=(q; q2)∞(q2; q2)∞(∑n≥0q4n2−nn−1∑j=−nq−2j2−2j −∑n≥0q4n2+nn−1∑j=−nq−2j2−2j+∑n≥0q4n2+3n+1n∑j=−nq−2j2 −∑n≥0q4n2+5n+2n∑j=−nq−2j2).As before, we proceed withW2(q) =(q; q2)∞(q2; q2)∞(( ∑r,s≥0r 6≡s (mod 2)−∑r,s<0r 6≡s (mod 2))q12r2+3rs+ 32r+ 12s2+ 32s+1+( ∑r,s≥0r≡s (mod 2)−∑r,s<0r≡s (mod 2))q12r2+3rs+ 32r+ 12s2+ 32s+1)=(q; q2)∞(q2; q2)∞((∑r,s≥0−∑r,s<0)q12r2+3rs+ 32r+ 12s2+ 32s+1)=q(q; q2)∞(q2; q2)∞f1,3,1(−q2,−q2, q).By Theorem 2.1, (2.1) and (2.2), we havef1,3,1(−q2,−q2, q) = 2q−1j(−1, q)m(−q, q8,−1) + θ1,2(−q2,−q2, q)and soW2(q) = 4m(−q, q8,−1) +2qθ1,2(−q2,−q2, q)j(−1, q).For (1.13), apply Lemma 1.1 and let b = q, c→∞ and q → q2 in (1.4) to getq-HYPERGEOMETRIC DOUBLE SUMS AS MOCK THETA FUNCTIONS 9W3(q) =∑n≥0(−1)n(q; q2)nqn2b′n(q2)=(q; q2)∞(q2; q2)∞∑n≥0(−1)nqn2a′n(q2)=(q; q2)∞(q2; q2)∞(∑n≥0q4n2a′2n(q2)−∑n≥0q4n2+4n+1a′2n+1(q2))=2(q; q2)∞(q2; q2)∞(∑n≥0q12n2−2nn−1∑j=−nq−4j2−4j −∑n≥0q12n2+2nn−1∑j=−nq−4j2−4j+∑n≥0q12n2+10n+3n∑j=−nq−4j2 −∑n≥0q12n2+14n+5n∑j=−nq−4j2).So,W3(q) =2(q; q2)∞(q2; q2)∞(( ∑r,s≥0r 6≡s (mod 2)−∑r,s<0r 6≡s (mod 2))q2r2+8rs+5r+2s2+5s+3+( ∑r,s≥0r≡s (mod 2)−∑r,s<0r≡s (mod 2))q2r2+8rs+5r+2s2+5s+3)=2(q; q2)∞(q2; q2)∞((∑r,s≥0−∑r,s<0)q2r2+8rs+5r+2s2+5s+3)=2q3(q; q2)∞(q2; q2)∞f1,2,1(−q7,−q7, q4).By Theorem 2.1, (2.1), (2.2) and (2.3), we havef1,2,1(−q7,−q7, q4) = 2q−3j(−q, q4)m(−q, q12,−1) + θ1,1(−q7,−q7, q4)and soW3(q) = 4m(−q, q12,−1) +2q3θ1,1(−q7,−q7, q4)j(−q, q4).Finally, for (1.14), apply Lemma 1.2 and let b, c→∞ in (1.4) to get10 JEREMY LOVEJOY AND ROBERT OSBURNW4(q) =∑n≥0qn2+nβ′n(q)=(1− q)(q)∞∑n≥0qn2+nα′n(q)=(1− q)(q)∞(∑n≥0q4n2+2nα′2n(q) +∑n≥0q4n2+6n+2α′2n+1(q))=1(q)∞(∑n≥0q8n2+5nn−1∑j=−nq−2j2−2j +∑n≥0q8n2+3nn∑j=−nq−2j2−∑n≥0q8n2+13n+5n∑j=−nq−2j2 −∑n≥0q8n2+11n+3n∑j=−n−1q−2j2−2j).After replacing n with−n−1 in the third and fourth sums, we let n = (r+s+1)/2, j = (r−s−1)/2in the first and fourth sums and n = (r + s)/2, j = (r − s)/2 in the second and third sums togetW4(q) =1(q)∞(( ∑r,s≥0r 6≡s (mod 2)−∑r,s<0r 6≡s (mod 2))q32r2+5rs+ 132r+ 32s2+ 132s+5+( ∑r,s≥0r≡s (mod 2)−∑r,s<0r≡s (mod 2))q32r2+5rs+ 32r+ 32s2+ 32s)=1(q)∞(2q13f3,5,3(−q25,−q29, q4) + f3,5,3(−q9,−q9, q4) + q11f3,5,3(−q25,−q25, q4))=1(q)∞f3,5,3(q3, q3, q)where in the last step we have used (2.6) and (2.7). By Theorem 2.1, (2.1), (2.2), (2.3) and(2.4), we havef3,5,3(q3, q3, q) = −2q−4j(q, q3)m(−q5, q48,−1)− 2q−2j(q, q3)m(−q11, q48,−1) + θ3,2(q3, q3, q)and soW4(q) = −2q−4m(−q5, q48,−1)− 2q−2m(−q11, q48,−1) +θ3,2(q3, q3, q)j(q, q3).Proof of Corollary 1.4. Equations (4.36) and (4.38) of [11] stateq-HYPERGEOMETRIC DOUBLE SUMS AS MOCK THETA FUNCTIONS 11S1(q) = −2q−1m(−q, q8,−1) +J3,8J22,8qJ21,8T1(q) = q−1m(−q, q8, q6).By (1.12), (2.3) and (2.5), the claim is equivalent to the identityJ3,8J22,8J21,8+2qθ1,2(−q2,−q2, q)j(−1, q)=−2j(q8, q24)3j(−q6, q8)j(q6, q8)j(−1, q8)j(−q7, q8).We have verified this identity using Garvan’s MAPLE program (see http://www.math.ufl.edu/~fgarvan/qmaple/theta-supplement/).AcknowledgementsThe authors would like to thank Song Heng Chan and Renrong Mao for their helpful com-ments.References[1] G.E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986), no.1, 113–134.[2] G.E. Andrews, q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics,Physics, and Computer Algebra, volume 66 of Regional Conference Series in Mathematics. American Math-ematical Society, Providence, RI, 1986.[3] G.E. Andrews, Bailey’s transform, lemma, chains and tree, Special functions 2000: current perspectiveand future directions (Tempe, AZ), 1–22, NATO Sci. Ser. II Math. Phys. Chem., 30, Kluwer Acad. Publ.,Dordrecht, 2001.[4] G.E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169(2007), 37–73.[5] G.E. Andrews and D. Hickerson, Ramanujan’s “lost” notebook. VII. The sixth order mock theta functions,Adv. Math. 89 (1991), no. 1, 60–105.[6] K. Bringmann and B. Kane, Multiplicative q-hypergeometric series arising from real quadratic fields, Trans.Amer. Math. Soc. 363 (2011), no. 4, 2191–2209.[7] K. Bringmann, J. Lovejoy and R. Osburn, Automorphic properties of generating functions for generalizedrank moments and Durfee symbols, Int. Math. Res. Not. (2010), no. 2, 238–260.[8] K. Bringmann and K. Ono, The f(q) mock theta function conjecture and partition ranks, Invent. Math. 165(2006), 243–266.[9] K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, Ann. of Math. 171 (2010), 419–449.[10] B. Gordon and R.J. McIntosh, Some eighth order mock theta functions, J. London Math. Soc. (2) 62 (2000),321–335.[11] D. Hickerson and E. Mortenson, Hecke-type double sums, Appell-Lerch sums, and mock theta functions (I),preprint available at http://arxiv.org/abs/1208.1421[12] K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Proceedings of the2008 Harvard-MIT Current Developments in Mathematics Conference, International Press, Somerville, MA,2009, 347–454.[13] S. O. Warnaar, 50 years of Bailey’s lemma, Algebraic combinatorics and applications (Gößweinstein, 1999),333–347, Springer, Berlin, 2001.[14] D. Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann),Astérisque 326 (2009), 143–164.[15] S. Zwegers, Mock Theta Functions, PhD Thesis, Universiteit Utrecht (2002).12 JEREMY LOVEJOY AND ROBERT OSBURNCNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCESchool of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, IrelandE-mail address: lovejoy@liafa.jussieu.frE-mail address: robert.osburn@ucd.ie

q-hypergeometric double sums as mock theta functions

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