Combinatorial proofs of q-series identities

We provide combinatorial proofs of some of the q-series identities considered by Andrews, Jimenez-Urroz and Ono [q-series identities and values of certain $L$-functions. Duke Math. J. 108 (2001), no. 3, 395--419].Comment: 14 pages. Submitted to Journal of Combinatorial Theory

A tiling approach to eight identities of Rogers

AbstractBeginning in 1893, L.J. Rogers produced a collection of papers in which he considered series expansions of infinite products. Over the years, his identities have been given a variety of partition-theoretic interpretations and proofs. These existing combinatorial techniques, however, do not highlight the similarities and the subtle differences seen in so many of these remarkable identities. It is the goal of this paper to present a new combinatorial approach that unifies numerous q-series identities. The eight identities of Rogers that appear in G.E. Andrews’ 1986 CBMS monograph on q-series will serve as a basis for the collection of identities studied in this paper

Combinatorial aspects of the theory of q-series

This thesis is concerned mainly with the interplay between identities involving power series (which are called q-series) and combinatorics, in particular the theory of partitions. The thesis includes new proofs of some q-series identities and some ideas about the generating functions for the rank and crank, a new proof of the triple product identity and a combinatorial proof of a q-elliptic identity

Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities

We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized $k$-product $q$-series identity is also obtained by employing the `tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired $q$-series identity. Several new recursive $q$-series identities were also established. The `tiling-method' holds promise for giving an aesthetically pleasing approach to prove old and new $q$-series identities.Comment: 24 page

In Praise of an Elementary Identity of Euler

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.Comment: Published versio

Split (n + t)-color partitions and 2-color F-partitions

Andrews [Generalized Frobenius partitions. Memoirs of the American Math. Soc., 301:1{44, 1984] defined the two classes of generalized F-partitions: F-partitions and k-color F-partitions. For many q-series and Rogers-Ramanujan type identities, the bijections are established between F-partitions and (n + t)-color partitions. Recently (n + t)-color partitions have been extended to split (n+t)-color partitions by Agarwal and Sood [Split (n+t)-color partitions and Gordon-McIntosh eight order mock theta functions. Electron. J. Comb., 21(2):#P2.46, 2014]. The purpose of this paper is to study the k-color F-partitions as a combinatorial tool. The paper includes combinatorial proofs and bijections between split (n + t)-color partitions and 2-color F-partitions for some generalized q-series. Our results further give rise to innate three-way combinatorial identities in conjunction with some Rogers-Ramanujan type identities for some particular cases

Hall-Littlewood polynomials and characters of affine Lie algebras

The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall-Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras C_n^{(1)}, A_{2n}^{(2)} and D_{n+1}^{(2)}. Through specialisation this yields generalisations for B_n^{(1)}, C_n^{(1)}, A_{2n-1}^{(2)}, A_{2n}^{(2)} and D_{n+1}^{(2)} of Macdonald's identities for powers of the Dedekind eta-function. These generalised eta-function identities include the Rogers-Ramanujan, Andrews-Gordon and G\"ollnitz-Gordon q-series as special, low-rank cases.Comment: 33 pages, proofs of several conjectures from the earlier version have been include

An open question of Corteel, Lovejoy and Mallet

Corteel, Lovejoy and Mallet concluded their paper \An extension to overpartitions of the Rogers-Ramanujan identities for even moduli" with an open question of investigating the combinatorial properties of a q-series with two additional parameters.We settle their question, unfortunately in the negative, by showing that the series yields only the known results in overpartitions. However; when one annihilates one of the parameters, the resulting series have nice integer partitions interpretations. Those series appeared in another publication as well. In particular, Corteel, Lovejoy, and Mallet's series involve an index d. This index unifies two classes of overpartition identities for d = 1 and d = 2, but does not give additional overpartition identities for d => 3. Upon setting one of the parameters zero, one does get regular partition identities for all d. The proofs are conventional, formal verifications for brevity, but we show how to make the proofs constructive

Double series representations for Schur's partition function and related identities

We prove new double summation hypergeometric $q$-series representations for several families of partitions, including those that appear in the famous product identities of G\"ollnitz, Gordon, and Schur. We give several different proofs for our results, using bijective partitions mappings and modular diagrams, the theory of $q$-difference equations and recurrences, and the theories of summation and transformation for $q$-series. We also consider a general family of similar double series and highlight a number of other interesting special cases.Comment: 19 page