Related papers: Rogers--Ramanujan Type Identities for Rank Two Par…
We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $\Gamma_0(4)$ with Kohnen's plus condition and…
Rogers-Ramanujan type identities occur in various branches of mathematics and physics. As a classic and powerful tool to deal with Rogers-Ramanujan type identities, the theory of Bailey's lemma has been extensively studied and generalized.…
A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Cz\'edli and E.T. Schmidt as the duals of the lattices…
We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $ L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r $ and rational functions…
The Rogers-Ramanujan identities have been studied from the viewpoints of combinatorics, number theory, affine Lie algebras, statistical mechanics, and quantum field theory. This note connects the Rogers-Ramanujan identities with the finite…
We derive exact matrix integral representations for different sums over partitions. The characteristic feature of all obtained matrix models is the presence of logarithmic (or, vice versa, exponential) terms in the potential. Our derivation…
We revisit the solvable lattice models described by Andrews Baxter and Forrester and their generalizations. The expressions for the local state probabilities were shown to be related to characters of the minimal models. We recompute these…
We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then…
We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of…
Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we…
We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of…
While examples of Ramanujan-type congruences are amply available via their relation to Hecke operators, it remains unclear which of them should be considered of combinatorial origin and which of them are mere artifacts of the connection…
We derive Rogers--Ramanujan type partition identities at the fundamental weight $\Lambda_0$ for the exceptional affine types $G_2^{(1)}$, $D_4^{(3)}$, $F_4^{(1)}$, $E_6^{(2)}$, $E_6^{(1)}$, $E_7^{(1)}$ and $E_8^{(1)}$. Our starting point is…
We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator…
The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in\mathbb N_0$) with parts congruent to $\pm 1 \pmod{5}$ (respectively $\pm 2 \pmod{5}$) and the number of partitions of $n$ with super-distinct…
We obtain using exponential quadratic sums, explicit expressions for the number of triple persymmetric matrices over F_2 of given rank. (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s)
The sums $\sum_{j = 0}^k {u_{rj + s}^{2n}z^j }$, $\sum_{j = 0}^k {u_{rj + s}^{2n-1}z^j }$, $\sum_{j = 0}^k {v_{rj + s}^{n}z^j }$ and $\sum_{j = 0}^k {w_{rj + s}^{n}z^j }$ are evaluated; where $n$ is any positive integer, $r$, $s$ and $k$…
Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced…
Let $\mathcal{R}$ be a free Lie conformal algebra of rank $2$ with $\mathbb{C}[\partial]$-basis $\{L,I\}$ and relations \begin{eqnarray*} \left[L_{\lambda} L\right]=(\partial+2 \lambda) (L+I),\ \left[L_{\lambda} I\right]=(\partial+\lambda)…
In a recent paper, Griffin, Ono and Warnaar present a framework for Rogers-Ramanujan type identities using Hall-Littlewood polynomials to arrive at expressions of the form \[\sum_{\lambda : \lambda_1 \leq m}…