aa r X i v : . [ m a t h . C O ] J un BILATERAL IDENTITIES OF THE ROGERS–RAMANUJAN TYPE
MICHAEL J. SCHLOSSER ∗ Dedicated to the memory of Srinivasa Ramanujan
Abstract.
We derive by analytic means a number of bilateral identities of theRogers–Ramanujan type. Our results include bilateral extensions of the Rogers–Ramanujan and the G¨ollnitz–Gordon identities, and of related identities by Ramanu-jan, Jackson, and Slater. We give corresponding results for multiseries including mul-tilateral extensions of the Andrews–Gordon identities, of Bressoud’s even modulusidentities, and other identities. The here revealed closed form bilateral and multilat-eral summations appear to be the very first of their kind. Given that the classicalRogers–Ramanujan identities have well-established connections to various areas inmathematics and in physics, it is natural to expect that the new bilateral and mul-tilateral identities can be similarly connected to those areas. This is supported byconcrete combinatorial interpretations for a collection of four bilateral companions tothe classical Rogers–Ramanujan identities. Introduction
For complex variables a and q with | q | < k ∈ Z ∪ ∞ , the q -shifted factorialsare defined as follows (cf. [18]):( a ; q ) k := k = 0 , Q kj =1 (1 − aq j − ) for k > , Q − kj =1 (1 − aq − j ) − for k < . The variable q is referred to as the base . For brevity, we use the compact notation( a , . . . , a m ; q ) k = ( a ; q ) k · · · ( a m ; q ) k , where m is a positive integer. Unless stated otherwise, all the summations in this paperconverge absolutely everywhere (subject to the condition | q | < Mathematics Subject Classification.
Primary 11P84; Secondary 05A17, 33D15.
Key words and phrases.
Rogers–Ramanujan type identities, q -series, basic hypergeometric series,bilateral summations, multilateral summations, combinatorial interpretations. ∗ Research partly supported by FWF Austrian Science Fund grant F50-08 within the SFB “Algo-rithmic and enumerative combinatorics”.
The first and second Rogers–Ramanujan identities, ∞ X k =0 q k ( q ; q ) k = 1( q, q ; q ) ∞ , (1.1a) ∞ X k =0 q k ( k +1) ( q ; q ) k = 1( q , q ; q ) ∞ , (1.1b)have a prominent history. They were first discovered and proved in 1894 by Rogers [33],and then independently rediscovered by the legendary Indian mathematician SrinivasaRamanujan some time before 1913 (cf. Hardy [23]). They were also independentlydiscovered and proved in 1917 by Schur [34]. About the pair of identities in (1.1)Hardy [25, p. xxxiv] remarked‘It would be difficult to find more beautiful formulae than the “Rogers-Ramanujan” identities, . . . ’It is not clear how Ramanujan originally was led to discover (1.1). Bhatnagar [12]describes a method to conjecture these identities. A basic hypergeometric proof of(1.1) was found by Watson [37] who observed that these identities can be obtainedfrom the (now called) Watson transformation by taking suitable limits and applyinginstances of Jacobi’s triple product identity.The Rogers–Ramanujan identities are deep identities which have found interpreta-tions in combinatorics, number theory, probability theory, statistical mechanics, repre-sentations of Lie algebras, vertex algebras, and conformal field theory [6, 11, 17, 31, 32].A recent highlight in the theory concerns the construction of these identities for higher-rank Lie algebras [22].A pair of identities similar to (1.1) are the first and second G¨ollnitz–Gordon identities, ∞ X k =0 ( − q ; q ) k ( q ; q ) k q k = 1( q, q , q ; q ) ∞ , (1.2a) ∞ X k =0 ( − q ; q ) k ( q ; q ) k q k ( k +2) = 1( q , q , q ; q ) ∞ . (1.2b)These appeared in a combinatorial study of partitions of numbers in unpublished workby G¨ollnitz in 1960 ([19], see also [20]) and were independently rediscovered in 1965by Gordon [21]. However, they were already more than 40 years earlier recorded byRamanujan in his lost notebook, see [7, p. 36–37, Entries 1.7.11–12], and were alsoindependently published in 1952 by Slater [36] as specific entries in her famous list of130 identities of the Rogers–Ramanujan type. The systematic study of such identitieshad been commenced by Bailey [8, 9] a few years earlier. A more complete list ofidentities of the Rogers–Ramanujan type was recently given by McLaughlin, Sills andZimmer [30]. Further such identities were given by Chu and Zhang [15]. McLaughlin, ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 3
Sills and Zimmer’s list is reproduced (with some typographical errors corrected) inAppendix A of Sills’ recent book [35] which provides an excellent introduction to theRogers–Ramanujan identities.The analytic identities in (1.1) and (1.2) admit partition-theoretic interpretations (cf.[5]). Because of the specific form of the q -products on the right-hand sides, the identitiesin (1.1), resp. (1.2), are often classified as mod 5 and mod 8 identities, respectively.Another identity intimately linked to Ramanujan’s name is the following summationformula (cf. [18, Appendix (II.29)]) ∞ X k = −∞ ( a ; q ) k ( b ; q ) k z k = ( q, az, q/az, b/a ; q ) ∞ ( b, z, b/az, q/a ; q ) ∞ , | b/a | < | z | < . (1.3)This identity, commonly known as Ramanujan’s ψ summation, is a bilateral extensionof the q -binomial theorem (cf. [18, Appendix (II.3)]) ∞ X k =0 ( a ; q ) k ( q ; q ) k z k = ( az ; q ) ∞ ( z ; q ) ∞ , | z | < , (1.4)which is the most fundamental identity in the theory of basic hypergeometric series.Hardy described (1.3), which Ramanujan had noted but did not publish, as “a remark-able formula with many parameters” [24, Eq. (12.12.2)]. Importantly, (1.3) containsJacobi’s triple product identity (3.1) as a limiting case, an identity which plays a keyrole in the standard proofs of identities of the Rogers–Ramanujan type (and which wealso make heavy use of in this paper).Knowing that the q -binomial theorem (1.4) extends to a bilateral summation, one canask the same question about the Rogers–Ramanujan and G¨ollnitz–Gordon identities in(1.1) and (1.2). While some authors have studied properties of bilateral series whichextend the series in (1.1) (see [2, 16, 27]), no closed form bilateral summations whichinclude the evaluations in (1.1) (or (1.2)) as special cases have yet been obtained.In this paper, we derive bilateral extensions of the Rogers–Ramanujan and G¨ollnitz–Gordon identities in (1.1) and (1.2) and provide a number of related results. Our mainresults for single series are given in Section 2, together with several noteworthy corollar-ies. The proofs of the main results of Section 2, namely Theorems 2.1, 2.4 and 2.6 aredeferred to Section 3. The proofs are analytic and involve a method similar to that usedby Watson in [37] to prove the classical Rogers–Ramanujan identities. In particular,we utilize suitable limiting cases of a bilateral basic hypergeometric transformation for-mula of Bailey in combination with special instances of Jacobi’s triple product identityto establish the respective identities. In Section 4 multiseries extensions of our resultsare given, which in particular include multilateral extensions of the Andrews–Gordonidentities among other multiseries identities. In Section 5 we provide combinatorialinterpretations of the four bilateral companions to the Rogers–Ramanujan identitiesgiven in Corollary 2.3. We end our paper with some concluding remarks. MICHAEL J. SCHLOSSER
All the identities in this paper were tested by Mathematica by performing powerseries expansion in q up to sufficiently high order (usually comparing the first 200coefficients, where feasible).2. Main results and corollaries in the single series case
Our first result is a bilateral extension of the two Rogers–Ramanujan identities in(1.1).
Theorem 2.1.
We have the following two bilateral summations: ∞ X k = −∞ q k ( zq ; q ) k z k = (1 /z ; q ) ∞ (1 /z , z q ; q ) ∞ ( q ; q ) ∞ × (cid:2) ( z q , z − q ; q ) ∞ + z − ( z q , z − q ; q ) ∞ (cid:3) , (2.1a) ∞ X k = −∞ q k ( k +1) ( zq ; q ) k z k = (1 /z ; q ) ∞ (1 /z , z q ; q ) ∞ ( q ; q ) ∞ × (cid:2) ( z q , z − q ; q ) ∞ + z − ( z q, z − q ; q ) ∞ (cid:3) . (2.1b)The z → z → Remark . As was kindly brought to the author’s attention by George Andrews afterbeing shown an earlier version of this paper, a (sporadic) result related to the series onthe left-hand side of (2.1b) was found by Andrews in 1970 [2, Thm. 3], namely:
Let g ( z ) = ( − z ; q ) ∞ ∞ X k = −∞ q k ( k − ( − z ; q ) k z k , (2.2a) then g ( z ) + g ( − z )2 = ( q , − z , − z − q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ . (2.2b)As consequence of Theorem 2.1, we obtain the following four bilateral summations: ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 5
Corollary 2.3 (Bilateral mod 25 identities) . ∞ X k = −∞ q k (5 k − ( q ; q ) k = ( q ; q ) ∞ ( q , q , q ; q ) ∞ ( q , q ; q ) ∞ , (2.3a) ∞ X k = −∞ q ( k − k − ( q ; q ) k = ( q ; q ) ∞ ( q , q , q ; q ) ∞ ( q, q ; q ) ∞ , (2.3b) ∞ X k = −∞ q k (5 k − ( q ; q ) k = ( q ; q ) ∞ ( q , q , q ; q ) ∞ ( q, q ; q ) ∞ , (2.3c) ∞ X k = −∞ q k (5 k +3) ( q ; q ) k = ( q ; q ) ∞ ( q , q , q ; q ) ∞ ( q , q ; q ) ∞ . (2.3d)Combinatorial interpretations of theses identities are given in Theorem 5.2.To deduce the bilateral identities in Corollary 2.3, first replace q by q in (2.1) andthen observe that the respective z = q − and z = q − cases of (2.1a) give (2.3b) and(2.3c), whereas the respective z = q − and z = q − cases of (2.1b) give (2.3a) and(2.3d).Our next result is a bilateral extension of the two G¨ollnitz–Gordon identities in (1.2). Theorem 2.4.
We have the following two bilateral summations: ∞ X k = −∞ ( − zq ; q ) k ( zq ; q ) k q k z k = ( − zq, /z ; q ) ∞ ( z q , /z ; q ) ∞ ( q ; q ) ∞ × (cid:2) ( z q , z − q ; q ) ∞ + z − ( z q , z − q ; q ) ∞ (cid:3) , (2.4a) ∞ X k = −∞ ( − zq ; q ) k ( zq ; q ) k q k ( k +2) z k = ( − zq, /z ; q ) ∞ ( z q , /z ; q ) ∞ ( q ; q ) ∞ × (cid:2) ( z q , z − q ; q ) ∞ + z − ( z q, z − q ; q ) ∞ (cid:3) . (2.4b)The z → z → Corollary 2.5 (Bilateral mod 32 identities) . ∞ X k = −∞ ( − q ; q ) k ( q ; q ) k +1 q ( k +2)(4 k +1) = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ ( q ; q ) ∞ , (2.5a) ∞ X k = −∞ ( − q ; q ) k ( q ; q ) k +1 q k (4 k +3) = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ ( q ; q ) ∞ , (2.5b) ∞ X k = −∞ ( − q ; q ) k ( q ; q ) k q k (4 k − = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q, q ; q ) ∞ ( q ; q ) ∞ , (2.5c) MICHAEL J. SCHLOSSER ∞ X k = −∞ ( − q ; q ) k ( q ; q ) k q k (4 k +7) = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ ( q ; q ) ∞ . (2.5d)To deduce the bilateral identities in Corollary 2.5, first replace q by q in (2.4) andthen observe that the respective z = q and z = q − cases of (2.4a) give (2.5b) and(2.5c), whereas the respective z = q and z = q − cases of (2.4b) give (2.5a) and (2.5d).Notice that Equations (2.5a) and (2.5b) can be obtained from each other by replacing q by − q . The same relation also holds for Equations (2.5c) and (2.5d).We would like to stress that the bilateral summations in Corollaries 2.3 and 2.5,which we believe are new (and also beautiful , in line with Hardy’s quote about (1.1)stated in the introduction), are not special cases of the following bilateral extension ofthe Lebesgue identity ∞ X k = −∞ ( a ; q ) k ( bq ; q ) k q ( k +12 ) b k = ( q , abq, q/ab, bq /a ; q ) ∞ ( bq, q/a ; q ) ∞ (2.6)(which can be obtained from [18, Appendix (II.30), c → ∞ followed by ( a, b ) ( ab, a )]).A noteworthy special case of (2.6) due to G¨ollnitz [20], which can be compared tothe G¨ollnitz–Gordon identities in (1.2), is obtained by letting ( a, b, q ) ( − q, , q ): ∞ X k =0 ( − q ; q ) k ( q ; q ) k q k ( k +1) = 1( q , q , q ; q ) ∞ . (2.7)Another noteworthy special case of (2.6) is obtained by letting ( a, b ) ( − q, ∞ X k =0 ( − q ; q ) k ( q ; q ) k q ( k +12 ) = ( q ; q ) ∞ ( q ; q ) ∞ . (2.8)which is identity (8) in Slater’s list.Other bilateral summations of the Rogers–Ramanujan type which we found are col-lected in the following theorem: Theorem 2.6.
We have the following four bilateral summations: ∞ X k = −∞ ( − z ; q ) k ( z q ; q ) k q ( k ) z k = ( − z ; q ) ∞ ( q ; q ) ∞ ( z ; q ) ∞ ( q /z ; q ) ∞ ( q , z , z − q ; q ) ∞ , (2.9a) ∞ X k = −∞ ( − z ; q ) k ( zq ; q ) k q k ( k +1) z k = ( q/z ; q ) ∞ ( − zq ; q ) ∞ ( z q , q /z , q ; q ) ∞ ( q , − z q , − z − q ; q ) ∞ , (2.9b) ∞ X k = −∞ ( − z ; q ) k ( z ; q ) k q k ( k − z k = ( q/z ; q ) ∞ ( − z ; q ) ∞ ( z , q /z , q ; q ) ∞ ( q ; q ) ∞ × (cid:2) ( − z q, − z − q ; q ) ∞ + z ( − z q , − z − q ; q ) ∞ (cid:3) , (2.9c) ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 7 ∞ X k = −∞ q k ( z ; q ) k +1 z k = ( q/z ; q ) ∞ ( z , q /z , q ; q ) ∞ ( q ; q ) ∞ × (cid:2) ( − z q , − z − q ; q ) ∞ + z ( − z q , − z − q ; q ) ∞ (cid:3) . (2.9d)The case q q , followed by z → q , of (2.9a) reduces to identity (25) in Slater’s list,which can be stated as ∞ X k =0 ( − q ; q ) k ( q ; q ) k q k = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ = ( q ; q ) ∞ ( q ; q ) ∞ ( q, q ; q ) ∞ . (2.10)The z → ∞ X k =0 ( − q ) k ( q ; q ) k q k ( k +1) = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ . (2.11)The z → q cases of (2.9b) and (2.9c) reduce to identities (50) and (29) in Slater’s list,which can be stated as ∞ X k =0 ( − q ; q ) k ( q ; q ) k +1 q k ( k +2) = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ , (2.12a)and ∞ X k =0 ( − q ; q ) k ( q ; q ) k q k = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ , (2.12b)respectively. The z → q case of (2.9c) reduces to identity (28) in Slater’s list, whichcan be stated as ∞ X k =0 ( − q ; q ) k ( q ; q ) k +1 q k ( k +1) = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ . (2.13)Multiplication of both sides of (2.9d) by (1 − z ) and letting z → X k ≥ q k ( q ; q ) k = 1( q , q , q ; q ) ∞ ( q , q ; q ) ∞ . (2.14)The z → q case of (2.9d) reduces to identity (38) in Slater’s list, which can be statedas X k ≥ q k ( k +1) ( q ; q ) k +1 = 1( q, q , q ; q ) ∞ ( q , q ; q ) ∞ . (2.15) MICHAEL J. SCHLOSSER
The z → − k by − k , to the identities X k ≥ ( − q ) k ( q ; q ) k q k = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ = 1( q, q ; q ) ∞ ( q, q ; q ) ∞ , (2.16a) X k ≥ ( − q ; q ) k ( q ; q ) k q k = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ = 1( q, q ; q ) ∞ ( q , q ; q ) ∞ . (2.16b)Equation (2.16a) is given by Slater as identity (24), while (2.16b) is due to Ismail andStanton [26, Thm. 7]. It is not difficult to transform the specific φ series (with van-ishing lower parameter) on the left-hand sides of Equations (2.16) by suitable instancesof the q -Pfaff transformation [18, Appendix (III.4)] to φ series, by which (2.16a) isseen to be equivalent to the q
7→ − q case of (2.11) and also to (2.10), while (2.16b) isthen seen to be equivalent to an identity by Ramanujan (cf. [7, p. 87, Entry 4.2.11])and also to the q
7→ − q case of (2.13).As consequence of Equation (2.9c), we obtain the following two bilateral summations: Corollary 2.7 (Bilateral mod 6 identities) . ∞ X k = −∞ ( q ; q ) k ( − q ; q ) k +1 ( − k q k (3 k +2) = ( q , q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ , (2.17a) ∞ X k = −∞ ( q ; q ) k ( − q ; q ) k ( − k q k (3 k − = ( q, q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ . (2.17b)To deduce the bilateral identities in Corollary 2.7, first replace q by q in (2.9c) andthen observe that the respective z = − q − and z = − q cases give (2.17a) and (2.17b).The identities in Corollary 2.7 become even nicer if the summation index k is replacedby − k : Corollary 2.7 ′ (Bilateral mod 6 identities) . ∞ X k = −∞ ( − q ; q ) k − ( q ; q ) k q k − = ( q , q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ , (2.18a) ∞ X k = −∞ ( − q ; q ) k ( q ; q ) k q k = ( q, q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ . (2.18b)Further, as consequence of Equation (2.9d), we obtain the following four bilateralsummations: ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 9
Corollary 2.8 (Bilateral mod 32 identities) . ∞ X k = −∞ q k (4 k − ( q ; q ) k ( − q ; q ) k = ( q , q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ ( q ; q ) ∞ , (2.19a) ∞ X k = −∞ q k (4 k +3) ( q ; q ) k +1 ( − q ; q ) k = ( q , q ; q ) ∞ ( q ; q ) ∞ ( q, q ; q ) ∞ ( q ; q ) ∞ , (2.19b) ∞ X k = −∞ q k (4 k − ( q ; q ) k ( − q ; q ) k = ( q , q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ ( q ; q ) ∞ , (2.19c) ∞ X k = −∞ q k (4 k +3) ( q ; q ) k ( − q ; q ) k +1 = ( q, q ; q ) ∞ ( q ; q ) ∞ ( q , q ; q ) ∞ ( q ; q ) ∞ . (2.19d)To deduce the bilateral identities in Corollary 2.8, first replace q by − q in (2.9d)and then observe that the respective z = q and z = q − cases give (2.19b) and (2.19c).The identities in (2.19a) and (2.19d) follow by replacing q by − q in (2.19c) and (2.19b),respectively.3. Derivations of the main results in the single series case
A rich source of material on basic hypergeometric series is Gasper and Rahman’sclassic textbook [18]. In particular, we refer to that book for standard notions (such asthat of a bilateral basic hypergeometric r ψ s series), and to Appendix I of that book forthe elementary manipulations of q -shifted factorials which we employ without explicitmention.An identity which we make crucial use of is Jacobi’s triple product identity (cf. [18,(II.28)]) ∞ X k = −∞ q ( k )( − z ) k = ( q, z, q/z ; q ) ∞ . (3.1)Our starting point for deriving bilateral summations of the Rogers–Ramanujan typeis the following transformation of a general bilateral ψ series into a multiple of avery-well-poised ψ series due to Bailey [10, (3.2)] (see also [18, Exercise 5.11, secondidentity]). ∞ X k = −∞ ( e, f ; q ) k ( aq/c, aq/d ; q ) k (cid:18) aqef (cid:19) k = ( q/c, q/d, aq/e, aq/f ; q ) ∞ ( aq, q/a, aq/cd, aq/ef ; q ) ∞ × ∞ X k = −∞ (1 − aq k )( c, d, e, f ; q ) k (1 − a )( aq/c, aq/d, aq/e, aq/f ; q ) k q k (cid:18) a qcdef (cid:19) k , (3.2)valid for | aq/cd | < | aq/ef | <
1. Bailey obtained this transformation by bilat-eralizing Watson’s transformation (cf. [18, (III.18)]) using the same method (replacing n by 2 n , shifting the summation index k k + n , suitably shifting parameters andtaking the limit n → ∞ ), applied by Cauchy [14] in his second proof of Jacobi’s tripleproduct identity.In (3.2) we now let f → ∞ and perform the simultaneous substitutions ( a, c, d, e ) ( az, az/b, az/c, a ). This yields the following transformation of a general ψ series intoa multiple of a very-well-poised ψ series. ∞ X k = −∞ ( a ; q ) k ( bq, cq ; q ) k q ( k +12 )( − z ) k = ( bq/az, cq/az, zq ; q ) ∞ ( azq, q/az, bcq/az ; q ) ∞ × ∞ X k = −∞ (1 − azq k )( az/b, az/c, a ; q ) k (1 − az )( bq, cq, zq ; q ) k q ( k ) (cid:0) − bczq (cid:1) k , (3.3)valid for | bcq/az | < Proof of Theorem 2.1.
In (3.3), we first let c →
0, perform the substitutions ( b, z ) ( z, bz/a ) and let a → ∞ . We obtain ∞ X k = −∞ q k ( bz ) k ( zq ; q ) k = ( q/b ; q ) ∞ ( bzq, q/bz ; q ) ∞ ∞ X k = −∞ (1 − bzq k )(1 − bz ) ( b ; q ) k ( zq ; q ) k q ( k ) (cid:0) − b z q (cid:1) k . (3.4)Now the b = z case of (3.4) reduces to ∞ X k = −∞ q k z k ( zq ; q ) k = (1 /z ; q ) ∞ z − ( z q, /z ; q ) ∞ ∞ X k = −∞ (1 + zq k ) q ( k ) (cid:0) − z q (cid:1) k , which after two applications of (3.1) yields (2.1a). Similarly, the b = zq case of (3.4)reduces to ∞ X k = −∞ q k ( k +1) z k ( zq ; q ) k = (1 /z ; q ) ∞ ( z q, /z ; q ) ∞ ∞ X k = −∞ (1 − z q k ) q ( k ) (cid:0) − z q (cid:1) k , which after two applications of (3.1) yields (2.1b). (cid:3) In the remaining proofs we only give brief sketches of details.
Proof of Theorem 2.4.
In (3.3), we first let c →
0, replace q by q and set ( a, b, z ) ( − zq, z, − zq − ). The result, after two applications of (3.1), is (2.4a). Now (2.4b) canreadily be obtained from (2.4a) by replacing z by − /zq and reversing the sum. (cid:3) Proof of Theorem 2.6.
The identity (2.9a) follows from (3.3) by making the substitution( a, b, c, z ) ( − z, zq − / , − zq − / , − z ), and applying (3.1). The identity (2.9b) followsfrom (3.3) by replacing q by q , setting ( a, b, c, z ) ( − z, z, zq − , − z ), and applying(3.1). The identity (2.9c) follows from (3.3) by replacing q by q , setting ( a, b, c, z ) ( − z, zq − , zq − , − zq − ), and applying (3.1) twice. The identity (2.9d) follows from ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 11 (3.3) by replacing q by q , setting ( b, c, z ) ( z, zq − , z /a ) followed by taking a → ∞ ,applying (3.1) twice and dividing both sides by (1 − z ). (cid:3) Multiseries extensions
Here we derive multiseries extensions of the results from Section 2. Throughout weassume r ≥
2. We write k = ( k , . . . , k r − ) and define k r := 0. Further, we defineΛ r − := { k ∈ Z r − | ∞ > k ≥ · · · ≥ k r − > −∞} in order to compactly specify the range of our multilateral summations.Our multiseries extensions of Theorems 2.1 and 2.4 are multilateral extensions of theAndrews–Gordon identities [3, 4], which, for integers r and i with r ≥ ≤ i ≤ r ,can be written as X ∞ >k ≥···≥ k r − ≥ q P rj =1 k j + P r − j = i k j ( q ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 = ( q i , q r +1 − i , q r +1 ; q r +1 ) ∞ ( q ; q ) ∞ , (4.1)and X ∞ >k ≥···≥ k r − ≥ ( − q − k ; q ) k q P r − j =1 k j +2 P r − j = i k j ( q ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 = ( q ; q ) ∞ ( q i − , q r +1 − i , q r ; q r ) ∞ ( q ; q ) ∞ . (4.2)These identities reduce to (1.1) and (1.2) for r = 2.In [13], Bressoud also gave an even modulus analogue of the Andrews–Gordon iden-tities in (4.1), namely X ∞ >k ≥···≥ k r − ≥ q P r − j =1 k j + P r − j = i k j ( q ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 = ( q i , q r − i , q r ; q r ) ∞ ( q ; q ) ∞ , (4.3)where 1 ≤ i ≤ r . The ( r, i ) = (2 ,
1) and ( r, i ) = (2 ,
2) cases of (4.3) are special cases ofthe q -binomial theorem.Our multilateral summations in this section are obtained by applying an analysisanalogous to the single series case. Our starting point is the following multiseriestransformation which is immediately obtained from a result by Agarwal, Andrews andBressoud [1, Theorem 3.1 with Equations (4.1) and (4.2)]: Proposition 4.1.
Let r and i be integers with r ≥ and ≤ i ≤ r . Further, let n bea nonnegative integer. Then, with k := n and k r := 0 , we have the following series transformation: X n ≥ k ≥···≥ k r − ≥ r Y j =1 ( b j , c j ; q ) k j ( q ; q ) k j − − k j i − Y j =1 ( a/b j c j ; q ) k j − − k j ( a/b j , a/c j ; q ) k j − × r Y j = i ( aq/b j c j ; q ) k j − − k j ( aq/b j , aq/c j ; q ) k j − r − Y j =1 (cid:18) ab j c j (cid:19) k j · q P r − j = i k i ! = n X k =0 ( a ; q ) k ( − k q ( k ) +( r +1 − i ) k ( q ; q ) k ( q ; q ) n − k ( a ; q ) n + k a rk Q rj =1 ( b j c j ) k × i − Y j =1 ( b j , c j ; q ) k ( a/b j , a/c j ; q ) k r Y j = i ( b j , c j ; q ) k ( aq/b j , aq/c j ; q ) k × " − q k ) aq k − (1 − aq k − ) r Y j = i b j c j (1 − aq k /b j )(1 − aq k /c j ) aq (1 − b j q k − )(1 − c j q k − ) . (4.4)This is (even in the i = r case) different from the multivariate Watson transformationby Andrews [4, Thm. 4] (which, if used as a starting point instead, would only serve toprove the extremal i = 1 and i = r cases of the multiseries identities we are after).By multilateralization, we now deduce the following transformation of multiseries. Corollary 4.2.
Assuming k := ∞ , we have for r ≥ and ≤ i ≤ r the followingtransformation: X k ∈ Λ r − Q r − j =1 ( b j , c j ; q ) k j Q r − j =1 ( q ; q ) k j − k j +1 i − Y j =1 ( a/b j c j ; q ) k j − − k j ( a/b j , a/c j ; q ) k j − × Q rj = i ( aq/b j c j ; q ) k j − − k j Q rj = i ( aq/b j , aq/c j ; q ) k j − r − Y j =1 (cid:18) ab j c j (cid:19) k j · q P r − j = i k j ! = ( q/b r , q/c r ; q ) ∞ ( a, q/a, aq/b r c r ; q ) ∞ × ∞ X k = −∞ q k +( r − i ) k a ( r +1) k Q rj =1 ( b j c j ) k i − Y j =1 ( b j , c j ; q ) k ( a/b j , a/c j ; q ) k r Y j = i ( b j , c j ; q ) k ( aq/b j , aq/c j ; q ) k × " − r Y j = i b j c j (1 − aq k /b j )(1 − aq k /c j ) aq (1 − b j q k − )(1 − c j q k − ) , (4.5) valid for (cid:12)(cid:12) q r − i Q r − j =1 ( a/b j c j ) (cid:12)(cid:12) < and (cid:12)(cid:12) q r − i Q r − j =1 ( a/b j +1 c j +1 ) (cid:12)(cid:12) < . ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 13
Remark . Notice that for i = r the expression in the big brackets on the right-handside of (4.5) simplifies to1 − bc (1 − aq k /b )(1 − aq k /c ) aq (1 − bq k − )(1 − cq k − ) = (1 − aq k − )(1 − bc/aq )(1 − bq k − )(1 − cq k − )(whereas the corresponding larger expression in the big brackets on the right-hand sideof (4.4) does not factorize for i = r ), where we replaced ( b r +1 , c r +1 ) by ( b, c ), and thetransformation in (4.5) is then seen to be an ( r − a replaced by a/q ) which alternatively could alsobe obtained by multilateralization of Andrews’ formula [4, Thm. 4]. Proof of Corollary 4.2.
To obtain (4.5) from (4.4), replace n by 2 n , shift the summationindices k , . . . , k r − (on the left-hand side) and k (on the right-hand side) by n , performthe substitutions a aq − n , b j b j q − n , c j c j q − n , for j = 1 , . . . , r , and let n → ∞ while appealing to Tannery’s theorem for taking termwise limits. (cid:3) All the multiseries identities of the Rogers–Ramanujan type in this section are derivedby means of the following lemma which extends Equation (3.3):
Lemma 4.4.
We have for r ≥ and ≤ i ≤ r the following transformation: X k ∈ Λ r − ( q − k /a ; q ) k q P r − j =1 k j + P r − j = i k j ( bq, cq ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 ( az ) P r − j =1 k j = ( zq, b/az, c/az ; q ) ∞ ( azq, /az, bc/az ; q ) ∞ × ∞ X k = −∞ (cid:18) q (2 r − ( k ) (cid:0) − a r − bcz r q r − i (cid:1) k ( a, azq/b, azq/c ; q ) k ( zq, bq, cq ; q ) k × (cid:20) − a i +1 − r z i +1 − r q i − r ) k (1 − bq k )(1 − cq k ) b c (1 − azq k /b )(1 − azq k /c ) (cid:21) (cid:19) . (4.6) Proof.
In Corollary 4.2 successively let b , . . . , b r − → ∞ and c , c , . . . , c r − → ∞ ,and perform the substitution ( a, b , b r , c r ) ( azq, a, azq/b, azq/c ). This establishes(together with some elementary manipulations of q -shifted factorials) the i = 2 , . . . , r cases of the Lemma directly. The i = 1 case can be established as follows: Start withthe i = 1 case of the right-hand side of (4.6) and split the sum according to the twoterms in the bracket. After shifting the summation index k by one in the second sum,the two sums can be combined and the resulting expression is seen to be equal to the i = r and z zq case of the right-hand side of (4.6). (For i = r the expression in thebracket factorizes as we know from Remark 4.3.) Thus the sum equals the left-handside of the i = r and z zq case of (4.6) which is the same as its i = 1 case with z left unchanged. (cid:3) For convenience, we write out the i = r case of Lemma 4.4 separately: Lemma 4.5.
We have for r ≥ the following transformation: X k ∈ Λ r − ( q − k /a ; q ) k q P r − j =1 k j ( az ) P r − j =1 k j ( bq, cq ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 = ( zq, bq/az, cq/az ; q ) ∞ ( azq, q/az, bcq/az ; q ) ∞ × ∞ X k = −∞ (1 − azq k )(1 − az ) ( a, az/b, az/c ; q ) k ( zq, bq, cq ; q ) k q (2 r − ( k ) (cid:0) − a r − bcz r − q r (cid:1) k . (4.7)From Lemma 4.4 (and its special case Lemma 4.5) we now readily deduce a numberof multilateral identities of the Rogers–Ramanujan type.We start with a multiseries generalization of Theorem 2.1. Theorem 4.6.
We have for r ≥ and ≤ i ≤ r the following multilateral summations: X k ∈ Λ r − q P r − j =1 k j + P r − j = i k j z P r − j =1 k j ( zq ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 = (1 /z ; q ) ∞ (1 /z , z q ; q ) ∞ ( q r +1 ; q r +1 ) ∞ × (cid:2) ( z r +1 q r +1 − i , z − r − q i ; q r +1 ) ∞ + z i − − r ( z r +1 q i , z − r − q r +1 − i ; q r +1 ) ∞ (cid:3) . (4.8) Proof.
In Lemma 4.4, first let c →
0, then perform the substitution ( b, z ) ( z, z /a )and let a → ∞ . After two applications of (3.1) the identity (4.8) is obtained. (cid:3) The z → Corollary 4.7.
We have for r ≥ and ≤ i ≤ r the following multilateral summations: X k ∈ Λ r − q (2 r +1) P r − j =1 k j − i P r − j =1 k j +(2 r +1) P r − j = i k j ( q r +1 − i ; q r +1 ) k r − Q r − j =1 ( q r +1 ; q r +1 ) k j − k j +1 = ( q i ; q r +1 ) ∞ ( q i (2 r +1) , q (2 r +1 − i )(2 r +3) , q (2 r +1) ; q (2 r +1) ) ∞ ( q i , q r +1 − i ; q r +1 ) ∞ , (4.9a) and q ( i − r +1 − i ) X k ∈ Λ r − q (2 r +1) P r − j =1 k j − r +1 − i ) P r − j =1 k j +(2 r +1) P r − j = i k j ( q i ; q r +1 ) k r − Q r − j =1 ( q r +1 ; q r +1 ) k j − k j +1 = ( q r +1 − i ; q r +1 ) ∞ ( q i (2 r +1) , q (2 r +1 − i )(2 r +1) , q (2 r +1) ; q (2 r +1) ) ∞ ( q i , q r +1 − i ; q r +1 ) ∞ . (4.9b) Proof.
First replace q by q r +1 in (4.6). Then the special case z = q − i gives (4.9a), whilethe special case z = q i , after some elementary manipulations (including a simultaneousshift of the summation indices by − (cid:3) Next we give a multiseries generalization of Theorem 2.4.
ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 15
Theorem 4.8.
We have for r ≥ and ≤ i ≤ r the following multilateral summations: X k ∈ Λ r − ( − q − k /z ; q ) k q P r − j =1 k j +2 P r − j = i k j ( zq ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 z P r − j =1 k j = ( − zq, /z ; q ) ∞ ( z q , /z ; q ) ∞ ( q r ; q r ) ∞ × h ( z r q r +1 − i , z − r q i − ; q r ) ∞ + z i − − r ( z r q i − , z − r q r +1 − i ; q r ) ∞ i . (4.10) Proof.
In Lemma 4.4, first let c →
0, replace q by q and set ( a, b, z ) ( − zq, z, − zq − ).After two applications of (3.1) the identity (4.10) is obtained. (cid:3) The z → Corollary 4.9.
We have for r ≥ and ≤ i ≤ r the following multilateral summations: X k ∈ Λ r − ( − q r − i − rk ; q r ) k q r P r − j =1 k j − i − P r − j =1 k j +4 r P r − j = i k j ( q r +1 − i ; q r ) k r Q r − j =1 ( q r ; q r ) k j − k j +1 = ( q i − ; q r ) ∞ ( q r (2 i − , q r (2 r +1 − i ) , q r ; q r ) ∞ ( q i − , q r +1 − i ; q r ) ∞ ( q r +1 − i ) ; q r ) ∞ , (4.11a) and q i − r − i ) X k ∈ Λ r − ( − q r +1 − i − rk ; q r ) k q r P r − j =1 k j +2(2 i − P r − j =1 k j +4 r P r − j = i k j ( q i − ; q r ) k r − Q r − j =1 ( q r ; q r ) k j − k j +1 = ( q r +1 − i ; q r ) ∞ ( q r (2 i − , q r (2 r +1 − i ) , q r ; q r ) ∞ ( q r − i , q r +1 − i ) ; q r ) ∞ ( q i − ; q r ) ∞ . (4.11b) Proof.
First replace q by q r in (4.8). Then the special case z = q − i gives (4.11a), whilethe special case z = q i − , after some elementary manipulations, gives (4.11b). (cid:3) Next we give a multilateral extension of the extremal i = r case of Bressoud’s evenmodulus analogue of the Andrews–Gordon identities (4.3). (For 1 ≤ i ≤ r − r = 2 case.) Theorem 4.10.
We have for r ≥ the following multilateral summation: X k ∈ Λ r − q P r − j =1 k j z P r − j =1 k j ( zq ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 = ( q ; q ) ∞ ( z r q r , z − r q r , q r ; q r ) ∞ ( zq ; q ) ∞ ( q/z ; q ) ∞ . (4.12) Proof.
In Lemma 4.5, first let c →
0, replace q by q and perform the substitution( b, c, z ) ( z , − z , z/a ), let a → ∞ and apply (3.1). (cid:3) For z = 1 (4.12) reduces to the i = r case of (4.3). For r = 2 (4.12) reduces to aspecial case of Ramanujan’s ψ summation (1.3).Next we give a multilateral generalization of Theorem 2.6. (Again, for reasons ofabsolute convergence, we are only able to apply Lemma 4.5, i.e. the i = r case ofLemma 4.4. The c = 1 cases of the latter could be applied to obtain multiseriesidentities which would be naturally bounded from below, such as the original Andrews–Gordon identities. However, in this work we are after multilateral identities.) Theorem 4.11.
We have for r ≥ the following multilateral summations: X k ∈ Λ r − ( − q − k /z ; q ) k q P r − j =1 k j ( k j − ( z q ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 z P r − j =1 k j = ( − z ; q ) ∞ ( q ; q ) ∞ ( z r − , z − r q r − , q r − ; q r − ) ∞ ( z ; q ) ∞ ( q /z ; q ) ∞ , (4.13a) X k ∈ Λ r − ( − q − k /z ; q ) k q P r − j =1 k j ( zq ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 z P r − j =1 k j = ( q/z ; q ) ∞ ( − zq ; q ) ∞ ( − z r − q r − , − z − r q r − , q r − ; q r − ) ∞ ( z q , q /z , q ; q ) ∞ , (4.13b) X k ∈ Λ r − ( − q − k /z ; q ) k q P r − j =1 k j ( k j − ( z ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 z P r − j =1 k j = ( q/z ; q ) ∞ ( − z ; q ) ∞ ( z , q /z , q ; q ) ∞ ( q r − ; q r − ) ∞ × (cid:2) ( − z r − q, − z − r q r − ; q r − ) ∞ + z r − ( − z r − q r − , − z − r q ; q r − ) ∞ (cid:3) , (4.13c) X k ∈ Λ r − q P r − j =1 k j z P r − j =1 k j ( z ; q ) k r − Q r − j =1 ( q ; q ) k j − k j +1 = ( q/z ; q ) ∞ ( q r ; q r ) ∞ ( z , q /z , q ; q ) ∞ × (cid:2) ( − z r q r − , − z − r q r +1 ; q r ) ∞ + z ( − z r q r +1 , − z − r q r − ; q r ) ∞ (cid:3) . (4.13d) Proof.
To prove the respective identities, apply Lemma 4.5, perform a specific sub-stitution of variables (as specified below), occasionally combined with taking a limit,and then apply one or two instances of Jacobi’s triple product identity (3.1). For(4.13a), take ( a, b, c, z ) ( − z, zq − , − zq − , − z ). For (4.13b), take ( a, b, c, z, q ) ( − z, z, − zq − , − z, q ). For (4.13c), take ( a, b, c, z, q ) ( − z, zq − , − zq − , − zq − , q ).Finally, for (4.13d), take ( b, c, z, q ) ( z, zq − , z /a, q ), divide both sides by (1 − z )and subsequently let a → ∞ . (cid:3) ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 17
All identities from Theorem 4.11 reduce to multiseries generalizations of correspond-ing unilateral identities discussed after Theorem 2.6. For instance, we have X k ∈ Λ r − q P r − j =1 k j ( k j + δ ) ( q ; q ) δ +2 k r − Q r − j =1 ( q ; q ) k j − k j +1 = ( − q r (1+ δ ) − , − q r (1 − δ )+1 , q r ; q r ) ∞ ( q ; q ) ∞ , (4.14)where δ = 0 ,
1, which generalizes (2.14) and (2.15), respectively. The δ = 0 case isobtained by multiplying both sides of (4.13d) by (1 − z ) and letting z →
1, while the δ = 1 case is obtained from (4.13d) by letting z → q . We leave other specializations ofidentities from Theorem 4.11 which generalize classical unilateral summations to thereader.If in (4.13c) we replace q by q r − and set z = − q − or z = − q , we obtain thefollowing two multilateral summations generalizing Corollary 2.7. Corollary 4.12.
For r ≥ we have the following multilateral summations: X k ∈ Λ r − ( q − r − k ; q r − ) k q r − P r − j =1 k j +4( r − P r − j =1 k j ( − q r − ; q r − ) k r − Q r − j =1 ( q r − ; q r − ) k j − k j +1 = ( q r − , q r − ) ∞ ( q r − , q r − r − , q r − ; q r − ) ∞ ( q ; q r − ) ∞ ( q r − , q r − ; q r − ) ∞ , (4.15a) X k ∈ Λ r − ( q r − − r − k ; q r − ) k q r − P r − j =1 k j − r − P r − j =1 k j ( − q ; q r − ) k r − Q r − j =1 ( q r − ; q r − ) k j − k j +1 = ( q, q r − ) ∞ ( q r − , q r − r − , q r − ; q r − ) ∞ ( q r − ; q r − ) ∞ ( q , q r − ; q r − ) ∞ . (4.15b)Finally, we have the following generalization of Corollary 2.8: Corollary 4.13.
For r ≥ we have the following multilateral summations: X k ∈ Λ r − q r P r − j =1 k j − r − P r − j =1 k j ( q ; q r ) k r − ( − q r +1 ; q r ) k r − Q r − j =1 ( q r ; q r ) k j − k j +1 = ( q r , q r − ; q r ) ∞ ( q r , q r (2 r − , q r ; q r ) ∞ ( q , q r − ; q r ) ∞ ( q r , q r − ; q r ) ∞ , (4.16a) X k ∈ Λ r − q r P r − j =1 k j +2(2 r − P r − j =1 k j ( q r − ; q r ) k r − ( − q r − ; q r ) k r − Q r − j =1 ( q r ; q r ) k j − k j +1 = ( q r , q r +1 ; q r ) ∞ ( q r , q r (2 r − , q r ; q r ) ∞ ( q, q r − ; q r ) ∞ ( q r , q r +1) ; q r ) ∞ , (4.16b) X k ∈ Λ r − q r P r − j =1 k j − r − P r − j =1 k j ( q r +1 ; q r ) k r − ( − q ; q r ) k r − Q r − j =1 ( q r ; q r ) k j − k j +1 = ( q r − , q r ; q r ) ∞ ( q r , q r (2 r − , q r ; q r ) ∞ ( q , q r − ; q r ) ∞ ( q r − , q r ; q r ) ∞ , (4.16c) X k ∈ Λ r − q r P r − j =1 k j +2(2 r − P r − j =1 k j ( q r − ; q r ) k r − ( − q r − ; q r ) k r − Q r − j =1 ( q r ; q r ) k j − k j +1 = ( q, q r ; q r ) ∞ ( q r , q r (2 r − , q r ; q r ) ∞ ( q r +1 , q r − ; q r ) ∞ ( q , q r ; q r ) ∞ . (4.16d)To deduce the multilateral identities in Corollary 4.13, first replace q by − q r in(4.13d) and then put z = q r − to deduce (4.16b) or z = q − r to deduce (4.16c). Theidentities in (4.16a) and (4.16d) follow by replacing q by − q in (4.16c) and (4.16b),respectively. 5. Combinatorial applications
In this section we provide first combinatorial interpretations for bilateral identitiesof the Rogers–Ramanujan type. We restrict to the identities in Corollary 2.3. Com-binatorial interpretations of other bilateral and multilateral Rogers–Ramanujan typeidentities shall be considered elsewhere.MacMahon [29] and Schur [34] were the first to interpret the Rogers–Ramanujanidentities in (1.1) combinatorially. Their interpretations use the notions of (number)partitions.A partition λ = ( λ , λ , . . . , λ l ) of a nonnegative integer n (shortly denoted by λ ⊢ n )is a decomposition of n into a sum of positive integer parts λ , λ , . . . , λ l , for somenonnegative integer l , such that n = λ + λ + · · · + λ l . The order of the parts doesnot matter. Without loss of generality, we may assume the parts to be ordered weaklydecreasing, i.e., we have λ ≥ λ ≥ · · · ≥ λ l . If λ j = m , we also say that the j -th partof λ has size m . The number l of parts of λ is called the length of λ . For n = 0 we musthave l = 0 and by definition there is exactly one partition of 0, the empty partition ∅ .If λ ⊢ n , then we call n the norm of λ . The combined norm of a pair ( λ, µ ) of partitionsis defined to be the sum of the norms of λ and µ . See Andrews’ book [5] for a thoroughaccount of partitions (including the explanation of other standard notions such as thatof the conjugate of a partition and the sum of two partitions, which we use below).It is easy to see that the sum on the left-hand side of (1.1a) is the generating functionfor partitions into different parts which differ by at least 2 while the product on theright-hand side of (1.1a) is the generating function for partitions into parts of sizecongruent to 1 or 4 modulo 5. Similarly the sum on the left-hand side of (1.1b) isthe generating function for partitions into different parts greater or equal to 2 whosedifference is at least 2 while the product on the right-hand side of (1.1b) is the generatingfunction for partitions into parts of size congruent to 2 or 3 modulo 5. Thus, incombinatorial terms the two classical Rogers–Ramanujan identities in (1.1) take thefollowing form: ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 19
Proposition 5.1 (MacMahon/Schur) . Let n be a nonnegative integer. A: The number of partitions of n into parts which differ by at least equals thenumber of partitions of n into parts of size congruent to or modulo . B: The number of partitions of n into parts greater or equal to which differ by atleast equals the number of partitions of n into parts of size congruent to or modulo . After having reviewed the classical case, we now turn to the four bilateral summationsin Corollary 2.3. In particular, consider the expression q k (5 k − ( q ; q ) k (5.1)which is the k -th term of the sum in (2.3a). For positive k , the q -series expansion of(5.1) clearly has nonnegative integer coefficients. When this expression is divided by( q ; q ) ∞ and k is replaced by − k , we have( − k q k (5 k +3)2 ( q k ; q ) ∞ (5.2)whose q -series expansion, depending on the parity of k , has either exclusively nonneg-ative or negative integer coefficients.The strategy is thus to divide both sides of the respective identities in Corollary 2.3by their numerator factors from the product sides. In addition, we split the bilateralsums into three parts, namely a first part with the terms having positive summationindex, a second part with the terms having nonpositive even summation index, and athird part with the terms having negative odd summation index. The latter, third partis moved to the other side of the equation.Concretely, we rewrite the bilateral identities in (2.3) in the following form:1( q ; q ) ∞ ( q , q , q ; q ) ∞ X k ≥ q k (5 k − ( q ; q ) k + 1( q , q , q ; q ) ∞ X k ≥ k even q k (5 k +3)2 ( q k ; q ) ∞ = 1( q , q ; q ) ∞ + 1( q , q , q ; q ) ∞ X k ≥ k odd q k (5 k +3)2 ( q k ; q ) ∞ , (5.3a) q − ( q ; q ) ∞ ( q , q , q ; q ) ∞ X k ≥ q ( k − k − ( q ; q ) k + q − ( q , q , q ; q ) ∞ X k ≥ k even q k (5 k +11)2 +3 ( q k ; q ) ∞ = 1( q, q ; q ) ∞ + q − ( q , q , q ; q ) ∞ X k ≥ k odd q k (5 k +11)2 +3 ( q k ; q ) ∞ , (5.3b) q − ( q ; q ) ∞ ( q , q , q ; q ) ∞ X k ≥ q k (5 k − ( q ; q ) k + q − ( q , q , q ; q ) ∞ X k ≥ k even q k (5 k +9)2 +2 ( q k ; q ) ∞ = 1( q, q ; q ) ∞ + q − ( q , q , q ; q ) ∞ X k ≥ k odd q k (5 k +9)2 +2 ( q k ; q ) ∞ , (5.3c)1( q ; q ) ∞ ( q , q , q ; q ) ∞ X k ≥ q k (5 k +3) ( q ; q ) k + 1( q , q , q ; q ) ∞ X k ≥ k even q k (5 k − ( q k ; q ) ∞ = 1( q , q ; q ) ∞ + 1( q , q , q ; q ) ∞ X k ≥ k odd q k (5 k − ( q k ; q ) ∞ . (5.3d)We now deliver combinatorial interpretations of these identities. We first analyze theexpressions appearing in (5.3a). For positive integer k , (5.1) is the generating functionfor partitions into exactly 5 k − k −
1) parts appear in ( k −
1) differentgroups of parts having multiplicity 5, and where parts from different groups differ byat least 2. This interpretation comes from interpreting 1 / ( q ; q ) k as the generatingfunction of partitions into at most 5 k − o = ( o , . . . , o l ) of length l = 5 k − k − , k − , k − , k − , . . . , , , , (cid:0) k +5 (cid:0) k (cid:1)(cid:1) = k (5 k −
3) which thus explains the contribution of the factor q k (5 k − in the numeratorof (5.1) and also explains the condition that the different parts of the partitions mustdiffer by at least 2.On the other hand, the expression in (5.2) without the factor ( − k can be seento be the generating function of partitions into parts congruent to 4 modulo 5 wherethe k th smallest part is marked (the marking is to make k unique as one then sumsover k ) and of size 5 k −
1, and the k − ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 21 particular, they have all the k − k − , k − , . . . , , · · · + (5 k −
1) = 4 k + 5 (cid:0) k (cid:1) = k (5 k + 3) / q ( k − k − ( q ; q ) k . For positive integer k , this is the generating function for partitions into exactly 5 k − k −
1) parts all have mul-tiplicity 5, and where parts from different groups differ by at least 2. This interpretationcomes from interpreting 1 / ( q ; q ) k as the generating function of partitions into at most5 k − o = ( o , . . . , o l ) of length l = 5 k − k − , k − , k − , k − , k − , . . . , , , , (cid:0) k −
1) + 5 (cid:0) k − (cid:1)(cid:1) + 5 k − k − k −
1) + 2.Further, the expression q k (5 k +11)2 +3 ( q k ; q ) ∞ (5.4)appearing in (5.3b) can be seen to be the generating function of partitions into partscongruent to 3 modulo 5 where the ( k + 1)st smallest part is marked and of size 5 k + 3,and the k smaller parts are all of different size (in particular, they have all the k differentsizes 5 k − , k − , . . . , , · · · + (5 k + 3) = 3( k + 1) + 5 (cid:0) k +12 (cid:1) = k (5 k +11)2 + 3.The details for the two other identities, (5.3c) and (5.3d), are similar.Altogether, we deduce the following partition-theoretic interpretation of the bilateralcompanions to the Rogers–Ramaunjan identities in (2.3), as rewritten in (5.3): Theorem 5.2.
Let • α be a partition into parts congruent to modulo or parts congruent to , , or modulo ; • β be a partition into parts congruent to modulo or parts congruent to , ,or modulo ; • γ be a partition into parts congruent to modulo or parts congruent to , ,or modulo ; • δ be a partition into parts congruent to modulo or parts congruent to , ,or modulo ; • ε and ζ be partitions into parts congruent to , , or modulo ; • η and ϑ be partitions into parts congruent to , , or modulo ; • ι be a partition into parts congruent to or modulo ; • κ be a partition into parts congruent to or modulo ; • λ be a partition into parts all greater or equal to where the largest part appearswith multiplicity and the other parts with multiplicity , and where parts ofdifferent size differ by at least ; • µ be a partition where the largest part appears with multiplicity and the otherparts with multiplicity , and where parts of different size differ by at least ; • ν be a partition where the largest part appears with multiplicity and the otherparts with multiplicity , and where parts of different size differ by at least ; • ξ be a partition into parts all greater or equal to where the largest part appearswith multiplicity and the other parts with multiplicity , and where parts ofdifferent size differ by at least ; • π be a (possibly empty) partition into parts congruent to modulo where forsome nonnegative even integer k the k th smallest part is marked and of size k − and the k − smaller parts are all of different size; • ̺ be a partition into parts congruent to modulo where for some positive oddinteger k the k th smallest part is marked and of size k − and the k − smallerparts are all of different size; • σ be a (possibly empty) partition into parts congruent to modulo where forsome nonnegative even integer k the ( k + 1) st smallest part is marked and ofsize k + 3 and the k smaller parts are all of different size; • τ be a partition into parts congruent to modulo where for some positive oddinteger k the ( k + 1) st smallest part is marked and of size k + 3 and the k smaller parts are all of different size; • ϕ be a (possibly empty) partition into parts congruent to modulo where forsome nonnegative even integer k + 1 the ( k + 1) st smallest part is marked andof size k − and the k smaller parts are all of different size; • χ be a partition into parts congruent to modulo where for some positive oddinteger k + 1 the ( k + 1) st smallest part is marked and of size k − and the k smaller parts are all of different size; • ψ be a (possibly empty) partition into parts congruent to modulo where forsome nonnegative even integer k the k th smallest part is marked and of size k − and the k − smaller parts are all of different size; • ω be a partition into parts congruent to modulo where for some positive oddinteger k the k th smallest part is marked and of size k − and the k − smallerparts are all of different size.Then A: the number of pairs of partitions ( α, λ ) with combined norm n plus the number ofpairs of partitions ( ε, π ) with combined norm n equals the number of partitions ι with norm n plus the number of pairs of partitions ( ζ , ̺ ) with combined norm n ; B: the number of pairs of partitions ( β, µ ) with combined norm n + 2 plus thenumber of pairs of partitions ( η, σ ) with combined norm n + 3 equals the number ILATERAL ROGERS–RAMANUJAN TYPE IDENTITIES 23 of partitions κ with norm n plus the number of pairs of partitions ( ϑ, τ ) withcombined norm n + 3 ; C: the number of pairs of partitions ( γ, ν ) with combined norm n + 2 plus thenumber of pairs of partitions ( η, ϕ ) with combined norm n + 2 equals the numberof partitions κ with norm n plus the number of pairs of partitions ( ϑ, χ ) withcombined norm n + 2 ; D: the number of pairs of partitions ( δ, ξ ) with combined norm n plus the number ofpairs of partitions ( ε, ψ ) with combined norm n equals the number of partitions ι with norm n plus the number of pairs of partitions ( ζ , ω ) with combined norm n . We leave it an open problem to give bijective proofs of Theorem 5.2 A–D.6.
Concluding remarks
In this paper, we derived a number of bilateral and multilateral identities of theRogers–Ramanujan type by analytic means. The closed form bilateral summationsexhibited here appear to be the very first of their kind. We expect that more identitiesof this kind can be found. Their very compact form and beauty suggests that theseobjects merit further study.In view of the well-established connections of the classical Rogers–Ramanujan iden-tities to various areas in mathematics and in physics (including combinatorics, numbertheory, probability theory, statistical mechanics, representations of Lie algebras, ver-tex algebras, and conformal field theory), we hope that similar connections can beestablished for the newly found bilateral identities. A first step in this direction wasachieved by providing explicit combinatorial interpretations for a specific collection offour bilateral Rogers–Ramanujan type identities.On the conceptual level the question arises whether the work in this paper tells usanything new about the classical case. On one hand it is interesting to observe that onebilateral identity may contain different unilateral identities of interest. Perfect exam-ples are the bilateral identities in (2.9b) and (2.9c) which each include three differentunilateral identities, as made explicit in the discussion after Theorem 2.6. On the otherhand we would like to emphasize that the derivations of our bilateral identities of theRogers–Ramanujan type do not require the combination of two unilateral sums intoa bilateral sum (such as by replacing the summation index k in the second sum by − − k ), which one usually requires, before applying Jacobi’s triple product identityin order to obtain the respective summations. 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