VEV of Q -operator in U(1) linear quiver 5d gauge theories
PPrepared for submission to JHEP
YerPhI/2018/01
VEV of Q -operator in U (1) linear quiver 5d gauge theories Gabriel Poghosyan
Yerevan Physics Institute,Alikhanian Br. 2, AM-0036 Yerevan, Armenia
E-mail: [email protected]
Abstract:
Linear quiver N = 1 5d gauge theory in Ω background is considered.It is shown that under certain restrictions on the VEV’s of the adjoint scalar fieldcorresponding to the first node, only the array of Young diagrams, such that thefirst diagram is a single column and the others are empty, contribute to the partitionfunction. Furthermore it is proved that this partition function in a simple way isrelated to the expectation values of Baxter’s Q operator (at specific discrete val-ues of the spectral parameter) in the gauge theory with the special node removed.Using known expression of the partition function in the U (1) quiver, Baxter’s T-Qdifference equations are established and explicit expressions for the VEV of the Q operator in terms of generalized q-deformed Appel’s functions is found. Finally thecorresponding expressions for the 4d limit are derived. Keywords:
AGT, Deformed Seiberg-Witten equation, Toda and Liouville fieldtheories a r X i v : . [ h e p - t h ] J a n ontents Q observable 44 Difference equation for Q and its solution 65 Reduction to dimensions 86 Acknowledgments 10A Proof of the equality (3.5) 10B Restriction on Young diagrams at the special node 12C Generalized Appel and hypergeometric functions 12 The 4d N = 2 gauge theories have natural uplift to the 5 dimensions. Embedding N = 2 gauge theory in Ω-background was instrumental in all developments related tothe instanton counting with the help of equivariant localization technics. In fact thegeometric meaning of Ω-background is more transparent in 5d theory compactified ona circle. One simply considers a 5d geometry fibered over a circle of circumference L so that the complex coordinates ( z , z ) of the (four real dimensional) fiber get rotatedalong the circle as: z → exp( iL(cid:15) ), z → exp( iL(cid:15) ) accompanied with suitable R -symmetry and gauge rotations [1, 2]. (cid:15) , are the Omega -background parameters. In5d setting we’ll use the notation T , = exp( − β(cid:15) , ), where β = iL and for technicalreasons it will be assumed that β has a tiny real positive part. The initial 4d theory isrecovered by sending R →
0. Furthermore, sending both Ω-background parameters (cid:15) , to 0, one gets the standard Seiberg-Witten theory [3, 4]. It is interesting thateven the case of U (1) gauge group, in contrast to the case without Ω-background, thetheory is non-trivial. A characteristic feature of this case, is that the instanton sumsbecome tractable, and for Nekrasov partition function one obtains closed formulae.– 1 –n this paper it is shown that not only the partition function, but also a more refinedquantity, namely the expectation value of the Q -observable can be computed inclosed form. It was shown in [5] that the analog of Baxters Q operator in purelygauge theory context naturally emerges in Necrasov-Shatashvili limit ( (cid:15) = 0) [6]as an entire function whose zeros are given in terms of an array of ”critical” Youngdiagrams, namely those, that determine the most important instanton configurationcontributing to the partition function. This observable encodes perfectly not onlyinformation about partition function (which is simply related to the total sum ofcolumn lengths of Young diagrams) but also the entire chiral ring [7] constructed from (cid:104) tr Φ J (cid:105) , J = 0 , , , . . . (Φ is the scalar of vector multiplet) which can be expressed interms of power sum symmetric functions of the column lengths. This is why it is notsurprising that the logarithmic differential of (shifted) ratio y ( x ) ∼ Q ( x ) /Q ( x + (cid:15) ) isthe direct analog of Seiberg-Witten differential: xd log( y ( x )) ∼ ω SW . Subsequently Q and y observables have been extended for theories with various matter and gaugecontents [5, 8–10]. In particular [10] interprets the equations satisfied by y ( x ) asdeformed character relations and also considers the 5d setup, while in [11] a relationbetween the T − Q difference equation and the AGT dual [12, 13] (quasi-classical)2d Toda conformal blocks with a fully degenerate insertion is found. The next step,namely extension to the case of generic Ω-background has been achieved in [14]and [15], where Dyson-Shwinger type equations (called qq -character relations) for y -observable are derived. For recent developments see also the series of papers byNekrasov[14, 16–19].In [20] the already mentioned link between Q observable and Toda conformalblocks with a degenerate field insertion remains valid for the case of generic Ω-background and, in AGT dual 2d CFT side, fully quantum conformal blocks aswell. The case of the gauge group SU (2) corresponding to the Liouville theorywas analyzed in much details and starting from second order the BPZ differentialequation [21] a difference-differential equation, generalizing conventional Baxters T − Q relation [22] was derived. In present paper simpler U (1) case in 5d setting isanalyzed. The corresponding T − Q difference equations as well as their solutionsin closed form are found. The solution is expressed in terms of generalized Appel’sfunction.The rest of material is organized as follows.In chapter 2 is a short review of 5d linear quiver gauge theory: the Nekrasov partitionfunction and important observables Q , y are introduced.In chapter 3 an extended quiver with specific parameters at the extra nod is intro-duced and its relation to the Q -observable is analyzed.Chapter 4 specializes to the case of U (1) r theory. Difference equations Q -observableare derived. Explicit expressions for the Q observable in terms of generalized Appeland hypergeometric functions are found.In chapter 5 through dimensional reduction, corresponding difference equations and– 2 –heir solutions for the 4d theory are found.In appendices A, B, C some technical details, used in the main text, are presented. The (instanton part of) partition function of the 5d, A r +1 linear quiver theory withgauge group U ( n ) is given by (see Fig.1 where the setup and the notations are brieflydescribed) Z = (cid:88) ( (cid:126)Y ,...,(cid:126)Y r ) Z Y q | (cid:126)Y | . . . q | (cid:126)Y r | r (2.1)The sum in (2.1) is over all possible r -tuples of arrays of n Young diagrams. | (cid:126)Y k | isthe total number of boxes in the k -th array of n Young diagrams and Z Y is definedas: Z Y = Z (cid:126)Y ,...,(cid:126)Y r ( (cid:126)a , (cid:126)a , . . . , (cid:126)a r +1 ) = n (cid:89) u,v =1 Z bf ( ø , a ,u | Y ,v , a ,v ) Z bf ( Y ,u , a ,u | Y ,v , a ,v ) . . . Z bf ( Y r,u , a r,u | ø , a r +1 ,v ) Z bf ( Y ,u , a ,u | Y ,v , a ,v ) . . . Z bf ( Y r,u , a r,u | Y r,v , a r,v ) (2.2)For a pair of Young diagrams λ , µ the bifundamental contribution is given by [23, 24] Z bf ( λ, a | µ, b ) = (cid:89) s ∈ λ (cid:16) − ab T − L µ ( s )1 T A λ ( s )2 (cid:17) (cid:89) s ∈ µ (cid:16) − ab T L λ ( s )1 T − A µ ( s )2 (cid:17) (2.3) A λ and L λ , known as the arm and leg lengths respectively, are defined as: if s isa box with coordinates ( i, j ) and λ i ( λ (cid:48) j ) is the length of i -th ( j -th) column (row),then: L λ ( s ) = λ (cid:48) j − i , A λ ( s ) = λ i − j (2.4) The important observable of main interest in this paper, the Q -observable, is definedas Q ( x, λ ) = (cid:89) ( i,j ) ∈ λ x − T i T j − x − T i − T j − (2.5)Of course an analogous observable with the roles of T and T exchanged can beintroduced as well. In 4d case β → (cid:15) → a ø U ( n ) (cid:126)a (cid:126)Y q U ( n ) (cid:126)a (cid:126)Y q (cid:126)a r (cid:126)Y r q r U ( n ) (cid:126)a r +1 ø Figure 1 . The linear quiver U(n) gauge theory: r circles stand for gauge multiplets; twosquares represent n anti-fundamental (on the left edge) and n fundamental (the right edge)matter multiplets while the line segments connecting adjacent circles represent the bi-fundamentals. q , . . . , q r are the exponentiated gauge couplings, the n -dimensional vectors (cid:126)a , . . . , (cid:126)a r +1 encode respective (exponentiated) masses/VEV’s and (cid:126)Y , . . . , (cid:126)Y r +1 are n -tuplesof young diagrams specifying fixed (ideal) instanton configurations. observable satisfies Baxter’s T-Q equation [5]: a difference equation introduced byBaxter in context of lattice integrable models [22]. Generalization for the case ofgeneric Ω background (in both 4d and 5d cases) is due to [14].An important role is played also by the observable y ( x, λ ) = Q ( x, λ ) Q ( x/T , λ ) ≡ (cid:89) ( i,j ) ∈ λ ( x − T i T j − )( x − T i − T j )( x − T i − T j − )( x − T i T j ) (2.6)In 4d Nekrasov-Shatashvili limit the logarithmic derivative of this observable gen-erates all expectation values (cid:104) φ J (cid:105) of the vector multiplet scalar. Besides, its ex-pectation value satisfies the (quantized analog of) Seiberg-Witten curve equation[5]. In generic Ω-background the corresponding equations (the so called qq-characterequations) were introduced and investigated in [14] (see also [15]). Q observable The expectation value of the Q -operator associated to the first node, by definition is Q ( x ) = Z − (cid:88) ( (cid:126)Y ,...,(cid:126)Y r ) n (cid:89) u =1 Q (cid:18) xa ,u , Y ,u (cid:19) Z Y q | (cid:126)Y | . . . q | (cid:126)Y r | r (3.1)It was noticed in [20] that such insertion of the operator Q is equivalent to addingan extra node with specific expectation values. Here this statement will be proved inmore general 5d setting. Note that a detailed proof in [20] was absent, so that alsothis gap automatically will be filled.Let’s look at a quiver with r + 1 nodes with expectation values at the additionalnode (denoted as ˜0) specified as (see Fig.2): a ˜0 ,u = a ,u T δ ,u . (3.2)– 4 –ue to the specific choice of (cid:126)a ˜0 , in order to give a nonzero contribution, the array of n diagrams associated with the special node ˜0 has to be severely restricted. Namely, thediagram Y ˜0 , should consist of a single column and the remaining n − Y ˜0 , , . . . , Y ˜0 ,n − must be empty. The proof of this statement is given in the Appendix B.There is a close relation between the Nekrasov partition function associated toabove described specific length r + 1 quiver and the expectation value of a particular Q operator in a generic quiver with r nodes. This relation is a consequence of theidentity Z (cid:126)Y ˜0 ,(cid:126)Y ,...,(cid:126)Y r ( (cid:126)a , (cid:126)a ˜0 , (cid:126)a , . . . , (cid:126)a r +1 ) q l ˜0 ( T q ) | (cid:126)Y | q | (cid:126)Y | . . . q | (cid:126)Y r | r = n (cid:89) u =1 Q (cid:18) a , a ,u T l , Y ,u (cid:19) (cid:16) a , a ,u T ; T (cid:17) l (cid:16) a , a ,u T ; T (cid:17) l Z (cid:126)Y ,...,(cid:126)Y r ( (cid:126)a , (cid:126)a , . . . , (cid:126)a r +1 ) q l ˜0 q | (cid:126)Y | . . . q | (cid:126)Y r | r , (3.3)where Y ˜0 ,u for u = 1 is a one column diagram with length l and the rest are emptydiagrams. The q-analog of Pochhammer’s symbol is defined as:( a ; q ) l = (1 − a )(1 − aq ) · · · (cid:0) − aq l − (cid:1) . (3.4)Inserting the definition (2.2) of Z (cid:126)Y and canceling out the common factors of q and Z bf , we see that (3.3) is equivalent to n (cid:89) u,v =1 (cid:18) Z bf ( ø , a ,u | Y ˜0 ,v , a ˜0 ,v ) Z bf ( Y ˜0 ,u , a ˜0 ,u | Y ,v , a ,v ) Z bf ( Y ˜0 ,u , a ˜0 ,u | Y ˜0 ,v , a ˜0 ,v ) (cid:19) T | (cid:126)Y | = n (cid:89) u =1 Q (cid:18) a , a ,u T l , Y ,u (cid:19) (cid:16) a ,u a ,u T , T (cid:17) l (cid:16) a , a ,u T ; T (cid:17) l n (cid:89) u,v =1 Z bf ( ø , a ,u | Y ,v , a ,v ) (3.5)The last equality is proven in Appendix A.Clearly, the eq. (3.3) shows that the VEV (3.1) at specific values x = x l x l = a , T l , l = 0 , , , . . . (3.6)is related to the partition function of the special quiver with the fixed instantonnumber | (cid:126)Y ˜0 | = l at the node ˜0 Q ( x l ) = Z − n (cid:89) u =1 (cid:16) a , a ,u T ; T (cid:17) l (cid:16) a ,u a ,u T ; T (cid:17) l × (cid:88) ( (cid:126)Y ,...,(cid:126)Y r ) Z (cid:126)Y ˜0 ,(cid:126)Y ,...,(cid:126)Y r ( (cid:126)a , (cid:126)a ˜0 , (cid:126)a , . . . , (cid:126)a r +1 ) ( T q ) | (cid:126)Y | q | (cid:126)Y | . . . q | (cid:126)Y r | r (3.7)– 5 – a ø q ˜0 U ( n ) (cid:126)a ˜0 (cid:126)Y ˜0 U ( n ) (cid:126)a (cid:126)Y T q U ( n ) (cid:126)a (cid:126)Y q (cid:126)a n (cid:126)Y r q r U ( n ) (cid:126)a r +1 ø Figure 2 . The quiver diagram with an extra node, labeled by ˜0, added. Note that thegauge coupling at the node 1 is chosen to be T q . Q and its solution From now on we’ll restrict ourselves to the simplest case of the quiver of U (1)’s. 5dNekrasov partition function of such linear quiver can be found using refined topo-logical vertex method [25], [26], [27–29] or through a direct instanton calculation(see e.g. [30] and references therein). The result can be represented as the infiniteproduct Z = ∞ (cid:89) l,s =0 r (cid:89) i =1 r (cid:89) j = i (cid:16) − a i − p i a j p j T l T s (cid:17) (cid:16) − a i p i a j +1 p j T l +11 T s +12 (cid:17)(cid:16) − a i p i a j p j T l T s (cid:17) (cid:16) − a i − p i a j +1 p j T l +11 T s +12 (cid:17) (4.1)where p i = a i (cid:89) l =1 q l (4.2)Applying the formula (4.1) for the special quiver discussed in Section 3, and forbrevity denoting the partition function of the special quiver simply as Z ( q ˜0 ), up tofactors independent of q ˜0 we get Z ( q ˜0 ) (cid:39) ∞ (cid:89) s =0 r (cid:89) i =0 − a p i a a i +1 q ˜0 T − δ i, T s − a p i a a i q ˜0 T s (4.3)Note now that in ratio Z ( q ˜0 ) /Z ( T − q ˜0 ) nearly all factors cancel out and one is leadto the relation Z ( q ˜0 ) r (cid:89) i =0 (cid:18) − a p i a a i +1 q ˜0 T − δ i, (cid:19) = Z ( T − q ˜0 ) r (cid:89) i =0 (cid:18) − a p i a a i q ˜0 T − (cid:19) (4.4)Expanding this equality in powers of q ˜0 and taking into account (3.7), we’ll geta linear relation (with rational in x l coefficients) among r + 2 quantities Q ( x l ), Q ( x l /T ),. . . , Q ( x l /T r +12 ). – 6 –irst let consider the simplest case r = 1. An easy computation allows us toestablish the equality Q ( x ) − (cid:18) q a T x − a a a ( x − a ) (cid:19) Q (cid:18) xT (cid:19) + q a ( x − a T ) ( T x − a ) a ( x − a )( x − a T ) Q (cid:18) xT (cid:19) = 0(4.5)which is valid for infinitely many values x = x l , l = 0 , , , . . . (see eq. (3.6)).An essential observation is in order here. Since Q ( x ) and hence the entire LHSof the eq. (4.5) restricted up to an arbitrary instanton order is a rational function of x , the equality must be valid also for generic values of x . It is not difficult to checkthat the q -hypergeometric function (see Appendix C for definition) Q ( x ) = ( q ; T ) ∞ (cid:16) q a T T a ; T (cid:17) ∞ φ (cid:18) a x , a T T a a x ; T , q (cid:19) (4.6)is a solution of (4.5). The ( x -independent) normalization coefficient in (4.6) is fixedfrom the asymptotic condition lim x →∞ Q ( x ) = 1 (4.7)In fact it is possible to argue that (4.6) is the only solution of (4.5) with correctasymptotic and rationality properties discussed above. Using the special n = 1 caseof the identity (C.6) the eq. (4.6) can be rewritten also as (this equality is referredas Heine’s first transformation [31]) Q ( x ) = (cid:0) a x ; T (cid:1) ∞ (cid:0) a x ; T (cid:1) ∞ φ (cid:32) a a , q , q a T T a ; T , a x (cid:33) . (4.8)The general case with an arbitrary r though more cumbersome, could be analyzedin the same way. The resulting difference equation reads: r +1 (cid:88) s =0 ( − ) s C s Q (cid:0) T − s x (cid:1) = 0 (4.9)where C s C s = xT s − ( O ( s − + T O ( s )+ ) − a O ( s − − a O ( s ) x − a s − (cid:89) n =0 x − a T n x − a T n (4.10)and O ( i ) , O ( i )+ are the coefficients of the expansions: r (cid:89) i =1 (cid:18) t + p i a i (cid:19) = ∞ (cid:88) s = −∞ O ( s ) t r − s (4.11) r (cid:89) i =1 (cid:18) t + p i a i +1 (cid:19) = ∞ (cid:88) s = −∞ O ( s )+ t r − s (4.12)– 7 –r, explicitly O ( s ) = (cid:88) ≤ c <... A Proof of the equality (3.5) Here we present the derivation of (3.5). We first derive two auxiliary identities.Denote a Young diagram λ with column lengths λ ≥ λ ≥ · · · as { λ , λ , . . . } . Thecorresponding row lengths we’ll indicate as λ (cid:48) ≥ λ (cid:48) ≥ · · · . In particular λ (cid:48) wouldbe the number of columns. We want to show that Z bf ( { l } , a | λ, b ) = Q (cid:16) ab T T l , λ (cid:17) T −| λ | Z bf ( ø , aT | λ, b ) (cid:16) ab T T ; T (cid:17) l (A.1)To prove (A.1) we divide and multiply the LHS by Z bf ( ø , aT | λ, b ), then insert thedefinitions of Z bf and the values of arm and leg lengths. We get Z bf ( { l } , a | λ, b ) = Z bf ( ø , aT | λ, b ) l (cid:89) j =1 (cid:16) − ab T − λ (cid:48) j T l − j (cid:17) λ (cid:89) j =1 λ (cid:48) j (cid:89) i =1 − ab T θ ( l − j ) − i T − λ i + j − ab T − i T − λ i + j , (A.2)where θ ( x ) is the Heaviside step function θ ( x ) = (cid:40) , if x ≥ , if x < λ ≤ l or λ > l .1. λ ≤ l . – 10 –n this case θ ( l − j ) = 1 and the double product in (A.2) cancels out. The remainingsingle product by a simple manipulation can be rewritten as Z bf ( ø , aT | λ, b ) l (cid:89) j = λ +1 (cid:16) − ab T T l − j (cid:17) λ (cid:89) j =1 λ (cid:48) j (cid:89) i =1 − ab T − i T l − j − ab T − i T l − j λ (cid:89) j =1 (cid:16) − ab T T l − j (cid:17) . (A.4)Notice that the middle double product is nothing but Q ( ab T T l , λ ) T −| λ | which con-cludes the first case.2. λ > l We split λ into two parts: λ top consisting of boxes with vertical coordinates j > l ,and the part λ down of lower lying boxes with j ≤ l . Now the part of the doubleproduct in (A.2) corresponding to the boxes of λ top survives. For the single productpart we do the same manipulation as in previous case. As a result we get Z bf ( ø , aT | Y, b ) l (cid:89) j =1 λ (cid:48) j (cid:89) i =1 − ab T − i T l − j − ab T − i T l − j l (cid:89) j =1 (1 − ab T T l − j ) (cid:89) ( i,j ) ∈ λ top − ab T − i T − λ i + j − ab T − i T − λ i + j . (A.5)It is easy to see that the product over λ top can be rewritten as (cid:89) ( i,j ) ∈ λ top − ab T − i T − λ i + j − ab T − i T − λ i + j = (cid:89) ( i,j ) ∈ λ top − ab T − i T l − j − ab T − i T l − j . (A.6)Thus the first double product in (A.5) (which is a product over the boxes of λ down )naturally combines with that of over λ top to give a product over entire λ . As a result,instead of (A.5) we may as well write Z bf ( ø , aT | λ, b ) (cid:16) ab T T ; T (cid:17) l (cid:89) ( i,j ) ∈ λ − ab T − i T l − j − ab T − i T l − j . (A.7)As before, the product over λ gives Q ( ab T l , λ ) T −| λ | , which concludes the proof of eq.(A.1).We’ll need also the simple identity Z bf ( ø , a | λ, b ) = (cid:89) ( i,j ) ∈ λ (cid:16) − ab T − i T − j (cid:17) (A.8)– 11 –ow the only thing that remains to be done is to make use of (A.1) and (A.8): n (cid:89) u,v =1 Z bf ( ø , a ,u | Y ˜0 ,v , a ˜0 ,v ) Z bf ( Y ˜0 ,u , a ˜0 ,u | Y ,v , a ,v ) Z bf ( Y ˜0 ,u , a ˜0 ,u | Y ˜0 ,v , a ˜0 ,v ) = n (cid:89) u,v =2 Z bf ( ø , a ,u | Y ,v , a ,v ) n (cid:89) u =2 Z bf ( ø , a ,u | Y , , a , ) Z bf ( Y , , a , T | Y ,u , a ,u ) Z bf ( Y , , a , T | ø , a ,u ) × Z bf ( ø , a , | Y , , a , T ) Z bf ( Y , , a , T | Y , , a , ) Z bf ( Y , , a , T | Y , , a , T )= T −| (cid:126)Y | n (cid:89) u =1 (cid:16) Q ( a ,u a ,u T l , Y ,u ) ( a ,u a ,u T ; T ) l ( a , a ,u T ; T ) l (cid:17) n (cid:89) u,v =1 Z bf ( ø , a ,u | Y ,v , a ,v ) (A.9) B Restriction on Young diagrams at the special node To prove that the diagram Y ˜0 , at the special node ˜0 should have at most one columnin order to have a nonzero contribution to the partition function, let us assume incontrary that Y ˜0 , has a non-empty second column with length l ≥ 1. This meansthat the box with coordinates ( i, j ) = (2 , l ) belongs to this diagram. Any term ofthe instanton sum corresponding to such choice includes a factor Z bf ( ø , a , | Y ˜0 , , a , T − ) = (cid:89) s ∈ Y ˜0 , (cid:18) − a , a , T − T L ø ( s )1 T − A Y ˜0 , ( s )2 (cid:19) (B.1)The arm and leg lengths of the box (2 , l ) are easy to calculate: L ø (2 , l ) = − A Y (2 , l ) = 0 and the corresponding factor in eq. (B.1) vanishes.In a similar way we can easily argue that all remaining n − Y ˜0 ,i , i = 2 , . . . , n must be empty. In fact, if any of this diagrams is non-empty (denote it as λ ), then Z Y will include a factor Z bf ( ø , a ,i | λ, a ,i ) = (cid:89) s ∈ λ (1 − T L ø ( s )1 T − A λ ( s )2 ) (B.2)In this product the factor corresponding to the top box (1 , λ ) of its first columnbecomes zero, since for this box L ø (1 , λ ) = − A λ (1 , λ ) = 0.Thus we have proven that at the special node the first diagram has at most onecolumn while the remaining diagrams are empty. C Generalized Appel and hypergeometric functions Appels functions and their q-analogues generalize ordinary hypergeometric and q-hypergeometric functions for the case with more than one arguments. Here are thedefinitions: – 12 – Appel’s function F and its generalization for the arbitrary number of variables: F ( a, b , b ; c ; x, y ) = ∞ (cid:88) m,n =0 ( a ) m + n ( b ) m ( b ) n ( c ) m + n m ! n ! x m y n (C.1) F ( k )1 ( a, b , ..., b k ; c ; x , .., x k ) = (cid:88) m ,...,m k ≥ ( a ) m + ... + m r ( b ) m ... ( b k ) m k ( c ) m + ... + m r m ! ...m k ! ( x ) m ... ( x k ) m k (C.2) • The corresponding q-analogs:Φ ( a, b , b ; c ; q ; x, y ) = ∞ (cid:88) m,n =0 ( a ; q ) m + n ( b ; q ) m ( b ; q ) n ( c ; q ) m + n ( q ; q ) m ( q ; q ) n x m y n (C.3) Φ ( n )1 ( a, b , b , ..., b n ; c ; q ; x , .., x n ) = ∞ (cid:88) m ,...,m n =0 ( a ; q ) m + ... + m r ( b ; q ) m ... ( b n ; q ) m n ( c ; q ) m + ... + m r ( q ; q ) m ... 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