Related papers: Ramanujan-type identities for Shimura curves
The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks…
We extend to the category of relative regular holonomic modules on a manifold $X$, parametrized by a curve $S$, the Hermitian duality functor (or conjugation functor) of Kashiwara. We prove that this functor is an equivalence with the…
The Rogers-Ramanujan identities and various analogous identities (Gordon, Andrews-Bressoud, Capparelli, etc.) form a family of very deep identities concerned with integer partitions. These identities (written in generating function form)…
We prove that the ancestor Gromov-Witten correlation functions of one-dimensional compact Calabi-Yau orbifolds are quasi-modular forms. This includes the pillowcase orbifold which can not yet be handled by using Milanov-Ruan's B-model…
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are…
In this paper, we show that the classical Cassini and Catalan identities for Fibonacci numbers arise naturally from a single quadratic theta-function identity of Ramanujan. Expanding the identity…
The generating functions of stationary descendent Gromov-Witten invariants of an elliptic curve are known to be Fourier expansions of quasimodular forms. When one restricts to the subspace of forms of a fixed weight $k$, there is an…
We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd…
We derive 10 new Ramanujan-Sato series of $1/\pi$ by using the method of Huber, Schultz and Ye. The levels of these series are 14, 15, 16, 20, 21, 22, 26, 35, 39.
Infinite series are evaluated through the manipulation of a series for $\cos(2t \sin^{-1}x)$ resulting from Clausen's Product. Hypergeometric series equal to an expression involving $\frac{1} {\pi}$ are determined. Techniques to evaluate…
We prove two partition identities which are dual to the Rogers-Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions),…
Ramanujan presented four identities for third order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. provided different proofs by…
Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for $1/\pi$. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by…
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…
Characters and linear combinations of characters that admit a fermionic sum representation as well as a factorized form are considered for some minimal Virasoro models. As a consequence, various Rogers-Ramanujan type identities are…
Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we…
Recently, a construction of minimal codes arising from a family of almost Ramanujan graphs was shown. Ramanujan graphs are examples of expander graphs that minimize the second-largest eigenvalue of their adjacency matrix. We call such…
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
We define and study multi-colored dimer models on a segment and on a circle. The multivariate generating functions for the dimer models satisfy the recurrence relations similar to the one for Fibonacci numbers. We give closed formulae for…
Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a…