Finite Rogers-Ramanujan Type Identities

Polynomial generalizations of all 130 of the identities in Slater’s list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new. The author has implemented much of the finitization process in a Maple package which is available for free download from the author’s website. 0 Introduction 0.1 Three approaches to finitization There are at least three avenues of approach that lead to finite Rogers-Ramanujan type identities. ∗The research contained herein comprises a substantial portion of the author’s doctoral dissertation, submitted in partial fulfillment of the requirements for the Ph.D. degree at the University of Kentucky. The doctoral dissertation was completed under the supervision of George E. Andrews, Evan Pugh Professor of Mathematics at the Pennsylvania State University. This research was partially supported by a grant provided to the author by Professor Andrews. the electronic journal of combinatorics 10 (2003), #R13 1 1. Combinatorics and models from statistical mechanics. This approach has been studied extensively by Andrews, Baxter, Berkovich, Forrester, McCoy, Schilling, Warnaar and others; see, e.g., [7], [15], [16], [18], [17], [27], [30], [31], [36], [63], [70], [71], [72]. 2. The Strong Bailey Lemma. This method is discussed in chapter 3 of Andrews’ q-series monograph [10]. 3. The method of nonhomogeneous q-difference equations. This method is introduced in [10, Chapter 9] and studied extensively herein. While these three methods sometimes lead to similar results, often the results are different. Even in the cases where the different methods lead to the same finitization, each method has its own inherent interest. For instance, from the statistical mechanics point of view, finitization makes it possible to consider q → q−1 duality, which in the case of Baxter’s hard hexagon model, allows one to neatly pass from one regime to another [7]. Finitizations arising as a result of the application of the strong form of Bailey’s Lemma give rise to important questions in computer algebra as in Paule ([52] and [53]). Finally, the method of q-difference equations has been studied combinatorially in [9]. It is this method that will be studied in depth in this present work. Granting the intrinsic merit of all of these approaches, a particularly interesting aspect of the third method stems from the fact that there is no known overarching theory which guarantees a given attempt at finitization will be successful. The fact that all of Slater’s list succumbed to this method is evidence in favor of the existence of such an overarching theory. Let us now begin to study this third method in detail. 0.2 Overview of this work In his monograph on q-series [10, Chapter 9], Andrews indicated a method (referred to herein as the “method of first order nonhomogeneous q-difference equations,” or more briefly as the “method of q-difference equations”) to produce sequences of polynomials which converge to the Rogers-Ramanujan identities and identities of similar type. By appropriate application of the q-binomial theorem, formulas for the polynomials can easily be produced for what the physicists call “fermionic representations” of the polynomials. The identities explored in [7] and [10] relate to Baxter’s solution of the hard hexagon model in statistical mechanics [25]. In [16], Andrews and Baxter suggest some ideas for how a computer algebra system can be employed to find what the physicists call “bosonic representations” of polynomials which converge to Rogers-Ramanujan type products. When we have both a fermionic and bosonic representation of a polynomial sequence which converges to a series-product identity, the series-product identity is said to have been finitized. In his Ph.D. thesis [60], Santos conjectured bosonic (but no fermionic) representations for polynomial sequences which converge to many of the identities in Lucy Slater’s paper on Rogers-Ramanujan Type Identities [68]. the electronic journal of combinatorics 10 (2003), #R13 2 This present work extends and unifies the results found in [7], [10, Chapter 9], [16] and [60]. Background material is presented in §1. In §2, it is proved that the method of q-difference equations can be used to algorithmically produce polynomial generalizations of Rogers-Ramanujan type series, and find fermionic representations of them. As in [16] and [60], bosonic representations need to be conjectured, but the methods and computer algebra tools discussed in §2 indicate how appropriate conjectures can be found efficiently. In §3, at least one finitization is presented for each of the 130 identities in Slater’s list [68]. In the case of some of the simpler identities in Slater’s list, the finitization found corresponds to a previously known polynomial identity, but in many of the cases, the identities found are new. Considerable care was taken to provide appropriate references for the previously known, and previously conjectured identities or pieces of identities. In each case, the bosonic representations can best be understood in terms of either Gaussian polynomials or q-trinomial coëfficients [15]. Particularly noteworthy is the discovery that bosonic representations of a number of the finitized Slater identities used a weighted combination of two different q-trinomial coëfficients, referred to herein as V(L,A; q) (see (1.23)). It turns out that this “V ” function enters naturally into the theory of q-trinomial coëfficients due to certain internal symmetries of the T0 and T1 q-trinomial coëfficients (1.33), although its existence had previously gone unnoticed. Section 4 contains a discussion of various methods for proving the polynomial identities conjectured by the method of q-difference equations. Particular emphasis is placed upon the algorithmic proof theory of Wilf and Zeilberger ([55], [76], [77], [78], [79], [80]). It is to be noted that the author has proved every identity in §3 using the “method of recurrence proof” discussed in Section 4, including the 1991 Santos conjectures, as well as new polynomial identities. Thus, all of the identities in Slater’s list [68] may now be viewed as corollaries of the polynomial identities presented in §3. Once a series-product identity is finitized, a q → q−1 duality theory can be discussed. In [7], Andrews describes the duality between various identities associated with the four regimes of the hard hexagon model. An extensive study of the duality relationships among the identities presented in §3 is undertaken in §5. A number of previously unknown multisum identities arise as a result of this duality study. In §6, a relaxed version of the finitization method of §2 is considered wherein we drop the requirement that the two-variable generalization of the Rogers-Ramanujan type series satisfy a first order nonhomogeneous q-difference equation. It is then demonstrated that this method can be used to find several identities due to Bressoud [32], as well as to find additional new finitizations of Rogers-Ramanujan type identities, at least one of which arises in the work of Warnaar [72]. Finally, the appendix is an annotated and cross-referenced version of Slater’s list of identities from [68]. Since Slater’s list of identities has been the source for further research for many mathematicians, my hope is that others will find this version of Slater’s list useful. the electronic journal of combinatorics 10 (2003), #R13 3 1 Background Material 1.1 q-Binomial coëfficients We define the infinite rising q-factorial (a; q)∞ as follows: (a; q)∞ := ∞ ∏