On Series Expansions of Capparelli's Infinite Product
Andrew V. Sills

TL;DR
This paper introduces two new series representations for the infinite product linked to Capparelli's Rogers-Ramanujan type identity, expanding the analytic understanding of these combinatorial identities and presenting related new identities.
Contribution
It provides novel series expansions for Capparelli's infinite product and introduces additional related identities, enhancing the analytic framework of Rogers-Ramanujan type identities.
Findings
Two new series representations for Capparelli's infinite product
Additional related identities and infinite families presented
Enhanced analytic understanding of Capparelli's conjecture
Abstract
Using Lie theory, Stefano Capparelli conjectured an interesting Rogers-Ramanujan type partition identity in his 1988 Rutgers Ph.D. thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able to provide a Lie theoretic proof. Most combinatorial Rogers-Ramanujan type identities (e.g. the G\"ollnitz-Gordon identities, Gordon's combinatorial generalization of the Rogers-Ramanujan identities, etc.) have an analytic counterpart. The main purpose of this paper is to provide two new series representations for the infinite product associated with Capparelli's conjecture. Some additional related identities, including new infinite families are also presented.
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On Series Expansions of
Capparelli’s Infinite Product
Andrew V. Sills
Department of Mathematics,
Rutgers University, Piscataway, NJ, USA 08854-8019
http://www.math.rutgers.edu/~asills
(September 9, 2003.)
Abstract
Using Lie theory, Stefano Capparelli conjectured an interesting Rogers-Ramanujan type partition identity in his 1988 Rutgers Ph.D. thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able to provide a Lie theoretic proof.
Most combinatorial Rogers-Ramanujan type identities (e.g. the Göllnitz-Gordon identities, Gordon’s combinatorial generalization of the Rogers-Ramanujan identities, etc.) have an analytic counterpart. The main purpose of this paper is to provide two new series representations for the infinite product associated with Capparelli’s conjecture. Some additional related identities, including new infinite families are also presented.
1 Introduction
In 1894, L.J. Rogers was the first to discover a pair of series–product identities which are now known as the Rogers-Ramanujan identities. They may be stated compactly as follows:
Rogers-Ramanujan Identities–Analytic Form. * For or ,*
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
and
[TABLE]
(Although the results in this paper may be considered purely from the point of view of formal power series, they also yield identities of analytic functions provided .)
A partition of an integer is a nonincreasing finite sequence of positive integers such that . The ’s are called the parts of the partition .
MacMahon [22] and Schur [27] independently saw that the Rogers-Ramanujan identities were in fact equivalent to the following partition theoretic statement:
Rogers-Ramanujan Identities–Combinatorial Form. * Let denote the number of partitions of into parts wherein and . Let denote the number of partitions of wherein all parts are congruent to modulo . Then for all integers and for or , .*
Over the years, many other analytic and combinatorial identities of Rogers-Ramanujan type were discovered, including the following analytic identity of Slater [28, p. 155, equations (36) and (34)] and its combinatorial counterpart due to Göllnitz [13], and rediscovered by Gordon [14]:
Slater’s mod 8 Identities. * For or ,*
[TABLE]
The Göllnitz-Gordon Partition Identities. * Let denote the number of partitions of wherein , , and if or is even. Let denote the number of partitions of into parts congruent to or modulo 8. Then for all integers and or .*
Following a program of research initiated by Lepowsky-Milne ([15], [16]), and Lepowsky-Wilson ([17],[18],[19],[20],[21]), Stefano Capparelli was able to conjecture a partition identity as a result of his studies of the standard level 3 modules associated with the Lie Algebra , and included this conjecture in his Ph.D. thesis [10]:
Capparelli’s Conjecture. * Let denote the number of partitions of wherein , , and if , then either and are are both multiples of three, or and . Let denote the number of partitions of into parts congruent to or modulo . Then for all integers .*
George Andrews, inspired by the combinatorial techniques of Wilf and Zeilberger [24], provided the first proof in [6]. Later, Lie-theoretic proofs were supplied by Tamba and Xie [29] and by Capparelli himself [11]. In [23], Meurman and Primc embed Capparelli’s conjecture in an infinite family of three-color partition identities.
In [1], Alladi, Andrews, and Gordon provided refinements to Capparelli’s conjecture along with a corresponding identity of generating functions. By replacing with , and setting , and in [1, p. 648–9, Lemma 2(b)], one can deduce the following analytic counterpart to Capparelli’s conjecture:
[TABLE]
The main goal of this paper is to present two additional analytic identities involving the infinite product , namely
[TABLE]
where is the Legendre symbol, and
[TABLE]
which will actually arise as a corollary to the following analytic identity, an “-generalization of an analytic counterpart of Capparelli’s conjecture”:
[TABLE]
In section 2, it will be revealed how identity (1.4) arises from two of the simplest possible Bailey pairs. Section 3 will be devoted to a derivation of the Bailey pair necessary to yield identity (1.6). Once this is accomplished, the identities (1.4) and (1.5) will be embedded in infinite family of identities:
[TABLE]
where .
[TABLE]
where . Notice that the case of (1.8) is equivalent to (1.5) since
[TABLE]
In section 4, some related identities will be noted. In section 5, we conclude with some related open questions.
2 Implications of two simple Bailey pairs
We will require the standard machinery of Bailey’s Lemma and Bailey pairs (see [8], [9], [5, Ch. 3]). Recall that two sequences of rational functions form a Bailey pair if for all ,
[TABLE]
and that for any Bailey pair , the identity
[TABLE]
holds (Andrews [5, p. 27, equation (3.33)]).
In the literature (see e.g. Andrews [5, section 3.5]) the implications of a particular Bailey pair (often called the “unit Bailey pair”) consisting of an extremely simple and its corresponding are considered. Here, in contrast, we consider Bailey pairs where the ’s are of an especially simple nature.
Theorem 2.1**.**
Suppose
[TABLE]
and
[TABLE]
where the Gaussian polynomial is defined by
[TABLE]
Then form a Bailey pair.
Proof.
Considereing (2.1) with ,
[TABLE]
∎
Corollary 2.2**.**
[TABLE]
Proof.
By Theorem 2.1 and (2.2) with ,
[TABLE]
by Jacobi’s triple product identity [12, p. 12, equation (1.6.1)]. ∎
Theorem 2.3**.**
If
[TABLE]
and
[TABLE]
then form a Bailey pair.
Proof.
[TABLE]
∎
Corollary 2.4**.**
[TABLE]
Proof.
By Theorem 2.3 and (2.2) with ,
[TABLE]
by Jacobi’s triple product identity [12, p. 12, equation (1.6.1)]. ∎
Theorem 2.5**.**
Identity (1.4) is valid.
Proof.
Essentially all we need to do is subtract the identity in Corollary 2.4 from the identity in Corollary 2.2. For the left hand side, observe that
[TABLE]
(i.e. the inner sum on sums over the -th row of the -Pascal triangle weighting consecutive summands in turn by the factors , , and [math],)
[TABLE]
For the right hand side, observe that
[TABLE]
∎
Remark 2.6**.**
Identity (1.4) can be rewritten as
[TABLE]
by splitting the inner sum on in the left hand side of (1.4) into even and odd , interchanging the order of summation and replacing by . In this formulation, both sums are truly infinite over all nonnegative and .
3 Another Bailey pair and its implications
Recall the standard notation for basic hypergeometric series
[TABLE]
Remark 3.1**.**
The real challenge here was to find an appropriate so that
- •
when is inserted into (2.1), the resulting expression is a (finite product multiplied by a) basic hypergeometric series which can be transformed appropriately, and
- •
when is inserted into (2.2) and is set to , the generating function
[TABLE]
results.
Once this is achieved, the power of Bailey’s lemma and Bailey chains allows us to derive a number of identities with little additional effort.
Theorem 3.2**.**
If
[TABLE]
and
[TABLE]
then \big{(}\alpha_{n}(a,q),\beta_{n}(a,q)\big{)} form a Bailey pair.
Proof.
[TABLE]
∎
Theorem 3.3**.**
Identity (1.6) is valid.
Proof.
Insert the Bailey pair in Theorem 3.2 into equation (2.2), and then replace by and by throughout. On the left hand side, interchange the order of summation bringing out in front of and replace by . Then, interchange the order of summation bringing in front of and replace by . ∎
Remark 3.4**.**
Andrews [7] pointed out that a direct proof (i.e. one that is independent of Bailey’s lemma) of (1.6) is possible. Start out with the left hand side of (1.6) and set so that the double sum is now on and . The inner sum on is
[TABLE]
which is summable by the -Chu-Vandermonde formula [12, p. 236, equation (II.6)]. This form can then be converted to the right hand side using a formula due to Euler [2, p. 19, equation (2.2.5)].
Corollary 3.5**.**
Identity (1.5) is valid.
Proof.
Set in identity (1.6), then apply Jacobi’s triple product identity [5, p. 63, (7.1)] to the right hand side, and simplify the resulting product. ∎
Now that Bailey pairs have been established, it is a simple matter to embed the analytic Capparelli identities into infinite families of identities using the notion of the “Bailey chain” ([3], [5, p. 28 ff.]):
Theorem 3.6**.**
Identity (1.7) is valid.
Proof.
Insert the Bailey pairs from Theorem 2.1 and Theorem 2.3 into equation (3.34) of Andrews [5, p. 30], with . Subtract the second equation from the first, interchange orders of summation and change summation variables as appropriate. ∎
Theorem 3.7**.**
Identity (1.8) is valid.
Proof.
Insert and into equation (3.34) of Andrews [5, p. 30], interchange orders of summation and change summation variables as appropriate. Finally, replace by and by throughout. ∎
4 Related identities
In order to obtain the -generalization of the analytic counterpart of Capparelli’s conjecture, the Bailey pair from Theorem 3.2 was inserted into equation (2.2), which is a limiting case of Bailey’s lemma [5, p. 25, Thm. 3.3]. We now require a different limiting case of Bailey’s lemma: If form a Bailey pair, then
[TABLE]
Now, inserting the Bailey pair from Theorem 3.2 into (4.1), we obtain the identity
[TABLE]
which, for , yields
[TABLE]
Note the extremely simple product on the right hand side of (4.3), which is the generating function for partitions into distinct odd parts.
The analogous identity relative to (1.4) is
[TABLE]
Of course, (4.2), (4.3), and (4.4) could easily be embedded in infinite families of identities via the Bailey chain.
5 Conclusion
While we now have in hand two series representations for the infinite product namely the left hand sides of (1.4) and (1.5), it is not clear exactly how the partitions are generated by them. Such an explanation would be most welcome.
Also, it should be noted that this infinite product has appeared in the literature in at least two other combinatorial contexts beside Capparelli’s conjecture: see Andrews’ Memoir on generalized Frobenius partitions [4, p. 10, equation (5.9)], and Propp’s paper on generalized Ferrers diagrams [25, p. 113, Thm. 4(a)], although in both of these cases the product contained the additional factor . It would be interesting to see a direct connection between Capparelli’s partitions and the combinatorial constructs of Andrews and Propp.
Acknowledgement
I thank George Andrews for suggesting that I look at this problem, and for his encouragement during the project.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] K. Alladi, G.E. Andrews, and B. Gordon, Refinements and generalizations of Capparelli’s conjecture on partitions, J. Algebra 174 (1995), 636–658.
- 2[2] G.E. Andrews, The Theory of Partitions , Encyclopedia of Mathematics and its Applications, vol. 2, Addison-Welsey, 1976. Reissued Cambridge Univ. Press, 1998.
- 3[3] G.E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math 114 (1984), 267–283.
- 4[4] G.E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc. 49 (1984), no. 301.
- 5[5] G.E. Andrews, q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra , CBMS Regional Conference Series in Mathematics, no. 66, Amer. Math. Soc., Providence, 1986.
- 6[6] G.E. Andrews, Schur’s Theorem, Capparelli’s conjecture and q 𝑞 q -trinomial coefficients, in “Proc. Rademacher Centenary Conf., 1992” pp. 141–154, Contemp. Math. 166, Amer. Math. Soc., Providence, 1994.
- 7[7] G.E. Andrews, private communication, August 18, 2003.
- 8[8] W.N. Bailey, Some identities in combinatory analysis, Proc. London Math Soc. (2), 49 (1947), 421–425.
