Abstract

1
Introduction
The Rogers-Ramanujan identities (see [1, 3]) : where α = 0 or 1, are among the most famous q-series identities in partitions and combinatorics. Since their discovery the Rogers-Ramanujan identities have been proved and generalized in various ways (see [1, 3, 4, 12] and the references cited there). In [12], by adapting a method of Macdonald for calculating partial fraction expansions of symmetric formal power series, Stembridge gave an unusual proof of Rogers-Ramanujan identities as well as fourteen other non trivial q-series identities of Rogers-Ramanujan type and their multiple analogs. Although it is possible to describe his proof within the setting of q-series, two summation formulas of Hall-Littlewood
11
polynomials were a crucial source of inspiration for such kind of identities. One of our or ginal motivations was to look for new multiple q-identities of Rogers-Ramanujan type through this approach, but we think that the new summation formulae of Hall-Littlewood polynomials are interesting for their own. Throughout this paper we will use the standard notations of q -series (see, for example, [5]). Set ( x )o := ( x ;q )o = 1 and for n ≥ 1
For n ≥ 0 and r ≥ 1, set
Let n ≥ 1 be a fixed integer and S n the group of permutations of the set {1, 2, . . , n}. Let X = {x 1, . . ,x n} be a set of indeterminates and q a parameter. For each partition λ = (λ 1, . . , λ n ) of length ≤ n, if m i := m i (λ) is the multiplicity of i in λ, then we also note λ by (1 m 1 2m 2 . .). Recall that the Hall-Littlewood polynomials P ( X , q ) are defined by [9, p.208] :
where the factor is added to ensure the coefficient of x λ 1 . . x λ n in P λ is 1. For a parameter α define the auxiliary function
Then it is well-known [9, p. 230] that the sums of P λ ( X , q ) over all partitions and even partitions are given by the following formulae :
2 For any sequence ξ ∈ {±1}π set Xξ = { xξ ¹,. . , xξ α } and denote by |ξ|− 1 the parts, Macdonald [9, p. 232] and Stembridge [12] have respectively general- ized (1) and (2) as follows :
Now, for parameters α , β define another auxiliary function
Then the following summation formulae similar to (1) and (2) for Hall- Littlewood polynomials hold true [9, p.232] :
where λ′ is the conjugate of λ and
In view of the numerous applications of (3) and (4) it is natural to seek such extensions for (5) and (6). However, as remarked by Stembridge [12, p. 475], in these other cases there arise complications which render doubtful the existence of expansions as explicit as those of (3) and (4). We noticed that these complications arise if one wants to keep exactly the same coefficients c λ (q) and d λ (q) as in (5) and (6) for the sums over bounded partitions. Actually we have the following Theorem 1  For k ≥ 1
して
side instead of the left-hand side and inspired by the similar formulae cor- responding to the case q = 0 of Hall-Littlewood polynomials [7], i.e., Schur functions. In the initial stage we made also the Maple tests using the pack- age ACE [13]. In the case q = 0, the right-hand sides of (3), (4), (7) and (8) can be written as quotients of determinants and the formulae reduce to the known identities of Schur functions [7] For any partition λ it will be convenient to adopt the following notation :
and to introduce the general q-binomial coefficients
with the convention that [ λ ] = 0 if λ  > n. If λ = (λ ) we recover the classical q-binomial coefficient. Finally, for any partition λ we denote by l(λ) the length of λ, i.e., the number of its positive parts, and n(λ) := The following is the key q-identity which allows to produce identities of Rogers-Ramanujan type.
orem 2 six multiple analogs of Rogers-Ramanujan type identities. In sec- tion 3 we give the proof of Theorem 1 and some consequences, and defer the elementary proof, i.e., without using the Hall-Littlewood polynomials, of Theorem 2 and other multiple q-series identities to section 4. To prove theo- rems 1, 2 and 4 (see section 3.3) we apply the generating function technique and the computation of residues, but theorem 4 can also be derived from theorem 1. In section 5 we will show how to derive some of our q-identities, which imply the six multianalogs of Rogers-Ramanujan type identities, from Andrews formula [3, Thm. 3.4], which was proved using Bailey's method. where
44
Multiple identities of Rogers-Ramanujan type
2
We need the Jacobi triple product identity [1, p.21] :

following identities of Rogers-Ramanujan type.

55
Proof. When z = q, we can rewrite (10) as follows :
For (12), letting a and b tend to 0 in (18) we obtain The right side of (12) follows then from (11) after simple manipulations. For (13), let a → 0 in (18) and multiply both sides by 1 − q − 2 . Identifying the coefficients of b we obtain :
The result follows from (11) after simple manipulations. Identity (14) follows from (18) with a = −q−1 and b → 0 and then by applying (11) with q replaced by q 4k +2 and x = −q 2k . For (15), we choose a = −1 in (18) and multiply both sides by 1 – then identify the coefficient of b . The identity follows then by applying (11) with q replaced by q4k+2 and x = -q2k-1 Identity (16) follows from (18) by taking a = −q− 1 and b = −1 and then applying (11) with q replaced by q 4k and x = −q 2k . For (17), we choose α = −1 and b → 0 in (18). The identity follows then by applying (11) with q replaced by q 4k +2 and x = −q 2k +1 → When k = 1 the above six identities reduce respectively to the following Rogers-Ramanujan type identities :

6

Note that (19), (20), (21) and (22) are already known, they correspond to Eqs. (39), (38), (29) and (28) in Slater's list [11], respectively. Identity (23) can be derived from the q-Kummer identity [5, p. 236] by the substitution q ← q², a = −1 and b = −q, but (24) seems to be new.
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Proof of Theorem 1 and consequences
3.1
Proof of identity (7)
For any statement A it will be convenient to use the true or false function χ(A), which is 1 if A is true and 0 if A is false. Consider the generating function
where the sum is over all partitions λ = (λ 1 , . . , λ n ) and the integers λ 0 ≥ λ 1 . Suppose λ = (μ 1 μ 2 · · μ 2 ), where μ 1 > μ 2 > ··· > μ k ≥ 0 and (r 1, . . , r k ) is a composition of n . Let S λ be the set of permutations of S n which fix λ. Each w ∈ S n/S n corresponds to a surjective mapping f : X  —→  {1,2,...,k}  such  that |f-1(i)| = ri. For any subset Y of X, let p(Y) denote the product of the elements of Y (in particular, p ( θ ) = 1). We can rewrite Hall-Littlewood functions as follows :
summed over all surjective mappings f : X  —→ {1,2,...,k} such that |f-1(i)| = r i . Furthermore, each such f determines a filtration of X :
77 according to the rule x i ∈ F l ⇐⇒ f(x i) ≤ l for 1 ≤ l ≤ k . Conversely, such a filtration F = (F 0 , F 1 , . . , F k ) determines a surjection f : X —→ {1,2,. . , k} uniquely. Thus we can write :
summed over all the filtrations F such that | F i | = r 1 + r 2 + · · + r i for 1 ≤ i ≤ k, and
where f is the function defined by F. Now let ν i = μ i − μ i+1 if 1 ≤ i ≤ k − 1 and ν k = μ k , thus ν i > 0 if i < k and ν k ≥ 0. Since the lengths of columns of λ are | F j | = r 1 + · ·  + r j with multiplicities ν j for 1 ≤ j ≤ k , we have
A filtration F is called even if |F j | is even for j ≥ 1. Furthermore, let μ 0 = λ 0 and ν 0 = μ 0 − μ 1 in the definition of S ( u ), so that ν 0 ≥ 0 and μρ = νρ + νι + · · + νκ . Define ϕ 2n (q) = (1 − q)(1 − q 3 ) ·· (1 − q 2n − 1 ) and c F (q) = Π i = 1 Ψ ( F i √ F i − 1 〈 q) for even filtrations F . Thus, since r j = m i ( λ ) for j ≥ 1, we have
Let F(X) be the set of filtrations of X. Summarizing we obtain
For any filtration F of X set

88
if F is even, and 0 otherwise. It follows from (28) that
Hence S(u) is a rational function of u with simple poles at 1/p(Y), where Y is a subset of X such that |Y| is even. We are now proceeding to compute the corresponding residue c(Y) at each pole u = 1/p(Y). Let us s tart with c(0). Writing λ 0 = λ 1 + k with k ≥ 0, we see that
It follows from (5) that
For the computations of other residues, we need some more notations. For any Y ⊆ X, let Y ′ = X \ Y and −Y = {x −1 : x i ∈ Y }. Let Y ⊆ X such that |Y| is even. Then
If Y # F , the corresponding summand is equal to 0. Thus we need only to consider the following filtrations F :
We may then split F into two filtrations F 1 and F 2  :
Then, writing v = p(Y)u and c = c r x c r 2 , we have

9

Using (29) and the result of c(Ø) , which can be written
we get
Each subset Y of X can be encoded by a sequence ξ ∈ {±1} n according to the rule : ξ i = 1 if x i̸ξ Y and ξ i = −1 if x i ∈ Y . Hence
Note also that
Now, extracting the coefficients of uk in the equation :
yields
10 Finally, substituting the value of c(Y) in the above formula we obtain (7). Remark.  Stembridge's formula (4) can be derived from Macdonald's (3) and Pieri's formula for Hall-Littlewood polynomials. Indeed, one of Pieri's formulas states that [9, p. 215] :
where the sum is over all partitions λ such that μ ⊆ λ with |λ/μ| = m and there is at most one cell in each row of the Ferrers diagram of λ/μ. It follows from (30) that noticing that λ determines in a unique way μ even by deleting a cell in X i -X i +1 each odd part of λ, and thus = 1. Finally we obtain the result, using the fact that Π ε (1 + x ξ ί )-1 = Π ε (1 + x ε )-1 × Π ε x (1-ξ ί )/2. It would be interesting to give a similar proof of (7) using (3) and another Pieri's formula [9, p. 218].
3.2
Proof of identity (8)
As in the proof of (7), we compute the generating function where the sum is over all partitions λ = (λ 1 , . . , λ n ) and integers λ 0 ≥ λ 1 . For any filtration F of X (cf. (25)) set
Thus, as r j = m j (λ), j ≥ 1, we have

11
In view of (26) we have
where
It follows that F(u) is a rational function of u and can be written as :
Extracting the coefficient of uk in the above identity yields
It remains to compute the residues. Writing λ 0 = λ 1 + r with r ≥ 0, then
it follows from (6) that
For computations of the other residues, set Y = X \ Y and define, for Y = F t , the two filtrations :
Then, writing v = p(Y)u and d F = d F 1 × d F 2 , we have

12
and B F (X, u)(1 – p(Y)u) can be written as
Rewriting (32) as
we get
Finally, the proof is completed by substituting the values of c(Y) in (31).
3.3
Some direct consequences on q-series
The following corollary of Theorem 1 will be useful in the proof of identities of Rogers-Ramanujan type. Theorem 4  For k ≥ 1,

13

In view of (9) we have
Replacing λ by 2λ and taking the conjugation in the left-hand side of (7) we obtain the left-hand side of (33). On the other hand, for any ξ ∈ {±1} n such that the number of ξ i = −1 is r, 0 ≤ r ≤ n, we have
which is readily seen to equal 0 unless ξ ∈ {−1} r × {1} n−r . Now, in the -ξέ)/2 latter case, we have Π x k (1
and [12, p. 476] :
Substituting these into the right side of (7) with r replaced by 2r we obtain the right side of (33). Next, by (9) we have
Similarly, in (8), replacing x i by zq i−1 (1 ≤ i ≤ n ) and invoking (35) we see that the left side of (8) reduces to that of (34). On the other hand, since
by (36), this is equal to zero unless ξ ∈ {−1}" × {1}"−r for some r, 0 ≤ r ≤ n. In the latter case, we have

14

4
Elementary approach to multiple q-identities
4.1
Preliminaries
Recall [1, pp. 36-37] that the binomial formula has the following q-analog :
polynomials [9, p. 215], but our proof is elementary.
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i.e., μ ⊆ λ, |λ/μ| = m and there is at most one cell in each column of the Ferrers diagram of λ/μ. Proof. Let l := l(μ) and μ 0 = n. Partition the set {1, 2, . . . , n} into l + 1 subsets : X i = {j | 1 ≤ j ≤ n and μ j = i} = {j | μ i+1 + 1 ≤ j ≤ μ i}, Using (45) to extract the coefficients of z m in the following identity :
r we define a partition λ = (λ 1 , λ 2 , . . .) by 1 ≤ i ≤ l + 1. λ ε = μ ε + r i − 1, Then λ/μ is a m-horizontal strip. So (47) can be written as
where the sum is over all partitions λ such that λ/μ is an m-horizontal strip. Now, since

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Lemma 2  There hold the following identities :
Proof. Identity (49) is due to Hall [6] and can be proved by using the q- binomial identity [8]. Stembridge [12] proved (50) using the q-binomial iden-
The identity (51) follows then from
→ which can be proved using the q-binomial formula [1, p. 36]. Remark. When n → ∞ the above identities reduce respectively to the following :

17
Also (52) and (54) are actually equivalent since the later can be derived from (52) by substituting q by q 2 and z by zq. The following is the q-Gauss sum [5, p.10] due to Heine :
Lemma 3  We have
Identity (57) follows then from (55). → Remark.  Formula (57) was derived in [12] from a more general formula of Hall-Littlewood polynomials.
4.2
Elementary proof of Theorem 4
We shall only prove (33) when n is even and leave the case when n is odd and (34) to the interested reader because their proofs are very similar. Consider the generating function of the left-hand side of (33) with n = 2r :
18 Now, each partition λ with parts bounded by r can be encoded by a pair of sequences ν = ( ν 0 , ν 1 , · · · , ν 1 and m = ( m 0 , · · · , m 1 ) such that λ = ( ν m o , . . . , ν m ι ), where r = ν 0 > ν 1 > · · · > ν 1 > 0 and ν i has multiplic- ity m i ≥ 1 for 1 ≤ i ≤ l and ν 0 = r has multiplicity m 0 ≥ 0. Using the notation :
where the sum is over all strict partitions ν = ( ν 0 , ν 1 , . . . , ν ι ) and
So ϕ( u ) is a rational fraction with simple poles at u − 1 for 0 ≤ p ≤ r . Let bp(z,r) be the corresponding residue of ϕ(u) at u − 1 for 0 ≤ p ≤ r. Then, it follows from (59) that
We shall first consider the cases where p = 0 or r . Using (58) and (51) we have
Now, by (59) and(60) we have
and
19 which, by setting μ i = r − ν l+1− i for 1 ≤ i ≤ l and μ 0 = r , can be written as
replaced by z-1q-2(2r-1). Il follows from (61) that
the corresponding summand in (60) is not zero only if ν j = p for some j, 0 ≤ j ≤ r . Furthermore, each such partition ν can be splitted into two strict partitions ρ = (ρ 0 ,ρ 1 , . . . ,ρ j − 1 ) and σ = (σ 0 , . . . ,σι− j ) such that ρ i = ν i − p for 0 ≤ i ≤ j − 1 and σ s = ν j+s for 0 ≤ s ≤ l − j . So we can write (60) as follows :
where for ρ = (ρ 0,ρι,...,ρι) with ρ 0 = p,
Comparing with (62) and (64) and using (61) and (65) we obtain
Finally, extracting the coefficients of u k in the equation
and using the values for bp (z,r) we obtain(33).
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4.3
Proof of Theorem 2

partition consisting of distinct parts ν 1 , ･ ･ , ν 1 of λ , so that ν 1 > ･ ･ > ν 1 > 0, and m = ( m 1 , . . . , m ι ) is the sequence of multiplicities of ν i for 1 ≤ i ≤ l . Therefore
sum, as a rational function of u , has a finite set of simple poles, which may occur at the points u r 1 for r ≥ 0. Therefore, each term is a linear combi- nation of partial fractions. Moreover, the sum of their expansions converges coefficientwise. So ϕαь has an expansion
where c r denotes the formal sum of partial fraction coefficients contributed by the terms of (67). It remains to compute these residues c, (r ≥ 0). By using (56) and (66), we get immediately
In view of (67), this yields the identity
21 Clearly, a summand in (67) has a non zero contribution to c r (r > 0) only if the corresponding partition ν has a part equal to r . For any partition ν such that ∃ j | ν j = r, set ρ i := ν i − r for 1 ≤ i < j and σ i := ν i+j for 0 ≤ i ≤ l − j, we then get two partitions ρ and σ , with σ i bounded by r . Multiplying (67) by (1 − u r u ) and setting u = 1/u r we obtain
we get
Now, the sum over ρ can be computed using (68) with α, b and z replaced

5
Proofs through Bailey’s method
A classical approach to identities of Rogers-Ramanujan type is based on Bailey's method (see [3, 14]). Recall that a pair of sequences ( α n , β n ) is a Bailey pair if there are two parameters x and q such that (see for example [3, p. 25-26]) :
If (α n , β n ) is a Bailey pair then Bailey's lemma [3, p. 25-26] states that ( α ′ , β ′ n ) is also a Bailey pair, where

22

In [2, 3] Andrews noticed that applying Bailey's lemma to the same Bailey pair iteratively leads to a Bailey chain, which yields almost straightforwardly multiple identities of Rogers-Ramanujan type. In what follows we shall briefly indicate how to derive our identity (18), from which we derived our six multisum identities (12)-(17), through this method. Our starting point is Theorem 3.4 of Andrews [3]. Indeed, letting N → ∞ and for i = 1, . . . , k − 1, letting bi → ∞ , ci → ∞ and setting bk = α− 1 and ck = b-1 in [3, Theorem 3.4], we obtain
where (α n, β n ) is a Bailey pair. Now, invoking the following Bailey pair (α n , β n ) [10, F(1)] : α 0 = β 0 = 1 and for n ≥ 1
and plugging it in (70) with x = 1 yields (18) after replacing q by q 2 . It is interesting to note that (23) and (24) are consequences of Bailey's lemma with Slater's pair (71), but they did not appear in [10, 11]. We note that Stembridge [12] derived his sixteen multianalogs of Rogers- Ramanujan type from the following specializations of his Theorem 3.4 :
and
23

In the same vein we can derive the above four identities from [3, Theorem 3.4]. For example, for (72) take x = 1 in (70) and use the Bailey pair B(1) of [10], and for (73) take x = q in (70) and use the Bailey pair B(3) of [10]. For (74) and (75), we need another specialization of [3, Theorem 3.4]. Letting N → ∞, bi → ∞ for i = 1, … ,k − 1 and setting bk = α− 1 and ci = −√xq for i = 1, . . , k in [3, Theorem 3.4] we obtain
where (α n, β n ) is a Bailey pair. Taking x = q in (76) and using the Bailey pair E(3) of Slater [10] yields (74). For (75), take x = 1 in (76) and use the following Bailey pair [10, p. 468] : α 0 = β 0 = 1 and for n ≥ 1
Recently, Bressoud, Ismail and Stanton [4] have pointed out that the sixteen multisum identities, but not the above four more general identities, in Stembridge [12] can be proved by means of change of base in Bailey pairs. Aknowledgments. We thank Jeremy Lovejoy for helpful comments about a previous version of this paper.
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[1] Andrews (G.E.), The theory of partitions , Encyclopedia of mathematics and its applications, Vol. 2 , Addison-Wesley, Reading, Massachusetts, 1976. [2] A NDREWS (G.E.), Multiple series Rogers-Ramanujan type identities, Pacific J. Math. , Vol. 114, No. 2, 267-283, 1984. [3] A NDREWS (G.E.), q -Series : Their Development and Application in Analysis, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series, Vol. 66, Amer. Math. Soc., Providence, 1986. [4] Bressoud (D.), Ismail (M.), and Stanton (D.), Change of Base in Bailey Pairs, The Ramanujan J., 4, 435-453, 2000. [5] G ASPER (G.) and R AHMAN (M.), Basic Hypergeometric Series , Encyclopedia. of mathematics and its applications, Vol. 35 , Cambridge University Press, Cambridge, 1990. [6] Hall (P.), A partition formula connected with Abelian groups, Comment. Math. Helv. 11, 126-129, 1938. [7] J OUHET (F.) and Z ENG (J.), Some new identities for Schur functions, Adv. Appl. Math., 27, 493-509, 2001. [8] M ACDONALD (I.G.), An elementary proof of a q -binomial identity, q -series and partitions (Minneapolis, MN, 1988), 73–75, IMA Vol. Math. Appl., 18, Springer, New York, 1989. [9] M ACDONALD (I.G.), Symmetric functions and Hall polynomials , Clarendon Press, second edition, Oxford, 1995. [10] S LATER (L. J.), A new proof of Rogers's transformations of infinite series, Proc. London Math. Soc. , 53 (2), 460-475, 1951. [11] Slater (L. J.), Further identities of the Rogers-Ramanujan Type, Proc. Lon- don Math. Soc. , 54 (2), 147-167 (1951-52). [12] S TEMBRIDGE (J. R.), Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities, Trans. Amer. Math. Soc. , 319 , no.2, 469-498, 1990. [13] VEIGNEAU S., ACE, an Algebraic Environment for the Computer algebra sys- tem MAPLE, http: file://phalanstere.univ-mlv.fr/ ace, 1998. [14] Warnaar (S. O.), 50 years of Bailey's lemma, Algebraic combinatorics and applications (Gößweinstein, 1999), 333–347, Springer, Berlin, 2001.