Partially ordered sets and the Rogers-Ramanujan identities
The object is to apply standard combinatorial techniques to partition identities. It is shown that series identities such as \[ \sum_{n=0}^\infty \frac{(-1)^k x^n q^{1/2n(3n+1)}(1-xq^{2n+1})}{\prod_{j=1}^n (1- q^j) \prod{r=n+1}^\infty (1- xq^r)} = 1\quad\text{(Sylvester)} \] and \[ \sum_{n=0}^\infty \frac{(-1)^k x^{2n} q^{1/2n(5n+1)}(1-x^2q^{4n+2})}{\prod_{j=1}^n (1- q^j) \prod_{r=n+1}^\infty (1- xq^r)} = \sum_{n=0}^\infty \frac{x^nq^{n^2}} {\prod_{j=1}^n (1- q^j)} \quad\text{(Rogers\ and\ Ramanujan)} \] allow proofs that rely on classical combinatorial arguments. In fact, Sylvester's identity is derived from a simple application of the inclusion-exclusion principle to the set of parts (not counting multiplicity) of a partition. The same technique allows a complete proof of Gordon's generalization of the Rogers-Ramanujan identities. In considering possible generalizations of our work, it is conjectured that \(W_k(N;m,n)\) defined by \[ 1+ \sum_{n=1}^\infty (aq)_{n-1}(1-aq^{2n})(-1)^n a^{kn} q^{1/2n((2k+1)n-1)} \begin{bmatrix} N \\ n\end{bmatrix} (aq^{N+1})_n^{-1} = \] \[ = (aq)_N \sum_{m,n=0}^\infty W_k(N;m,n)a^nq^n, \] where \((a)_n = (1 - a) (1 - aq) \cdots (1- aq^{n-1})\) and \(\begin{bmatrix} N \\ n\end{bmatrix} = (q)_N(q)_n^{-1} (q)_{N-n}^{-1}\) has a combinatorial interpretation related to the Alder polynomials.
Gordon's generalization of Rogers-Ramanujan identities
scientific article; zbMATH DE number 3467214