New Identities of Hall-Littlewood Polynomials and Applications

Frédéric Jouhet and Jiang Zeng

Institut Girard Desargues, Université Claude Bernard (Lyon 1) 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

              E-mail : {jouhet,zeng}@euler.univ-lyon1.fr

Abstract

Starting from Macdonald’s summation formula of Hall-Littlewood polynomials over bounded partitions and its even partition analogue, Stembridge (1990, Trans. Amer. Math. Soc., 319, no.2, 469-498) derived sixteen multiple q-identities of Rogers-Ramanujan type. Inspired by our recent results on Schur functions (2001, Adv. Appl. Math., 27, 493-509) and based on computer experiments we obtain two further such summation formulae of Hall-Littlewood polynomials over bounded partitions and derive six new multiple q-identities of Rogers-Ramanujan type.

1 Introduction

The Rogers-Ramanujan identities (see [1, 3]) :

∑_{n=0}^{∞} \frac{q^{n^{2}+an}}{(1 − q)(1 − q^{2}) ⋯ (1 − q^{n})} = ∏_{{n=1 \atop n≡±(a+1) (mod 5)}}^{∞}(1 − q^{n})^{−1},

where a = 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 polynomials were a crucial source of inspiration for such kind of identities. One of our original 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)_{0} := (x; q)_{0} = 1 and for n ≥ 1

(x)_{n} := (x; q)_{n} = ∏_{k=1}^{n}(1 − xq^{k−1}),

(x)_{∞} := (x; q)_{∞} = ∏_{k=1}^{∞}(1 − xq^{k−1}).

For n ≥ 0 and r ≥ 1, set

(a_{1}, ⋯, a_{r}; q)_{n} = ∏_{i=1}^{r}(a_{i})_{n} , (a_{1}, ⋯, a_{r}; q)_{∞} = ∏_{i=1}^{r}(a_{i})_{∞}.

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}} 2^{m_{2}} …). Recall that the Hall-Littlewood polynomials P_{λ}(X, q) are defined by [9, p.208] :

P_{λ}(X, q) = ∏_{i≥1} \frac{(1 − q)^{m_{i}}}{(q)_{m_{i}}} ∑_{w∈S_{n}} w (x_{1}^{λ_{1}} … x_{n}^{λ_{n}} ∏_{i<j} \frac{x_{i} − qx_{j}}{x_{i} − x_{j}} ) ,

where the factor is added to ensure the coefficient of x_{1}^{λ_{1}} … x_{n}^{λ_{n}} in P_{λ} is 1. For a parameter α define the auxiliary function

Ψ_{q}(X; α) := ∏_{i}(1 − x_{i})^{−1}(1 − αx_{i})^{−1} ∏_{j<k} \frac{1 − qx_{j}x_{k}}{1 − x_{j}x_{k}} .

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 :

∑_{λ} P_{λ}(X, q) = Ψ_{q}(X; 0),  (1)
∑_{λ} P_{2λ}(X, q) = Ψ_{q}(X; −1).  (2)

For any sequence ξ ∈ {±1}^{n} set X^{ξ} = {x_{1}^{ξ_{1}}, ⋯, x_{n}^{ξ_{n}}} and denote by |ξ|_{−1} the number of −1’s in ξ. Then, by summing P_{λ} over partitions with bounded parts, Macdonald [9, p. 232] and Stembridge [12] have respectively generalized (1) and (2) as follows :

∑_{λ_{1}≤k} P_{λ}(X, q) = ∑_{ξ∈{±1}^{n}} Ψ_{q}(X^{ξ}; 0) ∏_{i} x_{i}^{k(1−ξ_{i})/2},  (3)
∑_{{λ_{1}≤2k \atop λ even}} P_{λ}(X, q) = ∑_{ξ∈\{±1\}^{n}} Ψ_{q}(X^{ξ}; −1) ∏_{i} x_{i}^{k(1−ξ_{i})}.  (4)

Now, for parameters α, β define another auxiliary function

Φ_{q}(X; α, β) := ∏_{i} \frac{1 − αx_{i}}{1 − βx_{i}} ∏_{j<k} \frac{1 − qx_{j}x_{k}}{1 − x_{j}x_{k}} .

Then the following summation formulae similar to (1) and (2) for Hall-Littlewood polynomials hold true [9, p.232] :

∑_{λ′ even} c_{λ}(q) P_{λ}(X, q) = Φ_{q}(X; 0, 0),  (5)
∑_{λ} d_{λ}(q) P_{λ}(X, q) = Φ_{q}(X; q, 1),  (6)

where λ′ is the conjugate of λ and

c_{λ}(q) = ∏_{i≥1}(q; q^{2})_{m_{i}(λ)/2}, d_{λ}(q) = ∏_{i≥1} \frac{(q)_{m_{i}(λ)}}{(q^{2}; q^{2})_{[m_{i}(λ)/2]}} .

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,

∑_{{λ_{1}≤k \atop λ′ even}} c_{λ,k}(q)P_{λ}(X, q) = ∑_{{ξ∈\{±1\}^{n} \atop |ξ|_{−1} even}} Φ_{q}(X^{ξ}; 0, 0) ∏_{i} x_{i}^{k(1−ξ_{i})/2}, (7)

∑_{λ_{1}≤k} d_{λ,k}(q)P_{λ}(X, q) = ∑_{ξ∈{±1}^{n}} Φ_{q}(X^{ξ}; q, 1) ∏_{i} x_{i}^{k(1−ξ_{i})/2},  (8)

where

c_{λ,k}(q) = ∏_{i=1}^{k−1}(q; q^{2})_{m_{i}(λ)/2}, d_{λ,k}(q) = ∏_{i=1}^{k−1} \frac{(q)_{m_{i}(λ)}}{(q^{2}; q^{2})_{[m_{i}(λ)/2]}} .  (9)

Remark. We were led to such extensions by starting from the right-hand side instead of the left-hand side and inspired by the similar formulae corresponding 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 package 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 :

(x)_{λ} := (x; q)_{λ} = (x)_{λ_{1}−λ_{2}}(x)_{λ_{2}−λ_{3}} ⋯,

and to introduce the general q-binomial coefficients

\left[{n \atop λ}\right] := \frac{(q)_{n}}{(q)_{n−λ_{1}}(q)_{λ}} ,

with the convention that \left[{n \atop λ}\right] = 0 if λ_{1} > n. If λ = (λ_{1}) 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) := ∑_{i} \binom{λ_{i}}{2}.

The following is the key q-identity which allows to produce identities of Rogers-Ramanujan type.

Theorem 2 For k ≥ 1,

∑_{l(λ)≤k} z^{|λ|}q^{n(2λ)} \frac{(a, b; q^{−2})_{λ_{1}}}{(q^{2}; q^{2})_{λ}(q; q^{2})_{λ_{k}}} = \frac{(z; q^{2})_{∞}}{(abzq; q^{2})_{∞}}  (10)

× ∑_{r≥0} z^{kr}q^{(k+1)\binom{2r}{2}} \frac{(a, b; q^{−2})_{r}(aq^{2r+1}z, bq^{2r+1}z; q^{2})_{∞}}{(q)_{2r}(zq^{2r−1})_{∞}} (1 − zq^{4r−1}).

Here is an outline of this paper. in section 2 we first derive from Theorem 2 six multiple analogs of Rogers-Ramanujan type identities. In section 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 theorems 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.

2 Multiple identities of Rogers-Ramanujan type

We need the Jacobi triple product identity [1, p.21] :

J(x, q) := 1 + ∑_{r=1}^{∞}(−1)^{r}x^{r}q^{\binom{r}{2}}(1 + q^{r}/x^{2r}) = (q, x, q/x)_{∞}. (11)

For any partition λ set n_{2}(λ) = ∑_{i} λ_{i}^{2}. We derive then from Theorem 2 the following identities of Rogers-Ramanujan type. Theorem 3 For k ≥ 1,

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} = ∏_{n}(1 − q^{n})^{−1}  (12)

where n ≡ ±(2k + 1), ±(2k + 3), ±2, ±4, …, ±4k (mod 8k + 8);

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)−2λ_{1}}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} (1 − q^{2λ_{1}}) = \frac{(q^{2k−1}, q^{6k+9}; q^{8k+8})_{∞}}{∏_{n}(1 − q^{n})}  (13)

where n ≡ ±(2k + 5), ±2, …, ±4k, ±(4k + 2) (mod 8k + 8);

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)−λ_{1}^{2}}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} (−q; q^{2})_{λ_{1}}

= \frac{(−q; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} (q^{4k+2}, −q^{2k}, −q^{2k+2}; q^{4k+2})_{∞};  (14)

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)−λ_{1}^{2}−λ_{1}}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} (−1; q^{2})_{λ_{1}}(1 − q^{2λ_{1}})

= \frac{(−q^{2}; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} (q^{4k+2}, −q^{2k−1}, −q^{2k+3}; q^{4k+2})_{∞}; (15)

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)−2λ_{1}^{2}+λ_{1}}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} (−1; q^{2})_{λ_{1}}(−q; q^{2})_{λ_{1}}

= \frac{(−q)_{∞}}{(q)_{∞}} (q^{4k}, −q^{2k}, −q^{2k}; q^{4k})_{∞};  (16)

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)−λ_{1}^{2}+λ_{1}}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} (−1; q^{2})_{λ_{1}}

= \frac{(−q^{2}; q^{2})_{∞}}{(q^{2}; q^{2})_{∞}} (q^{4k+2}, −q^{2k+1}, −q^{2k+1}; q^{4k+2})_{∞}. (17)

Proof. When z = q, we can rewrite (10) as follows :

∑_{l(λ)≤k} q^{2n_{2}(λ)−2λ_{1}^{2}+2λ_{1}} \frac{(a^{−1}, b^{−1}; q^{2})_{λ_{1}}}{(q^{2}; q^{2})_{λ}(q; q^{2})_{λ_{k}}} (ab)^{λ_{1}}  (18)

= \frac{(aq^{2}, bq^{2}; q^{2})_{∞}}{(abq^{2}; q^{2})_{∞}(q^{2}; q^{2})_{∞}} (1 + ∑_{r≥1} q^{2kr^{2}+r} \frac{(a^{−1}, b^{−1}; q^{2})_{r}}{(aq^{2}, bq^{2}; q^{2})_{r}} (ab)^{r}(1 + q^{2r})) .

For (12), letting a and b tend to 0 in (18) we obtain

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} = (q^{2}; q^{2})_{∞}^{−1}J(−q^{2k+1}, q^{4k+4}).

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 :

∑_{l(λ)≤k} \frac{q^{2n_{2}(λ)−2λ_{1}}}{(q; q^{2})_{λ_{k}}(q^{2}; q^{2})_{λ}} (1 − q^{2λ_{1}}) = (q^{2}; q^{2})_{∞}^{−1} J(−q^{2k−1}, q^{4k+4}).

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 − q^{−2}, then identify the coefficient of b. The identity follows then by applying (11) with q replaced by q^{4k+2} and x = −q^{2k−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 a = −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 :

∑_{n=0}^{∞} \frac{q^{2n^{2}}}{(q)_{2n}} = ∏_{{n=1 \atop n≡±2,±3,±4,±5 (mod 16)}}^{∞} \frac{1}{1 − q^{n}} , (19)

∑_{n=0}^{∞} \frac{q^{2n^{2}+2n}}{(q)_{2n+1}} = ∏_{{n=1 \atop n≡±1,±4,±6,±7 (mod 16)}}^{∞} \frac{1}{1 − q^{n}} , (20)

∑_{n=0}^{∞} q^{n^{2}} \frac{(−q; q^{2})_{n}}{(q)_{2n}} = \frac{(q^{6}, q^{6}, q^{12}; q^{12})_{∞}}{(q)_{∞}} ,  (21)

∑_{n=0}^{∞} q^{n^{2}+n} \frac{(−q^{2}; q^{2})_{n}}{(q)_{2n+1}} = \frac{(q^{3}, q^{9}, q^{12}; q^{12})_{∞}}{(q)_{∞}} ,  (22)

1 + 2 ∑_{n≥1} q^{n} \frac{(−q)_{2n−1}}{(q)_{2n}} = \frac{(q^{4}, −q^{2}, −q^{2}; q^{4})_{∞}}{(q)_{∞}(q; q^{2})_{∞}} ,  (23)

1 + 2 ∑_{n≥1} q^{n(n+1)} \frac{(−q^{2}; q^{2})_{n−1}}{(q)_{2n}} = \frac{(q^{6}, −q^{3}, −q^{3}; q^{6})_{∞}}{(q)_{∞} (−q; q^{2})_{∞}} .  (24)

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 ← q^{2}, a = −1 and b = −q, but (24) seems to be new.

3 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

S(u) = ∑_{λ_{0},λ} χ(λ′ even) c_{λ,λ_{0}}(q)P_{λ}(X, q) u^{λ_{0}}

where the sum is over all partitions λ = (λ_{1}, …, λ_{n}) and the integers λ_{0} ≥ λ_{1}. Suppose λ = (μ_{1}^{r_{1}} μ_{2}^{r_{2}} … μ_{k}^{r_{k}}), where μ_{1} > μ_{2} > ⋯ > μ_{k} ≥ 0 and (r_{1}, …, r_{k}) is a composition of n.

Let S_{n}^{λ} be the set of permutations of S_{n} which fix λ. Each w ∈ S_{n}/S_{n}^{λ} corresponds to a surjective mapping f : X ⟶ {1, 2, …, k} such that |f^{−1}(i)| = r_{i}. For any subset Y of X, let p(Y) denote the product of the elements of Y (in particular, p(∅) = 1). We can rewrite Hall-Littlewood functions as follows :

P_{λ}(X, q) = ∑_{f} p(f^{−1}(1))^{μ_{1}} ⋯ p(f^{−1}(k))^{μ_{k}} ∏_{f(x_{i})<f(x_{j})} \frac{x_{i} − qx_{j}}{x_{i} − x_{j}} ,

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 :

F : ∅ = F_{0} ⊊ F_{1} ⊊ ⋯ ⊊ F_{k} = X,  (25)

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 :

P_{λ}(X, q) = ∑_{F} π_{F} ∏_{1≤i≤k} p(F_{i} \ F_{i−1})^{μ_{i}},  (26)

summed over all the filtrations F such that |F_{i}| = r_{1} + r_{2} + ⋯ + r_{i} for 1 ≤ i ≤ k, and

π_{F} = ∏_{f(x_{i})<f(x_{j})} \frac{x_{i} − qx_{j}}{x_{i} − x_{j}} ,

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

χ(λ′ even) = ∏_{j=1}^{k} χ(|F_{j}|even).  (27)

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 μ_{0} = ν_{0} + ν_{1} + ⋯ + ν_{k}. Define φ_{2n}(q) = (1 − q)(1 − q^{3}) ⋯ (1 − q^{2n−1}) and c_{F}(q) = ∏_{i=1}^{k} φ_{|F_{i}\F_{i−1}|}(q) for even filtrations F. Thus, since r_{j} = m_{μ_{j}}(λ) for j ≥ 1, we have

c_{λ, λ_{0}}(q) = c_{F}(q) (χ(ν_{k} = 0)φ_{|F_{k}\F_{k−1}|}(q) + χ(ν_{k} ≠ 0))^{−1}
× (χ(ν_{0} = 0)φ_{|F_{1}|}(q) + χ(ν_{0} ≠ 0))^{−1} .

Let F (X) be the set of filtrations of X. Summarizing we obtain

S(u) = ∑_{F∈F (X)} c_{F} π_{F} χ(F even) ∑_{ν_{1}>0}(u p(F_{1}))^{ν_{1}} ⋯ ∑_{ν_{k−1}>0}(u p(F_{k−1}))^{ν_{k−1}}

× ∑_{ν_{0}≥0} \frac{u^{ν_{0}}}{χ(ν_{0} = 0) φ_{|F_{1}|}(q) + χ(ν_{0} ≠ 0)}

× ∑_{ν_{k}≥0} \frac{u^{ν_{k}} p(F_{k})^{ν_{k}}}{χ(ν_{k} = 0) φ_{|F_{k}\F_{k−1}|}(q) + χ(ν_{k} ≠ 0)}.  (28)

For any filtration F of X set

A_{F}(X, u) = c_{F}(q) ∏_{|F_{j}| even} [\frac{p(F_{j})u}{1 − p(F_{j})u} + \frac{χ(F_{j} = X)}{φ_{|F_{j}\F_{j−1}|}(q)} + \frac{χ(F_{j} = ∅)}{φ_{|F_{1}|}(q)}]

if F is even, and 0 otherwise. It follows from (28) that

S(u) = ∑_{F∈F (X)} π_{F}A_{F}(X, u).

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 start with c(∅). Writing λ_{0} = λ_{1} + k with k ≥ 0, we see that

S(u) = ∑_{λ} χ(λ′ even) c_{λ}(q)P_{λ}(X, q)u^{λ_{1}} ∑_{k≥0} \frac{u^{k}}{χ(k = 0)φ_{m_{λ_{1}}} (q) + χ(k ≠ 0)}

= ∑_{λ} χ(λ′ even) c_{λ}(q)P_{λ}(X, q)u^{λ_{1}} (\frac{u}{1 − u} + \frac{1}{φ_{m_{λ_{1}}} (q)}) .

It follows from (5) that

c(∅) = [S(u)(1 − u)]_{u=1} = Φ_{q}(X; 0, 0).

For the computations of other residues, we need some more notations. For any Y ⊆ X, let Y′ = X \ Y and −Y = {x_{i}^{−1} : x_{i} ∈ Y }. Let Y ⊆ X such that |Y | is even. Then

c(Y) = [∑_{F} π_{F}A_{F}(X, u)(1 − p(Y)u)]_{u=p(−Y)} .  (29)

If Y ∉ F, the corresponding summand is equal to 0. Thus we need only to consider the following filtrations F :

∅ = F_{0} ⊊ ⋯ ⊊ F_{t} = Y ⊊ ⋯ ⊊ F_{k} = X 1 ≤ t ≤ k.

We may then split F into two filtrations F_{1} and F_{2} :

F_{1} : ∅ ⊊ −(Y \ F_{t−1}) ⊊ ⋯ ⊊ −(Y \ F_{1}) ⊊ −Y,
F_{2} : ∅ ⊊ F_{t+1} \ Y ⊊ ⋯ ⊊ F_{k−1} \ Y ⊊ Y′.

Then, writing v = p(Y)u and c_{F} = c_{F_{1}} × c_{F_{2}}, we have

π_{F}(X) = π_{F_{1}}(−Y)π_{F_{2}}(Y′) ∏_{x_{i}∈Y,x_{j}∈Y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} ,

and A_{F}(X, u)(1 − p(Y)u) is equal to

A_{F_{1}}(−Y, v)A_{F_{2}}(Y′, v)(1 − v) (\frac{v}{1 − v} + \frac{χ(Y = X)}{φ_{|Y \F_{t−1}|}(q)})

× (\frac{v}{1 − v} + \frac{1}{φ_{|Y \F_{t−1}|}(q)})^{−1} (\frac{v}{1 − v} + \frac{1}{φ_{|F_{t+1}\Y |}(q)})^{−1} .

Thus when u = p(−Y), i.e., v = 1,

[π_{F}(X)A_{F}(X, u)(1 − p(Y)u)]_{u=p(−Y)} =
[π_{F_{1}}(−Y)A_{F_{1}}(−Y, v)(1 − v)π_{F_{2}}(Y′)A_{F_{2}}(Y′, v)(1 − v)]_{v=1}

× ∏_{x_{i}∈Y,x_{j}∈Y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} .

Using (29) and the result of c(∅), which can be written

[∑_{F} π_{F}A_{F}(X, u)(1 − u)]_{u=1} = Φ_{q}(X; 0, 0),

we get

c(Y) = Φ_{q}(−Y; 0, 0)Φ_{q}(Y′; 0, 0) ∏_{x_{i}∈Y,x_{j}∈Y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} .

Each subset Y of X can be encoded by a sequence ξ ∈ {±1}^{n} according to the rule : ξ_{i} = 1 if x_{i} ∉ Y and ξ_{i} = −1 if x_{i} ∈ Y. Hence

c(Y) = Φ_{q}(X^{ξ}; 0, 0).

Note also that

p(Y) = ∏_{i} x_{i}^{(1−ξ_{i})/2}, p(−Y) = ∏_{i} x_{i}^{(ξ_{i}−1)/2}.

Now, extracting the coefficients of u^{k} in the equation :

S(u) = ∑_{{Y ⊆X \atop |Y | even>0}} \frac{c(Y)}{1 − p(Y)u},

yields

∑_{{λ_{1}≤k \atop λ′ even}} c_{λ,k}(q)P_{λ}(X, q) = ∑_{{Y ⊆X \atop |Y | even}} c(Y)p(Y)^{k}.

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] :

P_{μ}(X, q)e_{m}(X) = ∑_{λ} ∏_{i≥1} \left[{λ′_{i} − λ′_{i+1} \atop λ′_{i} − μ′_{i}}\right] P_{λ}(X, q),  (30)

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

∑_{{μ_{1}≤2k \atop μ even}} P_{μ}(X, q) ∑_{m≥0} e_{m}(X) = ∑_{λ_{1}≤2k+1} P_{λ}(X, q),

noticing that λ determines in a unique way μ even by deleting a cell in each odd part of λ, and thus \left[{λ′_{i}−λ′_{i+1} \atop λ′_{i}−μ′_{i}}\right] = 1. Finally we obtain the result,

using the fact that ∏_{i}(1 + x_{i}^{ξ_{i}})^{−1} = ∏_{i}(1 + x_{i})^{−1} × ∏_{i} x_{i}^{(1−ξ_{i})/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

F (u) = ∑_{λ_{0},λ} d_{λ,λ_{0}}(q)P_{λ}(X; q) u^{λ_{0}}

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

d_{F}(q) = ∏_{i=1}^{k} ψ_{|F_{i}\F_{i−1}|}(q), where ψ_{n}(q) = (q)_{n} ∏_{j=1}^{[n/2]}(1 − q^{2j})^{−1}.

Thus, as r_{j} = m_{μ_{j}}(λ), j ≥ 1, we have

d_{λ, λ_{0}}(q) = d_{F}(q) (χ(ν_{k} = 0)ψ_{|F_{k}\F_{k−1}|}(q) + χ(ν_{k} ≠ 0))^{−1}
× (χ(ν_{0} = 0)ψ_{|F_{1}|}(q) + χ(ν_{0} ≠ 0))^{−1} .

In view of (26) we have

F (u) = ∑_{F∈F (X)} π_{F}B_{F}(X, u),

where

B_{F}(X, u) = d_{F} ∏_{j} [\frac{p(F_{j})u}{1 − p(F_{j})u} + \frac{χ(F_{j} = X)}{ψ_{|F_{j}\F_{j−1}|}(q)} + \frac{χ(F_{j} = ∅)}{ψ_{|F_{1}|}(q)}] .

It follows that F (u) is a rational function of u and can be written as :

F (u) = \frac{c(∅)}{1 − u} + ∑_{{Y ⊆X \atop |Y |>0}} \frac{c(Y)}{1 − p(Y)u}.

Extracting the coefficient of u^{k} in the above identity yields

∑_{λ_{1}≤k} d_{λ,k}(q)P_{λ}(X, q) = ∑_{Y ⊆X} c(Y)p(Y)^{k}.  (31)

It remains to compute the residues. Writing λ_{0} = λ_{1} + r with r ≥ 0, then

F (u) = ∑_{λ} d_{λ}(q)P_{λ}(X, q)u^{λ_{1}} ∑_{r≥0} \frac{u^{r}}{χ(r = 0)ψ_{m_{λ_{1}}} (q) + χ(r ≠ 0)}

= ∑_{λ} d_{λ}(q)P_{λ}(X, q)u^{λ_{1}} (\frac{u}{1 − u} + \frac{1}{ψ_{m_{λ_{1}}} (q)}) ,

it follows from (6) that

c(∅) = (F (u)(1 − u)) |_{u=1} = Φ_{q}(X; q, 1).  (32)

For computations of the other residues, set Y′ = X \ Y and define, for Y = F_{t}, the two filtrations :

F_{1} : ∅ ⊊ −(Y \ F_{t−1}) ⊊ ⋯ ⊊ −(Y \ F_{1}) ⊊ −Y,
F_{2} : ∅ ⊊ F_{t+1} \ Y ⊊ ⋯ ⊊ F_{k−1} \ Y ⊊ Y′.

Then, writing v = p(Y)u and d_{F} = d_{F_{1}} × d_{F_{2}}, we have

π_{F}(X) = π_{F_{1}}(−Y)π_{F_{2}}(Y′) ∏_{x_{i}∈Y,x_{j}∈Y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} ,

and B_{F}(X, u)(1 − p(Y)u) can be written as

B_{F_{1}}(−Y, v)B_{F_{2}}(Y′, v)(1 − v) (\frac{v}{1 − v} + \frac{χ(Y = X)}{ψ_{|Y \F_{t−1}|}})

× (\frac{v}{1 − v} + \frac{1}{ψ_{|Y \F_{t−1}|}(q)})^{−1} (\frac{v}{1 − v} + \frac{1}{ψ_{|F_{t+1}\Y |}(q)})^{−1} .

Rewriting (32) as

[∑_{F} π_{F}B_{F}(X, u)(1 − u)]_{u=1} = Φ_{q}(X; q, 1),

we get

c(Y) = [∑_{F} π_{F}B_{F}(X; u)(1 − p(Y)u)]_{u=p(−Y)}
= Φ_{q}(−Y; q, 1)Φ_{q}(Y′; q, 1) ∏_{x_{i}∈Y,x_{j}∈Y′} \frac{1 − qx_{i}^{−1}x_{j}}{1 − x_{i}^{−1}x_{j}} .

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,

∑_{l(λ)≤k} \frac{(q; q^{2})_{λ}}{(q; q^{2})_{λ_{k}}} z^{|λ|}q^{n(2λ)} \left[{n \atop 2λ}\right] = (z; q^{2})_{n} ∑_{r≥0} z^{kr}q^{(k+1)\binom{2r}{2}}

× \left[{n \atop 2r}\right] \frac{1 − zq^{4r−1}}{(zq^{2r−1})_{n+1}} .  (33)

∑_{l(λ)≤k} ∏_{i=1}^{k−1} \frac{(q)_{λ_{i}−λ_{i+1}}}{(q^{2}; q^{2})_{[(λ_{i}−λ_{i+1})/2]}} z^{|λ|}q^{n(λ)} \left[{n \atop λ}\right] = (z^{2}; q^{2})_{n} ∑_{r≥0} z^{kr}q^{r+(k+1)\binom{r}{2}}

× \left[{n \atop r}\right] \frac{(1 − zq^{−1})(1 − z^{2}q^{2r−1})(1 − zq^{n})}{(1 − zq^{r−1})(1 − zq^{r})(z^{2}q^{r−1})_{n+1}}.  (34)

Proof. We know [9, p. 213] that if x_{i} = z^{1/2}q^{i−1} (1 ≤ i ≤ n) then :

P_{λ′}(X, q) = z^{|λ|/2}q^{n(λ)} \left[{n \atop λ}\right] .  (35)

In view of (9) we have

c_{(2λ)′,k}(q) = \frac{(q; q^{2})_{λ}}{(q; q^{2})_{λ_{k}}} .

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

Φ_{q}(X^{ξ}; 0, 0) = Ψ_{q}(X^{ξ}; −1) ∏_{i}(1 − x_{i}^{2ξ_{i}}),  (36)

which is readily seen to equal 0 unless ξ ∈ {−1}^{r} × {1}^{n−r}. Now, in the latter case, we have ∏_{i} x_{i}^{k(1−ξ_{i})/2} = z^{kr/2}q^{k\binom{r}{2}},

∏_{i=1}^{n}(1 − x_{i}^{2ξ_{i}}) = (−1)^{r}z^{−r}q^{−2\binom{r}{2}}(z; q^{2})_{n},  (37)

and [12, p. 476] :

Ψ_{q}(X^{ξ}; −1) = (−1)^{r}z^{r}q^{3\binom{r}{2}} \left[{n \atop r}\right] \frac{1 − zq^{2r−1}}{(zq^{r−1})_{n+1}} .  (38)

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

d_{λ′,k}(q) = ∏_{i=1}^{k−1} \frac{(q)_{λ_{i}−λ_{i+1}}}{(q^{2}; q^{2})_{[(λ_{i}−λ_{i+1})/2]}} .

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

Φ_{q}(X^{ξ}; q, 1) = Φ_{q}(X^{ξ}; 0, 0) ∏_{i=1}^{n} \frac{1 − qx_{i}^{ξ_{i}}}{1 − x_{i}^{ξ_{i}}} ,

by (36), this is equal to zero unless ξ ∈ {−1}^{r}×{1}^{n−r} for some r, 0 ≤ r ≤ n. In the latter case, we have

∏_{i=1}^{n} \frac{1 − qx_{i}^{ξ_{i}}}{1 − x_{i}^{ξ_{i}}} = q^{r} \frac{1 − zq^{−1}}{1 − zq^{r−1}} \frac{1 − zq^{n}}{1 − zq^{r}} ,  (39)

and invoking (36), (37) and (38) with z replaced by z^{2},

Φ_{q}(X^{ξ}; 0, 0) = q^{\binom{r}{2}} \left[{n \atop r}\right] (1 − z^{2}q^{2r−1})\frac{(z^{2}; q^{2})_{n}}{(z^{2}q^{r−1})_{n+1}}  (40)

Plunging these into the right side of (8) yields that of (34). □

When n → +∞, since \left[{n \atop λ}\right] → \frac{1}{(q)_{λ}}, equations (33) and (34) reduce respectively to :

∑_{l(λ)≤k} \frac{z^{|λ|}q^{n(2λ)}}{(q^{2}; q^{2})_{λ}(q; q^{2})_{λ_{k}}} = (z; q^{2})_{∞} ∑_{r≥0} \frac{z^{kr}q^{(k+1)\binom{2r}{2}}}{(q)_{2r}(zq^{2r−1})_{∞}} (1 − zq^{4r−1}), (41)

∑_{l(λ)≤k} \frac{z^{|λ|}q^{n(λ)}}{(q)_{λ_{k}} ∏_{i=1}^{k−1}(q^{2}; q^{2})_{[(λ_{i}−λ_{i+1})/2]}}  (42)

= (z^{2}; q^{2})_{∞} ∑_{r≥0} z^{kr}q^{r+(k+1)\binom{r}{2}} \frac{1 − zq^{−1}}{(q)_{r}(1 − zq^{r−1})} \frac{1 − z^{2}q^{2r−1}}{(1 − zq^{r})(z^{2}q^{r−1})_{∞}} .

Furthermore, setting z = q in (41) and (42) we obtain respectively (11) and

∑_{l(λ)≤k} \frac{q^{|λ|+n(λ)}}{(q)_{λ_{k}} ∏_{i=1}^{k−1}(q^{2}; q^{2})_{[(λ_{i}−λ_{i+1})/2]}} = \frac{1}{(q; q^{2})_{∞}} .  (43)

4 Elementary approach to multiple q-identities

4.1 Preliminaries

Recall [1, pp. 36-37] that the binomial formula has the following q-analog :

(z)_{n} = ∑_{m=0}^{n} \left[{n \atop m}\right](−1)^{m}z^{m}q^{m(m−1)/2}.  (44)

Since the elementary symmetric functions e_{r}(X) (0 ≤ r ≤ n) satisfy

(1 + x_{1}z)(1 + x_{2}z) ⋯ (1 + x_{n}z) = ∑_{r=0}^{n} e_{r}(X)z^{r},

it follows from (44) that for integers i ≥ 0 and j ≥ 1 :

e_{r}(q^{i}, q^{i+1}, …, q^{i+j−1}) = q^{ir}e_{r}(1, q, …, q^{j−1}) = q^{ir+\binom{r}{2}}\left[{j \atop r}\right].  (45)

The following result can be derived from the Pieri’s rule for Hall-Littlewood polynomials [9, p. 215], but our proof is elementary.

Lemma 1 For any partition μ such that μ_{1} ≤ n there holds

q^{\binom{m}{2}+n(μ)}\left[{n \atop m}\right]\left[{n \atop μ}\right] = ∑_{λ} q^{n(λ)}\left[{n \atop λ}\right] ∏_{i≥1} \left[{λ_{i} − λ_{i+1} \atop λ_{i} − μ_{i}}\right],  (46)

where the sum is over all partitions λ such that λ/μ is an m-horizontal strip, 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}}, 0 ≤ i ≤ l.

Using (45) to extract the coefficients of z^{m} in the following identity :

(1 + z)(1 + zq) ⋯ (1 + zq^{n−1}) = ∏_{i=0}^{l} ∏_{j∈X_{i}}(1 + zq^{j−1}),

we obtain

q^{\binom{m}{2}}\left[{n \atop m}\right] = ∑_{r} ∏_{i=0}^{l} q^{r_{i} μ_{i+1}+\binom{r_{i}}{2}}\left[{μ_{i} − μ_{i+1} \atop r_{i}}\right],  (47)

where r = (r_{0}, r_{1}, …, r_{l}) is a composition of m. For any such composition r we define a partition λ = (λ_{1}, λ_{2}, …) by

λ_{i} = μ_{i} + r_{i−1}, 1 ≤ i ≤ l + 1.

Then λ/μ is a m-horizontal strip. So (47) can be written as

q^{\binom{m}{2}}\left[{n \atop m}\right] = ∑_{λ} ∏_{i=0}^{l} q^{(λ_{i+1}−μ_{i+1})μ_{i+1}+\binom{λ_{i+1}−μ_{i+1}}{2}}\left[{μ_{i} − μ_{i+1} \atop μ_{i} − λ_{i+1}}\right],  (48)

where the sum is over all partitions λ such that λ/μ is an m-horizontal strip. Now, since

(λ_{i+1} − μ_{i+1})μ_{i+1} + \binom{λ_{i+1} − μ_{i+1}}{2} + \binom{μ_{i+1}}{2} = \binom{λ_{i+1}}{2}, 0 ≤ i ≤ l,

and \left[{n \atop μ}\right] ∏_{i=0}^{l} \left[{μ_{i}−μ_{i+1} \atop μ_{i}−λ_{i+1}}\right] and \left[{n \atop λ}\right] ∏_{i≥1} \left[{λ_{i}−λ_{i+1} \atop λ_{i}−μ_{i}}\right] are equal because they are both equal to

\frac{(q)_{n}}{(q)_{n−λ_{1}}(q)_{λ_{1}−μ_{1}}(q)_{μ_{1}−λ_{2}} ⋯ (q)_{μ_{l}}} ,

multiplying (48) by q^{n(μ)}\left[{n \atop μ}\right] yields (46). □

Lemma 2 There hold the following identities :

∑_{λ} z^{|λ|}q^{2n(λ)}\left[{n \atop λ}\right] = \frac{1}{(z)_{n}} ,  (49)

∑_{λ} z^{|λ|}q^{n(λ)}\left[{n \atop λ}\right] = \frac{(−z)_{n}}{(z^{2})_{n}} ,  (50)

∑_{λ}(q, q^{2})_{λ} z^{|λ|}q^{n(2λ)}\left[{n \atop 2λ}\right] = \frac{(z; q^{2})_{n}}{(z)_{n}} .  (51)

Proof. Identity (49) is due to Hall [6] and can be proved by using the qbinomial identity [8]. Stembridge [12] proved (50) using the q-binomial identity. Now, writing

\frac{(z^{2}; q^{2})_{n}}{(z^{2})_{n}} = (z)_{n} \frac{(−z)_{n}}{(z^{2})_{n}}

and applying successively (44), (50) and (46) we obtain

\frac{(z^{2}; q^{2})_{n}}{(z^{2})_{n}} = ∑_{μ,m}(−1)^{m}z^{m+|μ|}q^{\binom{m}{2}+n(μ)}\left[{n \atop m}\right]\left[{n \atop μ}\right]

= ∑_{μ,m}(−1)^{m}z^{m+|μ|} ∑_{λ: λ/μ=m−hs} ∏_{i≥1} \left[{λ_{i} − λ_{i+1} \atop λ_{i} − μ_{i}}\right] q^{n(λ)}\left[{n \atop λ}\right]

= ∑_{λ} z^{|λ|}q^{n(λ)}\left[{n \atop λ}\right] ∏_{i≥1} ∑_{r_{i}≥0}(−1)^{r_{i}} \left[{λ_{i} − λ_{i+1} \atop r_{i}}\right].

The identity (51) follows then from

∑_{j=0}^{m}(−1)^{j}\left[{m \atop j}\right] = \{ \begin{matrix}(q; q^{2})_{n} & if m = 2n, \\ 0 & if m is odd, \end{matrix}

which can be proved using the q-binomial formula [1, p. 36]. □

Remark. When n → ∞ the above identities reduce respectively to the following :

∑_{λ} \frac{z^{|λ|}q^{2n(λ)}}{(q)_{λ}} = \frac{1}{(z)_{∞}} ,  (52)

∑_{λ} \frac{z^{|λ|}q^{n(λ)}}{(q)_{λ}} = \frac{(−z)_{∞}}{(z^{2})_{∞}} ,  (53)

∑_{λ} \frac{z^{|λ|}q^{n(2λ)}}{(q^{2}; q^{2})_{λ}} = \frac{1}{(zq; q^{2})_{∞}} .  (54)

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 :

2ϕ_{1} ({a, b \atop x}; q; \frac{x}{ab}) := ∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{(q)_{n}(x)_{n}} (\frac{x}{ab})^{n} = \frac{(x/a, x/b)_{∞}}{(x, x/ab)_{∞}} .  (55)

Lemma 3 We have

∑_{λ} z^{|λ|}q^{n(2λ)} \frac{(a, b; q^{−2})_{λ_{1}}}{(q^{2}; q^{2})_{λ}} = \frac{(azq, bzq; q^{2})_{∞}}{(zq, abzq; q^{2})_{∞}} .  (56)

Proof. Substituting q^{2} by q and z by zq, the identity is equivalent to

∑_{λ} z^{|λ|}q^{2n(λ)} \frac{(a, b; q^{−1})_{λ_{1}}}{(q)_{λ}} = \frac{(az, bz)_{∞}}{(z, abz)_{∞}} .  (57)

Now, writing k = λ_{1} and μ = (λ_{2}, λ_{3}, ⋯), and using (49) we get

∑_{λ} z^{|λ|}q^{2n(λ)} \frac{(a, b; q^{−1})_{λ_{1}}}{(q)_{λ}} = ∑_{k≥0} z^{k}q^{k(k−1)} \frac{(a, b; q^{−1})_{k}}{(q)_{k}} ∑_{μ} z^{|μ|}q^{2n(μ)}\left[{k \atop μ}\right]

= ∑_{k≥0}(abz)^{k} \frac{(a^{−1}, b^{−1})_{k}}{(q)_{k}(z)_{k}} .

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 :

φ(u) = ∑_{k≥0} u^{k} ∑_{l(λ)≤k} \frac{(q; q^{2})_{λ}}{(q; q^{2})_{λ_{k}}} z^{|λ|}q^{n(2λ)} \left[{2r \atop 2λ}\right]

= ∑_{λ} u^{l(λ)}z^{|λ|}q^{n(2λ)}(q; q^{2})_{λ} \left[{2r \atop 2λ}\right] ∑_{k≥0} \frac{u^{k}}{(q; q^{2})_{λ_{k+l(λ)}}}

= ∑_{λ} u^{l(λ)}z^{|λ|}q^{n(2λ)}(q; q^{2})_{λ} \left[{2r \atop 2λ}\right] (\frac{u}{1 − u} + \frac{1}{(q; q^{2})_{λ_{l(λ)}}} ) . (58)

Now, each partition λ with parts bounded by r can be encoded by a pair of sequences ν = (ν_{0}, ν_{1}, ⋯, ν_{l}) and m = (m_{0}, ⋯, m_{l}) such that λ = (ν_{0}^{m_{0}}, …, ν_{l}^{m_{l}}), where r = ν_{0} > ν_{1} > ⋯ > ν_{l} > 0 and ν_{i} has multiplicity m_{i} ≥ 1 for 1 ≤ i ≤ l and ν_{0} = r has multiplicity m_{0} ≥ 0. Using the notation :

< α >= \frac{α}{1 − α}, u_{i} = z^{i}q^{i(2i−1)} for i ≥ 0,

we can then rewrite (58) as follows :

φ(u) = ∑_{ν}(q; q^{2})_{ν} \left[{2r \atop 2ν}\right] (< u > +\frac{1}{(q; q^{2})_{ν_{l}}} )

× ∑_{m} ((u_{r}u)^{m_{0}} + \frac{χ(m_{0} = 0)}{(q; q^{2})_{r−ν_{1}}}) ∏_{i=1}^{l}(u_{ν_{i}}u)^{m_{i}}

= ∑_{ν} \frac{(q)_{2r}}{(q^{2}; q^{2})_{ν}} B_{ν},  (59)

where the sum is over all strict partitions ν = (ν_{0}, ν_{1}, …, ν_{l}) and

B_{ν} = (< u > +\frac{1}{(q; q^{2})_{ν_{l}}} ) (< u_{r}u > +\frac{1}{(q; q^{2})_{r−ν_{1}}} ) ∏_{i=1}^{l} < u_{ν_{i}}u > .

So φ(u) is a rational fraction with simple poles at u_{p}^{−1} for 0 ≤ p ≤ r. Let b_{p}(z, r) be the corresponding residue of φ(u) at u_{p}^{−1} for 0 ≤ p ≤ r. Then, it follows from (59) that

b_{p}(z, r) = ∑_{ν} \frac{(q)_{2r}}{(q^{2}; q^{2})_{ν}} [B_{ν}(1 − u_{p}u)]_{u=u_{p}^{−1}} .  (60)

We shall first consider the cases where p = 0 or r. Using (58) and (51) we have

b_{0}(z, r) = [φ(u)(1 − u)]_{u=1} = \frac{(z; q^{2})_{2r}}{(z)_{2r}} .  (61)

Now, by (59) and(60) we have

b_{0}(z, r) = ∑_{ν} \frac{(q)_{2r}}{(q^{2}; q^{2})_{ν}} (< u_{r} > +\frac{1}{(q; q^{2})_{r−ν_{1}}} ) ∏_{i=1}^{l} < u_{ν_{i}} >, (62)

and

b_{r}(z, r) = ∑_{ν} \frac{(q)_{2r}}{(q^{2}; q^{2})_{ν}} (< 1/u_{r} > +\frac{1}{(q; q^{2})_{ν_{l}}} ) ∏_{i=1}^{l} < u_{ν_{i}}/u_{r} >,  (63)

which, by setting μ_{i} = r − ν_{l+1−i} for 1 ≤ i ≤ l and μ_{0} = r, can be written as

b_{r}(z, r) = ∑_{μ} \frac{(q)_{2r}}{(q^{2}; q^{2})_{μ}} (< 1/u_{r} > +\frac{1}{(q; q^{2})_{r−μ_{1}}} ) ∏_{i=1}^{l} < u_{r−μ_{i}}/u_{r} > . (64)

Comparing (64) with (62) we see that b_{r}(z, r) is equal to b_{0}(z, r) with z replaced by z^{−1}q^{−2(2r−1)}. Il follows from (61) that

b_{r}(z, r) = b_{0}(z^{−1}q^{−2(2r−1)}, r) = (z; q^{2})_{2r}q^{r(2r−1)} \frac{1 − zq^{4r−1}}{(zq^{2r−1})_{2r+1}} .  (65)

Consider now the case where 0 < p < r. Clearly, for each partition ν, 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}, …, σ_{l−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 :

b_{p}(z, r) = \left[{2r \atop 2p}\right] ∑_{ρ} \frac{(q)_{2r−2p}}{(q^{2}; q^{2})_{ρ}} F_{ρ}(p) × ∑_{σ} \frac{(q)_{2p}}{(q^{2}; q^{2})_{σ}} G_{σ}(p)

where for ρ = (ρ_{0}, ρ_{1}, …, ρ_{l}) with ρ_{0} = r − p,

F_{ρ}(p) = (< u_{r}/u_{p} > +\frac{1}{(q; q^{2})_{r−p−ρ_{1}}} ) ∏_{i=1}^{l(ρ)} < u_{ρ_{i}+p}/u_{p} >,

and for σ = (σ_{0}, …, σ_{l}) with σ_{0} = p,

G_{σ}(p) = (< 1/u_{p} > +\frac{1}{(q; q^{2})_{σ_{l}}} ) ∏_{i=1}^{l(σ)} < u_{σ_{i}}/u_{p} > .

Comparing with (62) and (64) and using (61) and (65) we obtain

b_{p}(z, r) = \left[{2r \atop 2p}\right] b_{0}(zq^{4p}, r − p) b_{p}(z, p)
= \left[{2r \atop 2p}\right] (z; q^{2})_{2r}q^{\binom{2r}{2p}} \frac{1 − zq^{4p−1}}{(zq^{2p−1})_{2r+1}} .

Finally, extracting the coefficients of u^{k} in the equation

φ(u) = ∑_{p=0}^{r} \frac{b_{p}(z, r)}{1 − u_{p}u},

and using the values for b_{p}(z, r) we obtain(33).

4.3 Proof of Theorem 2

Consider the generating function of the left-hand side of (10) :

φ_{ab}(u) := ∑_{k≥0} u^{k} ∑_{l(λ)≤k} z^{|λ|}q^{n(2λ)} \frac{(a, b; q^{−2})_{λ_{1}}}{(q^{2}; q^{2})_{λ}(q; q^{2})_{λ_{k}}}

= ∑_{λ} ∑_{k≥0} u^{k+l(λ)}z^{|λ|}q^{n(2λ)} \frac{(a, b; q^{−2})_{λ_{1}}}{(q^{2}; q^{2})_{λ}(q; q^{2})_{λ_{l(λ)+k}}}

= ∑_{λ} u^{l(λ)}z^{|λ|}q^{n(2λ)} \frac{(a, b; q^{−2})_{λ_{1}}}{(q^{2}; q^{2})_{λ}} (\frac{u}{1 − u} + \frac{1}{(q; q^{2})_{λ_{l(λ)}}} ) , (66)

where the sum is over all the partitions λ. As in the elementary proof of Theorem 4, we can replace any partition λ by a pair (ν, m), where ν is a strict partition consisting of distinct parts ν_{1}, ⋯, ν_{l} of λ, so that ν_{1} > ⋯ > ν_{l} > 0, and m = (m_{1}, …, m_{l}) is the sequence of multiplicities of ν_{i} for 1 ≤ i ≤ l. Therefore

φ_{ab}(u) = ∑_{ν, m} \frac{(a, b; q^{−2})_{ν_{1}}}{(q^{2}; q^{2})_{ν}} (\frac{u}{1 − u} + \frac{1}{(q; q^{2})_{ν_{l}}} ) ∏_{i=1}^{l}(u_{ν_{i}}u)^{m_{i}}

= ∑_{ν} \frac{(a, b; q^{−2})_{ν_{1}}}{(q^{2}; q^{2})_{ν}} (< u > +\frac{1}{(q; q^{2})_{ν_{l}}} ) ∏_{i=1}^{l} < u_{ν_{i}}u >, (67)

where the sum is over all the strict partitions ν. Each of the terms in this 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 combination of partial fractions. Moreover, the sum of their expansions converges coefficientwise. So φ_{ab} has an expansion

φ_{ab}(u) = ∑_{r≥0} \frac{c_{r}}{1 − uz^{r}q^{r(2r−1)}} ,

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} (r ≥ 0). By using (56) and (66), we get immediately

c_{0} = [φ_{ab}(u)(1 − u)]_{u=1} = \frac{(azq, bzq; q^{2})_{∞}}{(zq, abzq; q^{2})_{∞}} .

In view of (67), this yields the identity

∑_{ν} \frac{(a, b; q^{−2})_{ν_{1}}}{(q^{2}; q^{2})_{ν}} ∏_{i=1}^{l} < u_{ν_{i}} >= \frac{(azq, bzq; q^{2})_{∞}}{(zq, abzq; q^{2})_{∞}} .  (68)

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

c_{r} = ∑_{ρ} \frac{(a, b; q^{−2})_{ρ_{1}+r}}{(q^{2}; q^{2})_{ρ}} ∏_{i=1}^{j−1} < u_{r+ρ_{i}}/u_{r} >

× ∑_{σ} \frac{1}{(q^{2}; q^{2})_{σ}} (< 1/u_{r} > +\frac{1}{(q; q^{2})_{σ_{l−j}}} ) ∏_{i=1}^{l−j} < u_{σ_{i}}/u_{r} > .

In view of (63) the inner sum over σ is equal to b_{r}(z, r)/(q)_{2r}, applying (65), we get

c_{r} = (z; q^{2})_{2r}q^{\binom{2r}{2}} \frac{1 − zq^{4r−1}}{(zq^{2r−1})_{2r+1}} \frac{(a, b; q^{−2})_{r}}{(q)_{2r}}

× ∑_{ρ} \frac{(aq^{−2r}, bq^{−2r}; q^{−2})_{ρ_{1}}}{(q^{2}; q^{2})_{ρ}} ∏_{i=1}^{j−1} < u_{r+ρ_{i}}/u_{r} > .

Now, the sum over ρ can be computed using (68) with a, b and z replaced by aq^{−2r}, bq^{−2r} and zq^{4r}, respectively. After simplification, we obtain

c_{r} = q^{\binom{2r}{2}} \frac{(z; q^{2})_{∞}}{(zq^{2r−1})_{∞}} \frac{(a, b; q^{−2})_{r}(azq^{2r+1}, bzq^{2r+1}; q^{2})_{∞}}{(q)_{2r}(abzq; q^{2})_{∞}} (1 − zq^{4r−1}),

which completes the proof.

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]) :

β_{n} = ∑_{r=0}^{n} \frac{α_{r}}{(q)_{n−r}(xq)_{n+r}}  ∀n ≥ 0.  (69)

If (α_{n}, β_{n}) is a Bailey pair then Bailey’s lemma [3, p. 25-26] states that (α′_{n}, β′_{n}) is also a Bailey pair, where

α′_{n} = \frac{(ρ_{1})_{n}(ρ_{2})_{n}(xq/ρ_{1}ρ_{2})^{n}}{(xq/ρ_{1})_{n}(xq/ρ_{2})_{n}} α_{n}

and

β′_{n} = ∑_{j≥0} \frac{(ρ_{1})_{j}(ρ_{2})_{j}(xq/ρ_{1}ρ_{2})^{j}}{(q)_{n−j}(xq/ρ_{1})_{n}(xq/ρ_{2})_{n}} β_{j}.

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 b_{i} → ∞, c_{i} → ∞ and setting b_{k} = a^{−1} and c_{k} = b^{−1} in [3, Theorem 3.4], we obtain

\frac{(xq, abxq)_{∞}}{(axq, bxq)_{∞}} ∑_{l(λ)≤k} q^{n_{2}(λ)−λ_{1}^{2}+λ_{1}}x^{|λ|}(a^{−1}, b^{−1})_{λ_{1}}(ab)^{λ_{1}} \frac{(q)_{λ_{k}}}{(q)_{λ}} β_{λ_{k}}  (70)

= ∑_{n≥0} q^{(k−1)n^{2}+n}x^{kn} \frac{(a^{−1}, b^{−1})_{n}(ab)^{n}}{(axq, bxq)_{n}} α_{n},

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

α_{n} = q^{n^{2}}(q^{n/2} + q^{−n/2}),  β_{n} = \frac{1}{(q^{1/2}, q)_{n}} ,  (71)

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 :

\frac{(q, abq)_{∞}}{(aq, bq)_{∞}} ∑_{l(λ)≤k} q^{n_{2}(λ)−λ_{1}^{2}+λ_{1}}(ab)^{λ_{1}} \frac{(a^{−1}, b^{−1})_{λ_{1}}}{(q)_{λ}}  (72)

= ∑_{n≥0} q^{(k+ \frac{1}{2} )n^{2}+ \frac{1}{2} n}(−ab)^{n} \frac{(a^{−1}, b^{−1})_{n}}{(aq, bq)_{n}} (1 + q^{n}),

\frac{(q, abq^{2})_{∞}}{(aq^{2}, bq^{2})_{∞}} ∑_{l(λ)≤k} q^{n_{2}(λ)+|λ|−λ_{1}^{2}+λ_{1}}(ab)^{λ_{1}} \frac{(a^{−1}, b^{−1})_{λ_{1}}}{(q)_{λ}}  (73)

= ∑_{n≥0} q^{(k+ \frac{1}{2} )n^{2}+(k+ \frac{3}{2} )n}(−ab)^{n} \frac{(a^{−1}, b^{−1})_{n}}{(aq^{2}, bq^{2})_{n}} (1 − q^{2n+1}),

\frac{(−aq, q)_{∞}}{(−q, aq^{2})_{∞}} ∑_{l(λ)≤k} q^{\frac{1}{2} (n_{2}(λ)+|λ|−λ_{1}^{2}+λ_{1})}(−a)^{λ_{1}} \frac{(a^{−1})_{λ_{1}}}{(q)_{λ}}  (74)

= ∑_{n≥0} q^{\frac{k+1}{2} (n^{2}+n)}a^{n} \frac{(a^{−1})_{n}}{(aq^{2})_{n}} (1 − q^{2n+1}),

\frac{(−aq^{1/2}, q)_{∞}}{(−q^{1/2}, aq)_{∞}} ∑_{l(λ)≤k} q^{\frac{1}{2} (n_{2}(λ)−λ_{1}^{2}+λ_{1})}(−a)^{λ_{1}} \frac{(a^{−1})_{λ_{1}}}{(q)_{λ}}  (75)

= ∑_{n≥0} q^{\frac{k+1}{2} n^{2}}a^{n} \frac{(a^{−1})_{n}}{(aq)_{n}} (1 + q^{n}).

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 → ∞, b_{i} → ∞ for i = 1, ⋯, k − 1 and setting b_{k} = a^{−1} and c_{i} = −\sqrt{xq} for i = 1, ⋯, k in [3, Theorem 3.4] we obtain

\frac{(xq, −a\sqrt{xq})_{∞}}{(axq, −\sqrt{xq})_{∞}} ∑_{l(λ)≤k} q^{\frac{1}{2} (n_{2}(λ)−λ_{1}^{2}+λ_{1})}x^{\frac{1}{2} |λ|}(a^{−1})_{λ_{1}}(−a)^{λ_{1}}  (76)

×\frac{(−\sqrt{xq}, q)_{λ_{k}}}{(q)_{λ}} β_{λ_{k}} = ∑_{n≥0} q^{\frac{1}{2} ((k−1)n^{2}+n)}x^{\frac{1}{2} kn} \frac{(a^{−1})_{n}(−a)^{n}}{(axq)_{n}} α_{n},

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

α_{n} = (−1)^{n}q^{n^{2}}(q^{n/2} + q^{−n/2}),  β_{n} = \frac{1}{(−q^{1/2}, q)_{n}} .  (77)

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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