q–ENGEL SERIES EXPANSIONS AND SLATER’S IDENTITIES

GEORGE E. ANDREWS^{†}, ARNOLD KNOPFMACHER, PETER PAULE^{‡}, AND HELMUT PRODINGER

Abstract. We describe the q–Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater’s list. In our study Al-Salam/Ismail and Santos polynomials play a crucial rôle.

Dedicated to the memory of John Knopfmacher 1937–1999

1. Introduction

In 1987 John Knopfmacher conceived of the idea of representing formal Laurent series as sums of reciprocals of polynomials. The initial motivation was some old results on representations for real numbers by sums of reciprocals of integers, due originally to Lambert, Engel and Sylvester. John together with Arnold had previously investigated various extensions of the real number representations (see the article by Kalpazidou and Ganatsiou [10] in this issue). However the development of analogous expansions for Laurent Series turned out to have some unexpected benefits. Around the time of publication of [11] it was noticed that a number of famous expansions including those of Euler and the Rogers–Ramanujan identities were, in fact, special cases of the q–Engel expansion. This led to the interesting project of using the q–Engel algorithm (described below) to provide new proofs of these identities. This gave rise to a first paper by George, Arnold and John [4] in which the Rogers– Ramanujan identities and some identities of Euler were given new inductive proofs using the q–Engel algorithm discovered by John. Subsequently George, Arnold and Peter have continued these investigations leading to the further publications [3, 6, 5]. To explain our new results we begin by recalling the q–Engel expansion [11, 12] for the field ℒ = ℂ((q)) of formal Laurent series over the complex numbers, ℂ. If

A = ∑_{n=ν}^{∞} L_{n}q^{n} with L_{ν} ≠ 0,

we call ν = ν(A) the order of A and we define the norm of A to be

∥A∥ = 2^{−ν(A)} .

Date: March 9, 2001.

1991 Mathematics Subject Classification. Primary: 11; Secondary: 11.

Key words and phrases. Engel series, q–series, Al-Salam/Ismail polynomials, identities.

†Partially supported by National Science Foundation Grant DMS-9206993 .

‡Partially supported by grant F1305 of the Austrian FWF.

In addition, we define the integral part of A by

[A] = ∑_{ν≤n≤0} L_{n}q^{n} .  (1)

Engel (c. f. [14, §34]) originally defined a series expansion for real numbers. In [11], this concept was extended to ℒ in the following way:

Theorem 1. [q–Engel Expansion Theorem ([11, th. 1. 4]).] Every A ∈ ℒ has a finite or convergent (relative to the above norm) series expansion of the form

A = a_{0} + ∑_{n=1}^{∞} \frac{1}{a_{1}a_{2} ⋯ a_{n}} ,  (2)

where a_{n} ∈ ℂ[q^{−1}], a_{0} = [A],

ν(a_{n}) ≤ −n , and ν(a_{n+1}) ≤ ν(a_{n}) − 1 .  (3)

The series (2) is unique for A, and it is finite if and only if A ∈ ℂ(q). In addition, if

a_{0} + ∑_{j=1}^{n} \frac{1}{a_{1} ⋯ a_{j}} = \frac{p_{n}}{q_{n}} , where q_{n} = a_{1}a_{2} ⋯ a_{n} ,

then

∥A − \frac{p_{n}}{q_{n}} ∥ ≤ \frac{1}{2^{n+1}∥q_{n}∥}

and

ν(A − \frac{p_{n}}{q_{n}} ) = −ν(q_{n+1}) ≥ \frac{(n + 1)(n + 2)}{2} .

In fact, the a_{n} (the “digits”) are for n ≥ 1 given recursively by

a_{n} = [\frac{1}{A_{n}} ]  (4)

where A_{0} = A, a_{0} = [A], A_{1} = A − a_{0}, and for n ≥ 1

A_{n+1} = a_{n}A_{n} − 1.  (5)

To illustrate for example how the first Rogers–Ramanujan identity represents a q–Engel expansion, we write it as

∏_{n=0}^{∞} \frac{1}{(1 − q^{5n+1})(1 − q^{5n+4})} = 1 + ∑_{n=1}^{∞} \frac{q^{n^{2}}}{(1 − q)(1 − q^{2}) ⋯ (1 − q^{n})}

= 1 + ∑_{n=1}^{∞} \frac{1}{a_{1}a_{2} ⋯ a_{n}} ,

where a_{n} = (1 − q^{n})/q^{2n−1} for n ≥ 1. In Section 2, we will in fact make use of a slight variation of the q–Engel algorithm in which (5) is replaced by

A_{n+1} = q(a_{n}A_{n} − 1).  (6)

Corresponding to (2) we have instead the modified expansion,

A = a_{0} + ∑_{n=1}^{∞} \frac{q^{−n}}{a_{1}a_{2} ⋯ a_{n}} .  (7)

An explicit treatment of this modified q–Engel expansion can be found in [6].

In this paper we use the Engel approach e. g. to derive in a new way the representation

lim_{n→∞} U_{n}(0) = (q; q^{2})_{∞} ∑_{n≥0} \frac{q^{2n^{2}}}{(q; q)_{2n}}

with (x; q)_{n} := (1−x)(1−xq) … (1−xq^{n−1}) for the limit of specialized Al-Salam/Ismail polynomials defined below. On the other hand, it has already been shown [7] that the Santos polynomials converge as follows

S_{n} → S_{∞} := ∏_{k≥1, k≡±2, ±3, ±4, ±5 (mod 16)} \frac{1}{1 − q^{k}} .

We link the Al-Salam/Ismail polynomials to the Santos polynomials and shed in this way new light on formulæ related to two identities due to Slater one of which being of the form

∑_{n≥0} \frac{q^{2n^{2}}}{(q; q)_{2n}} = ∏_{k≥1, k≡±2, ±3, ±4, ±5 (mod 16)} \frac{1}{1 − q^{k}} .

One of the main results is Theorem 2 which embeds a new generating function relation for Al-Salam/Ismail polynomials into a q–Engel context. Finally, in Section 3, we discuss Al-Salam/Ismail and Santos polynomials in the context of identities of Garrett/Ismail/Stanton type.

2. Al-Salam/Ismail polynomials and Slater’s identities (38) and (39)

The Al-Salam and Ismail polynomials U_{n}(x; a, b | q) are defined by [1]

U_{−1}(x; a, b | q) = 0,  U_{0}(x; a, b | q) = 1,
U_{n}(x; a, b | q) = x(1 + aq^{n−1})U_{n−1}(x; a, b | q) − bq^{n−2}U_{n−2}(x; a, b | q), n ≥ 1.

Al-Salam and Ismail gave the explicit representation

U_{n}(x; a, b | q) = ∑_{k=0}^{⌊n/2⌋} \frac{(−a; q)_{n−k}(q; q)_{n−k}x^{n−2k}}{(−a; q)_{k}(q; q)_{k}(q; q)_{n−2k}} (−b)^{k}q^{k(k−1)}

= ∑_{k=0}^{⌊n/2⌋} (−aq^{k}; q)_{n−2k}x^{n−2k}\left[{n − k \atop k}\right]_{q}(−b)^{k}q^{k(k−1)};

here we used the Gaussian polynomials

\left[{n \atop k}\right]_{q} = \frac{(q; q)_{n}}{(q; q)_{k}(q; q)_{n−k}} .

We specialize and consider U_{n}(1; −q^{2α+1}, −q^{2α+2} | q^{2}), but simply write U_{n}(α) for that. Let us also rewrite the recursion

U_{−1}(α) = 0, U_{0}(α) = 1,
U_{n}(α) = (1 − q^{2n+2α−1})U_{n−1}(α) + q^{2n+2α−2}U_{n−2}(α), n ≥ 1

and the explicit formula

U_{n}(α) = ∑_{k=0}^{⌊n/2⌋} \left[{n − k \atop k}\right]_{q^{2}}q^{2k^{2}+2kα}(q^{2k+1+2α}; q^{2})_{n−2k}.

In our treatment, α will be either 0 or 1, and we try to treat both cases simultaneously whenever possible. The recursion can be rewritten as

\frac{U_{n}(α)}{(q^{1+2α}; q^{2})_{n}} = \frac{U_{n−1}(α)}{(q^{1+2α}; q^{2})_{n−1}} + \frac{q^{2n−2+2α}}{(q^{1+2α}; q^{2})_{n}} U_{n−2}(α)

and summed:

\frac{U_{n}(α)}{(q^{1+2α}; q^{2})_{n}} = 1 + ∑_{k=1}^{n} \frac{q^{2k−2+2α}}{(q^{1+2α}; q^{2})_{k}} U_{k−2}(α).

In the limit n → ∞,

\frac{U_{∞}(α)}{(q^{1+2α}; q^{2})_{∞}} = 1 + ∑_{k≥1} \frac{q^{2k−2+2α}}{(q^{1+2α}; q^{2})_{k}} U_{k−2}(α).

Theorem 2. If one applies the (modified) Engel algorithm to U_{∞}(α)/(q^{1+2α}; q^{2})_{∞} the quantities A_{n}(α) are given by

A_{n}(α) = ∑_{k≥2} \frac{q^{2kn+2α−1}U_{k−2}(α)}{(q^{2n−1+2α}; q^{2})_{k}} = ∑_{j≥0} \frac{q^{2j^{2}+2αj+4nj+4n+2α−1}}{(q^{2n−1+2α}; q^{2})_{j+1}(q^{2n}; q^{2})_{j+1}} ;

the digits a_{n}(α) are given by

a_{n}(α) = \frac{(1 − q^{2n−1+2α})(1 − q^{2n})}{q^{4n−1+2α}} for n ≥ 1

and a_{0}(α) = 1.

Proof. First, let us prove that the two expressions given for A_{n}(α) are indeed equal:

∑_{k≥2} \frac{q^{2kn+2α−1}U_{k−2}(α)}{(q^{2n−1+2α}; q^{2})_{k}}

= ∑_{k≥0} \frac{q^{2(k+2)n+2α−1}}{(q^{2n−1+2α}; q^{2})_{k+2}} ∑_{0≤2j≤k} \left[{k − j \atop j}\right]_{q^{2}}q^{2j^{2}+2jα}(q^{2j+1+2α}; q^{2})_{k−2j}

= ∑_{j,k≥0} \frac{q^{2(k+2j+2)n+2α−1}}{(q^{2n−1+2α}; q^{2})_{k+2j+2}} \left[{k + j \atop j}\right]_{q^{2}}q^{2j^{2}+2jα}(q^{2j+1+2α}; q^{2})_{k}

= ∑_{j,k≥0} \frac{q^{2(2j+2)n+2α−1+2nk+2j^{2}+2jα}}{(q^{2n−1+2α}; q^{2})_{2j+2}(q^{2n+4j+3+2α}; q^{2})_{k}} \frac{(q^{2j+2}; q^{2})_{k}}{(q^{2}; q^{2})_{k}} (q^{2j+1+2α}; q^{2})_{k}

= ∑_{j≥0} \frac{q^{2(2j+2)n+2α−1+2j^{2}+2jα}}{(q^{2n−1+2α}; q^{2})_{2j+2}} ∑_{k≥0} \frac{(q^{2j+2}; q^{2})_{k}(q^{2j+1+2α}; q^{2})_{k}q^{2nk}}{(q^{2}; q^{2})_{k}(q^{2n+4j+3+2α}; q^{2})_{k}}

(the inner sum can be computed by q–Gauss [2, p.20])

= ∑_{j≥0} \frac{q^{2(2j+2)n+2α−1+2j^{2}+2jα}}{(q^{2n−1+2α}; q^{2})_{2j+2}} \frac{(q^{2n+2α+2j+1}; q^{2})_{∞}(q^{2n+2j+2}; q^{2})_{∞}}{(q^{2n+2α+4j+3}; q^{2})_{∞}(q^{2n}; q^{2})_{∞}}

= ∑_{j≥0} \frac{q^{4jn+4n+2α−1+2j^{2}+2jα}}{(q^{2n−1+2α}; q^{2})_{2j+2}} \frac{(q^{2n+2α+2j+1}; q^{2})_{j+1}}{(q^{2n}; q^{2})_{j+1}}

= ∑_{j≥0} \frac{q^{4jn+4n+2α−1+2j^{2}+2jα}}{(q^{2n−1+2α}; q^{2})_{j+1}(q^{2n}; q^{2})_{j+1}} ,

as desired. Let us now prove the announced formula for the digits. This is particularly easy since the j = 0 term is the reciprocal of a_{n}(α), and therefore

∑_{j≥0} \frac{q^{2j^{2}+2αj+4nj+4n+2α−1}}{(q^{2n−1+2α}; q^{2})_{j+1}(q^{2n}; q^{2})_{j+1}} = \frac{1}{a_{n}} + O(q^{8n+1+4α}) = \frac{1}{a_{n}} (1 + O(q^{4n+2α+2})).

The recursion is also simple, since

q(a_{n}(α)A_{n}(α) − 1)

= q(a_{n}(α) ∑_{j≥0} \frac{q^{2j^{2}+2αj+4nj+4n+2α−1}}{(q^{2n−1+2α}; q^{2})_{j+1}(q^{2n}; q^{2})_{j+1}} − 1)

= qa_{n}(α) ∑_{j≥1} \frac{q^{2j^{2}+2αj+4nj+4n+2α−1}}{(q^{2n−1+2α}; q^{2})_{j+1}(q^{2n}; q^{2})_{j+1}}

= ∑_{j≥1} \frac{q^{2j^{2}+2αj+4nj+1}}{(q^{2n+1+2α}; q^{2})_{j}(q^{2n+2}; q^{2})_{j}}

= ∑_{j≥0} \frac{q^{2(j+1)^{2}+2α(j+1)+4n(j+1)+1}}{(q^{2n+1+2α}; q^{2})_{j+1}(q^{2n+2}; q^{2})_{j+1}}

= ∑_{j≥0} \frac{q^{2j^{2}+2αj+4(n+1)j+4(n+1)+2α−1}}{(q^{2n+1+2α}; q^{2})_{j+1}(q^{2n+2}; q^{2})_{j+1}}

= A_{n+1}(α),

as desired.

Thus our Engel proof established the following relations.

Theorem 3. For α = 0, 1 we have

lim_{n→∞} U_{n}(α) = (q; q^{2})_{∞} ∑_{n≥0} \frac{q^{2n^{2}+2nα}}{(q; q)_{2n+α}} .

This is a new proof of known results due to Al-Salam and Ismail [1].

Slater’s identities (39) and (38)^{1} [16] are the formulæ

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

∑_{n≥0} \frac{q^{2n^{2}+2n}}{(q; q)_{2n+1}} = ∏_{k≥1, k≡±1, ±4, ±6, ±7 (mod 16)} \frac{1}{1 − q^{k}} .

Let us consider the Santos polynomials in the representation

S_{n} = ∑_{0≤2j≤n} q^{2j^{2}} \left[{n \atop 2j}\right]_{q},

T_{n} = ∑_{0≤2j≤n−1} q^{2j^{2}+2j}\left[{n \atop 2j + 1}\right]_{q}.

Originally these polynomials were studied in [15] and were discussed further in [7]. As pointed out ibid. they possess the following limit property,

S_{n} → S_{∞} := ∏_{k≥1, k≡±2, ±3, ±4, ±5 (mod 16)} \frac{1}{1 − q^{k}} ,

T_{n} → T_{∞} := ∏_{k≥1, k≡±1, ±4, ±6, ±7 (mod 16)} \frac{1}{1 − q^{k}} .

In [6], the quantities S_{∞} and T_{∞} were undergone an Engel treatment; see also Section 3 below.

1Observe the order!

After observing these limits, Slater’s identities (39) and (38) are immediate by taking n → ∞ in the polynomial representations of S_{n} and T_{n} above. However, the Engel approach encoded by Theorem 2 admits another link to Slater’s identities which is briefly explained as follows. Although it is not shown explicitly in [7], it is easy to verify that the polynomials S_{n} and T_{n} satisfy the following defining recurrences

S_{n} = S_{n−1} + q^{n}T_{n−1},  S_{0} = 1,
T_{n} = T_{n−1} + q^{n−1}S_{n−1},  T_{0} = 0.

As a direct consequence of these recurrences we see that

S_{∞} = 1 + ∑_{j≥1} q^{j}T_{j−1} and (1 − q) T_{∞} = 1 + ∑_{j≥1} q^{2j}T_{j−1}.

So, by Theorem 2 we obtain an alternative statement being equivalent to Slater’s identities; namely,

\frac{1}{q}A_{1}(α) = ∑_{j≥1} q^{(1+α)j}T_{j−1} (α = 0, 1)

where A_{1}(α) is chosen as in Theorem 2. As we shall see below, this fact is easily established, and—again as a by-product of the Engel context of Theorem 2—one obtains the more general relation

A_{n}(α) = ∑_{j≥1} q^{(2n−1+α)j+1}T_{j−1} (α = 0, 1)

where A_{n}(α) is chosen as in Theorem 2. However, the proof for n = 1 is at the same level of complexity as for n, and so we do the computation for general n.

∑_{j≥1} q^{(2n−1+α)j+1}T_{j−1} = ∑_{j≥1} q^{(2n−1+α)j+1} ∑_{0≤2k≤j−2} q^{2k^{2}+2k}\left[{j − 1 \atop 2k + 1}\right]_{q}

= ∑_{j,k≥0} q^{(2n−1+α)(j+2k+2)+1+2k^{2}+2k}\left[{j + 2k + 1 \atop 2k + 1}\right]_{q}

= ∑_{k≥0} q^{(2n−1+α)(2k+2)+1+2k^{2}+2k} ∑_{j≥0} q^{(2n−1+α)j}\left[{j + 2k + 1 \atop 2k + 1}\right]_{q}

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

For α = 0 we have

∑_{k≥0} \frac{q^{4nk+4n−1+2k^{2}}}{(q^{2n−1}; q)_{2k+2}} = ∑_{j≥0} \frac{q^{2j^{2}+4nj+4n−1}}{(q^{2n−1}; q^{2})_{j+1}(q^{2n}; q^{2})_{j+1}} ,

and for α = 1

∑_{k≥0} \frac{q^{4nk+4n+1+2k^{2}+2k}}{(q^{2n}; q)_{2k+2}} = ∑_{j≥0} \frac{q^{2j^{2}+2j+4nj+4n+1}}{(q^{2n+1}; q^{2})_{j+1}(q^{2n}; q^{2})_{j+1}} ,

as it should. (For other values of α it does not work!) Finally we will establish that the Al-Salam/Ismail sequences U_{n}(α)/(q; q^{2})_{n} converge about twice as fast to S_{∞} resp. T_{∞} as S_{n} resp. T_{n}. For that we need two simple facts:

\left[{A \atop B}\right]_{q} = \frac{(1 − q^{A})(1 − q^{A−1}) … (1 − q^{A−B+1})}{(q; q)_{B}}

= \frac{1}{(q; q)_{B}} (1 − q^{A−B+1} + O(q^{A−B+2}))

and

\frac{1}{(q^{c}; q)_{n}} = 1 + q^{c} + O(q^{c+1}).

Therefore

S_{n} = ∑_{j≥0} \left[{n \atop 2j}\right]_{q}q^{2j^{2}} = 1 + ∑_{j≥1} \frac{q^{2j^{2}}}{(q; q)_{2j}} (1 − q^{n−2j+1} + O(q^{n−2j+2}))

= ∑_{j≥0} \frac{q^{2j^{2}}}{(q; q)_{2j}} − q^{n+1} + O(q^{n+2}).

Similarly

T_{n} = ∑_{j≥0} \left[{n \atop 2j + 1}\right]_{q}q^{2j^{2}+2j} = \frac{1 − q^{n}}{1 − q} + ∑_{j≥1} \frac{q^{2j^{2}+2j}}{(q; q)_{2j+1}} (1 − q^{n−2j} + O(q^{n−2j+1}))

= ∑_{j≥0} \frac{q^{2j^{2}+2j}}{(q; q)_{2j+1}} − q^{n} + O(q^{n+1}).

On the other hand,

\frac{U_{n}(0)}{(q; q^{2})_{n}} = \frac{1}{(q; q^{2})_{n}} ∑_{j≥0} \left[{n − j \atop j}\right]_{q^{2}}q^{2j^{2}}(q^{2j+1}; q^{2})_{n−2j}

= ∑_{j≥0} \left[{n − j \atop j}\right]_{q^{2}}q^{2j^{2}} \frac{1}{(q; q^{2})_{j}(q^{2n−2j+1}; q^{2})_{j}}

= 1 + ∑_{j≥1} \frac{1}{(q^{2}; q^{2})_{j}} (1 − q^{2n−4j+2} + O(q^{2n−4j+4}))\frac{q^{2j^{2}}}{(q; q^{2})_{j}}

× (1 + q^{2n−2j+1} + O(q^{2n−2j+3}))

= ∑_{j≥0} \frac{q^{2j^{2}}}{(q; q)_{2j}} − q^{2n} + O(q^{2n+1}).

Similarly,

\frac{U_{n}(1)}{(q; q^{2})_{n}} = \frac{1}{(q; q^{2})_{n}} ∑_{j≥0} \left[{n − j \atop j}\right]_{q^{2}}q^{2j^{2}+2j}(q^{2j+3}; q^{2})_{n−2j}

= \frac{1 − q^{2n+1}}{1 − q} + ∑_{j≥1} \left[{n − j \atop j}\right]_{q^{2}}q^{2j^{2}+2j} \frac{1}{(q; q^{2})_{j+1}(q^{2n−2j+3}; q^{2})_{j−1}}

= \frac{1 − q^{2n+1}}{1 − q} + ∑_{j≥1} \frac{1}{(q^{2}; q^{2})_{j}} (1 − q^{2n−4j+2} + O(q^{2n−4j+4}))\frac{q^{2j^{2}+2j}}{(q; q^{2})_{j+1}}

× (1 + q^{2n−2j+3} + O(q^{2n−2j+5}))

= ∑_{j≥0} \frac{q^{2j^{2}+2j}}{(q; q)_{2j+1}} − q^{2n+1} + O(q^{2n+2}).

3. Identities of Garrett/Ismail/Stanton type

In [8] Garrett et al. presented a new parameterized generalization of the celebrated Rogers–Ramanujan identities. As a by-product of an Engel study, Andrews et al. [5] derived a polynomial version of it which in the limit coincides with the Garrett/Ismail/ Stanton result. In [9] Ismail et al. have put the polynomial version from [5] into the context of orthogonal polynomials, in particular, of the Al-Salam/Ismail polynomials U_{n}. It turned out that not only the Rogers–Ramanujan identities but also other entries listed by Slater [16] give rise to this type of generalization. For instance, in [9] it was shown that for m ≥ 0,

(−1)^{m}q^{m^{2}+m}U_{n}(m + 1) = U_{m}(0) U_{m+n}(1) − U_{m−1}(1) U_{m+n+1}(0).  (8)

Above we have proved that

\frac{U_{∞}(0)}{(q; q^{2})_{∞}} = S_{∞} and \frac{U_{∞}(1)}{(q; q^{2})_{∞}} = T_{∞}.  (9)

So after dividing both sides of (8) by (q; q^{2})_{∞} we obtain in the limit n → ∞ an identity of Garrett/Ismail/Stanton type; namely for m ≥ 0,

(−1)^{m}q^{m^{2}+m} ∑_{k≥0} \frac{q^{2k^{2}+2(m+1)k}}{(q; q)_{2k+1}(q^{2k+3}; q^{2})_{m}} = U_{m}(0) T_{∞} − U_{m−1}(1) S_{∞}.  (10)

The experimental use of Engel, a computer algebra implementation of the q–Engel expansion algorithm, led Andrews et al. [6] to the discovery of an identity similar to (10) but using Santos polynomials instead; namely for m ≥ 0,

q^{m}(q; q^{2})_{m} ∑_{k≥0} \frac{q^{2k^{2}+2(m+1)k}}{(q; q)_{2k+1}} = S_{m} T_{∞} − T_{m} S_{∞}.  (11)

The corresponding polynomial version, the counterpart to (8), reads as follows. For m ≥ 0,

q^{m}(q; q^{2})_{m} ∑_{k≥0} \left[{n \atop 2k + 1}\right]_{q}q^{2k^{2}+2(m+1)k} = S_{m} T_{m+n} − T_{m} S_{m+n}.  (12)

Once found, such identities most often find quite elementary proofs. Nevertheless, the theme of this section is to sketch a framework that helps to explain and to derive identities of this type. We also stress the fact that this approach can be ideally supplemented by computer algebra packages like Engel and q–versions of Zeilberger’s (“fast”) algorithm, as for instance qZeil [13]. Such packages not only can help in proving, but also in finding such identities.

All what follows is motivated by techniques from orthogonal polynomials as used e. g. in [9]. However, we will not enter this theory (e. g., numerator or associated polynomials) but rather restrict ourselves to recall a few basic facts from the general theory of difference equations. Let 𝔽 be a suitable field, as e. g. 𝔽 = ℂ(q) where q is an indeterminate. Let 𝔽^{⋆} denote the non-zero elements of 𝔽. Let us fix two coefficient sequences α = (α_{n})_{n≥0} and β = (β_{n})_{n≥0} with elements in 𝔽^{⋆}. Consider the recurrence equation

x_{n} = α_{n−1}x_{n−1} + β_{n−2}x_{n−2} (n ≥ 2).  (13)

Suppose the 𝔽–sequences a = (a_{n})_{n≥0} and b = (b_{n})_{n≥0} are solutions of (13). For n ≥ 0 we define the discrete Wronskian as usual by

W_{n}(a, b) := |\begin{matrix}a_{n} & b_{n} \\ a_{n+1} & b_{n+1} \end{matrix}| .

As a matter of fact, the solutions a and b are linearly independent over 𝔽 if and only if W_{n}(a, b) ≠ 0 for all n ≥ 0. But this is equivalent to W_{0}(a, b) ≠ 0 since

W_{n}(a, b) = (−1)^{n}β_{n−1}β_{n−2} … β_{0} W_{0}(a, b) (n ≥ 0).

Combining these facts one can easily prove the following theorem.

Theorem 4. Let α, β, a, and b be sequences as above where a and b are linearly independent solutions of (13). Let m be a non-negative integer and let c(m) = (c_{n}(m))_{n≥0} satisfy

z_{n} = α_{m+n−1} z_{n−1} + β_{m+n−2} z_{n−2} (n ≥ 2).  (14)

Then

c_{n}(m) = u_{m} a_{m+n} + v_{m} b_{m+n} (n ≥ 0)

where

u_{m} = −\frac{c_{1}(m) b_{m} − c_{0}(m) b_{m+1}}{W_{m}(a, b)}

and

v_{m} = \frac{c_{1}(m) a_{m} − c_{0}(m) a_{m+1}}{W_{m}(a, b)} .

Now we apply Theorem 4 in order to prove the polynomial identities (8) and (11).

Example 1. Let m be a non-negative integer. The Al-Salam/Ismail polynomials U_{n}(m) satisfy (14) with α_{n} = 1 − q^{n} and β_{n} = q^{n}. So in view of Theorem 4 we can take a = (a_{n})_{n≥0} with a_{n} = U_{n}(0) and c(m) = (c_{n}(m))_{n≥0} with c_{n}(m) = U_{n}(m). But we also need a linearly independent solution b = (b_{n})_{n≥0}. Let us try b_{n} = U_{n−1}(1) since then a and b are both solutions of (13). It turns out that b chosen this way is also linearly independent since W_{0}(a, b) = U_{0}(0)U_{0}(1) − U_{1}(0)U_{−1}(1) = 1; therefore we can invoke Theorem 4. It is easily checked that

W_{m}(a, b) = (−1)^{m}q^{2m}q^{2m−2} … q^{2}W_{0}(a, b) = (−1)^{m}q^{m^{2}+m},

and

c_{0}(m) = U_{0}(m) = 1 and c_{1}(m) = U_{1}(m) = 1 − q^{2m+1}.

For the coefficients we obtain

u_{m} = −(−1)^{m}q^{−m^{2}−m}((1 − q^{2m+1})U_{m−1}(1) − U_{m}(1)) = (−1)^{m}q^{−m^{2}+m}U_{m−2}(1)

where the last equation is by (14), and similarly,

v_{m} = (−1)^{m}q^{−m^{2}−m}((1 − q^{2m+1})U_{m}(0) − U_{m+1}(0)) = −(−1)^{m}q^{−m^{2}+m}U_{m−1}(0).

Now Theorem 4 yields identity (11) with m replaced by m − 1.

Example 2. As an easy consequence of the mixed recurrences for the Santos polynomials S_{n} and T_{n} one immediately derives that both polynomials are solutions of

x_{n} = (1 + q) x_{n−1} − q(1 − q^{2n−3}) x_{n−2} (n ≥ 2).  (15)

In view of (13) we have α_{n} = 1 + q and β_{n} = −q(1 − q^{2n+1}). Setting a_{n} = S_{n} and b_{n} = T_{n} we see that the corresponding sequences a and b are linearly independent since W_{0}(a, b) = S_{0}T_{1} − T_{0}S_{1} = 1. In addition, for any non-negative integer m the sequence c(m) = (c_{n}(m))_{n≥0} with

c_{n}(m) = ∑_{k≥0} \left[{n \atop 2k + 1}\right]_{q}q^{2k^{2}+2(m+1)k}

satisfies

z_{n} = (1 + q) z_{n−1} − q(1 − q^{2(m+n)−3}) z_{n−2} (n ≥ 2).  (16)

This e. g. can be proven automatically with the package qZeil. Summarizing, we are in the position to invoke Theorem 4. It is easily checked that

W_{m}(a, b) = q^{m}(1 − q^{2m−1})(1 − q^{2m−3}) … (1 − q)W_{0}(a, b) = q^{m}(q; q^{2})_{m},

and

c_{0}(m) = 0 and c_{1}(m) = 1.

Hence by Theorem 4, identity (8) is proved.

Finally we demonstrate the applicability of Theorem 4 by deriving a related identity which to our knowledge is new. Example 3. By using the package qZeil one finds that

c_{n}(m) = ∑_{k≥0} \left[{n \atop 2k}\right]_{q}q^{2k^{2}+2mk}

is also a solution of (16); this heuristic procedure will be described in a forthcoming paper. Thus we can take a and b as in Example 2 which gives the same expression for W_{m}(a, b). The only difference to the previous situation is that now

c_{0}(m) = 1 and c_{1}(m) = 1.

We obtain for the coefficients

u_{m} = −\frac{q^{−m}}{(q; q^{2})_{m}} (T_{m} − T_{m+1}) = \frac{1}{(q; q^{2})_{m}} S_{m}

where the last equation is by the mixed Santos recurrence, and similarly,

v_{m} = \frac{q^{−m}}{(q; q^{2})_{m}} (S_{m} − S_{m+1}) = −\frac{q}{(q; q^{2})_{m}} T_{m}.

Theorem 1 now implies that for m ≥ 0,

(q; q^{2})_{m} ∑_{k≥0} \left[{n \atop 2k}\right]_{q}q^{2k^{2}+2mk} = S_{m} S_{m+n} − q T_{m} T_{m+n}.  (17)

The corresponding limiting version, i. e. for n → ∞, reads as follows. For m ≥ 0,

(q; q^{2})_{m} ∑_{k≥0} \frac{q^{2k^{2}+2mk}}{(q; q)_{2k}} = S_{m} S_{∞} − q T_{m} T_{∞}.  (18)

References

[1] W. Al-Salam and M. Ismail. Orthogonal polynomials associated with the Rogers–Ramanujan continued fraction. Pacific J. Math., 104:269–283, 1983.

[2] G. E. Andrews. The Theory of Partitions, volume 2 of Encyclopedia of Mathematics and its Applications. Addison–Wesley, 1976, reissued: Cambridge University Press, Cambridge, 1985.

[3] G. E. Andrews and A. Knopfmacher. An algorithmic approach to discovering and proving q– series identities. Algorithmica , 29:34–43, 2001.

[4] G. E. Andrews, A. Knopfmacher, and J. Knopfmacher. Engel expansions and the Rogers– Ramanujan identities. Journal of Number Theory, 80:273–290, 2000.

[5] G. E. Andrews, A. Knopfmacher, and P. Paule. An infinite family of Engel expansions of Rogers–Ramanujan type. Advances in Applied Math., 25:2–11, 2000.

[6] G. E. Andrews, A. Knopfmacher, P. Paule, and B. Zimmermann. Engel expansions of q–series by computer algebra. To appear in: Kluwer book series “Developments in Mathematics,” 2001.

[7] G. E. Andrews and J. Santos. Rogers–Ramanujan type identities for partitions with attached odd parts. Ramanujan Journal, 1:91–99, 1997.

[8] T. Garrett, M. Ismail, and D. Stanton, Variants of the Rogers–Ramanujan identities, Advances in Applied Mathematics, 23 (1999), 274–299.

[9] M. Ismail, H. Prodinger, and D. Stanton, Schur’s Determinants and Partition Theorems, Séminaire Lotharingien de Combinatoire, B44a (2000), 10 pp.

[10] S. Kalpazidou and G. Ganatsiou. Knopfmacher expansions in number theory. Quaestiones Mathematicæ (this volume), 2001.

[11] A. Knopfmacher and J. Knopfmacher. Inverse polynomial expansions of Laurent series. Constr. Approx., 4:379–389, 1988.

[12] A. Knopfmacher and J. Knopfmacher. Inverse polynomial expansions of Laurent series ii. J. Comp. and Appl. Math., 28:249–257, 1989.

[13] P. Paule and A. Riese, A Mathematica q–Analogue of Zeilberger’s Algorithm Based on an Algebraically Motivated Approach to q–Hypergeometric Telescoping, pp. 179–210 in: Fields Institute Communications, Vol. 14, Amer. Math. Soc., Providence, 1997.

[14] O. Perron. Irrationalzahlen. Chelsea, 1951.

[15] J. Santos. Computer algebra and identities of the Rogers–Ramanujan type. Ph. D. thesis, Penn State University, 1991.

[16] L. Slater. Further identities of the Rogers–Ramanujan type. Proc. London Math. Soc., 54:147– 167, 1952.

(G. E. A.) Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U. S. A.

E-mail address: andrews@math.psu.edu

(A. K.) The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Private Bag 3, WITS 2050, South Africa

E-mail address: arnoldk@cam.wits.ac.za
URL: http://www.wits.ac.za/science/number theory/arnold.htm

(P. P.) Research Institute for Symbolic Computation, Johannes Kepler University Linz, A-4040 Linz, Austria

E-mail address: Peter.Paule@risc.uni-linz.ac.at
URL: http://www.risc.uni-linz.ac.at/research/combinat/

(H. P.) The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Private Bag 3, WITS 2050, South Africa

E-mail address: helmut@cam.wits.ac.za
URL: http://www.wits.ac.za/helmut/index.htm