Ramanujan Identities Research Articles

Topological strings, quiver varieties, and Rogers–Ramanujan identities

Motivated by some recent works on BPS invariants of open strings/knot invariants, we guess there may be a general correspondence between the Ooguri–Vafa invariants of toric Calabi–Yau 3-folds and cohomologies of Nakajima quiver varieties. In this short note, we provide a toy model to explain this correspondence. More precisely, we study the topological open string model of $${\mathbb {C}}^3$$ with one Aganagic–Vafa brane $${\mathcal {D}}_\tau $$ , and we show that, when $$\tau \le 0$$ , its Ooguri–Vafa invariants are given by the Betti numbers of certain quiver variety. Moreover, the existence of Ooguri–Vafa invariants implies an infinite product formula. In particular, we find that the $$\tau =1$$ case of such infinite product formula is closely related to the celebrated Rogers–Ramanujan identities.

Elementary proofs of two theta function identities of Ramanujan and corresponding Lambert series identities

We first present a simple proof of two theta function identities of Ramanujan by employing certain relations satisfied by Ramanujan’s cubic continued fraction. Then we establish an elementary proof of the corresponding Lambert series identities. Finally,

Multiple Lerch Zeta Functions and an Idea of Ramanujan

In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalizing an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.

Asymptotic Behavior of Odd-Even Partitions

Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions.

Garden of Eden partitions in the sand pile and related models

A Garden of Eden state in a dynamical system is one with no preimage. For operations on partitions known as the sand pile model and its generalizations, we characterize and enumerate these Garden of Eden partitions. In addition to motivation from the models themselves, there are also connections to the Rogers–Ramanujan identities and partitions with short sequences studied by Andrews.

Generalized Rogers–Ramanujan identities for twisted affine algebras

The characters of parafermionic conformal field theories are given by the string functions of affine algebras, which are either twisted or untwisted algebras. Expressions for these characters as generalized Rogers–Ramanujan algebras have been established for the untwisted affine algebras. However, we study the identities for the string functions of the twisted affine Lie algebras. A conjecture for the string functions was proposed by Hatayama et al., for the unit fields, which expresses the string functions as Rogers–Ramanujan type sums. Here we propose to check the Hatayama et al. conjecture, using Lie algebraic theoretic methods. We use Freudenthal’s formula, which we computerized, to verify the identities for all the algebras at low rank and low level. We find complete agreement with the conjecture.

A solution to a problem of Frechette and Locus

In a recent paper, Frechette and Locus examined and found expressions for the infinite product Dm(q) := ∏ t=1∞(1 − qmt)∕(1 − qt) in terms of products of q-series of the Rogers–Ramanujan type coming from Hall–Littlewood polynomials, when m ≡ 0,1,2(mod4). These q-series were originally discovered in 2014 by Griffin, Ono, and Warnaar in their work on the framework of the Rogers–Ramanujan identities. Extending this framework, Rains and Warnaar also recently discovered more q-series and their corresponding infinite products. Frechette and Locus left open the case where m ≡ 3(mod4). Here we find such an expression for the infinite products for m ≡ 3(mod4) by making use of the new q-series obtained by Rains and Warnaar.

Born–Infeld solitons, maximal surfaces, and Ramanujan’s identities

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one-parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan's Identities and the Weierstrass-Enneper representation of maximal surfaces, we derive further non-trivial identities.

Overpartitions into distinct parts without short sequences

In the first part of this paper we introduce overpartitions into distinct parts without k-sequences. When k=1 these are the partitions into parts differing by at least two which occur in the Rogers–Ramanujan identities. For general k we compute a three-variable double sum q-hypergeometric generating function and give asymptotic estimates for the number of such overpartitions. When k=2 we obtain several more double sum generating functions as well as a combinatorial identity. In the second part of the paper, we establish arithmetic and combinatorial properties of some related q-hypergeometric double sums.

On a new transformation formula for bilateral $$_6{\psi }_6$$ 6 ψ 6 series and applications

In this paper, we derive a new transformation formula for bilateral \(_6\psi _6\) series using Roger’s \(_6\phi _5\) summation and then use it to deduce the well-known Bailey’s \(_6\psi _6\) summation formula. Also, we deduce some identities involving theta functions and new identities analogous to identities of Ramanujan. Further, we give formulas for finding the number of representations of an integer as sums of squares and sums of triangular numbers.

Ghost series and a motivated proof of the Andrews–Bressoud identities

We present what we call a “motivated proof” of the Andrews–Bressoud partition identities for even moduli. A “motivated proof” of the Rogers–Ramanujan identities was given by G.E. Andrews and R.J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a “motivated proof” of the somewhat analogous Göllnitz–Gordon–Andrews identities has been found. In the present work, we introduce “shelves” of formal series incorporating what we call “ghost series,” which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that these new series will provide insight into the ongoing program of vertex-algebraic categorification of the various “motivated proofs.”

Erratum for Michael J. Griffin, Ken Ono, and S. Ole Warnaar, “A framework of Rogers--Ramanujan identities and their arithmetic properties,” Duke Math. J., Volume 165, Number 8 (2016), 1475--1527

Erratum for Michael J. Griffin, Ken Ono, and S. Ole Warnaar, “A framework of Rogers--Ramanujan identities and their arithmetic properties,” Duke Math. J., Volume 165, Number 8 (2016), 1475--1527

On rational matrix exact covering systems of [formula omitted] and its applications to Ramanujan's forty identities

For a nonsingular integer matrix B, the set of cosets of the quotient module Zn/BZn forms an exact covering system (ECS) of Zn. In this paper, we use the Smith normal form to obtain another type of matrix ECS with rational entries which we call rational matrix ECS. Using rational matrix ECS of Z2, we prove eight identities in Ramanujan's list of forty identities for the Rogers–Ramanujan functions, as well as some other identities.

On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

We shall study the differential equation $${y'^2} = {T_n}\left( y \right) - \left( {1 - 2{\mu ^2}} \right)$$ where μ 2 is a constant, T n (x) are the Chebyshev polynomials with n = 3, 4, 6. The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on 2 F 1(1/4, 3/4; 1; z), 2 F 1(1/3, 2/3; 1; z), 2 F 1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.

Ramanujan's integral identities of the Riemann Ξ-function and the Lerch transcendent

We give simple proofs of Ramanujan's integral identities of the Riemann Ξ-function and derive a general identity involving the Lerch transcendent. A corresponding modular type relation is also derived.

A framework of Rogers–Ramanujan identities and their arithmetic properties

The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{\lambda \lambda_1\leq m}} q^{a|\lambda|} P_{2\lambda}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_{\lambda}(x_1,x_2,\dots;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac--Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of $\textrm{A}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.

Vertex operators and principal subspaces of level one for [formula omitted

We consider two different methods of associating vertex algebraic structures with the level 1 principal subspaces for Uq(slˆ2). In the first approach, we introduce certain commutative operators and study the corresponding vertex algebra and its module. We find combinatorial bases for these objects and show that they coincide with the principal subspace bases found by B.L. Feigin and A.V. Stoyanovsky. In the second approach, we introduce the, so-called nonlocal q_-vertex algebras, investigate their properties and construct the nonlocal q_-vertex algebra and its module, generated by Frenkel–Jing operator and Koyama's operator respectively. By finding the combinatorial bases of their suitably defined subspaces, we establish a connection with the sum sides of the Rogers–Ramanujan identities. Finally, we discuss further applications to quantum quasi-particle relations.

Some identities involving convolutions of Dirichlet characters and the Möbius function

In this paper, we present some identities involving convolutions of Dirichlet characters and the Mobius function, which are related to a well known identity of Ramanujan, Hardy and Littlewood.

Basic series identities and combinatorics

Analogous to P.A. MacMahon’s combinatorial interpretations of the Rogers–Ramanujan identities, we interpret two basic series identities combinatorially in two different ways—using split $$(n+t)$$ -color partitions and the modified lattice paths. This leads to two new 3-way combinatorial identities. We conclude by posing three significant open problems.

On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form

This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective of physics and mathematics. The first part is concerned with (refined) topological string theory and the direct integration of the holomorphic anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as the generating functions of various enumerative invariants, is provided by the so-called propagator. We derive a universal expression for the propagator for geometries that have mirror curves of genus two which is given by the derivative of the logarithm of Igusa's cusp form of weight 10. In addition, we illustrate our findings by solving the refined topological string on the resolutions of the three toric orbifolds of order three, five and six. In the second part, we give explicit expressions for lowering and raising operators on Siegel modular forms, and define almost holomorphic Siegel modular forms based on them. Extending the theory of Fourier-Jacobi expansions to almost holomorphic Siegel modular forms and building up on recent work by Pitale, Saha, and Schmidt, we can show that there is no analogue of the almost holomorphic elliptic second Eisenstein series. In the case of genus 2, we provide an almost meromorphic substitute for it. This, in particular, leads us to a generalization of Ramanujan's differential equation for the second Eisenstein series. The two parts are intertwined by the observation that the meromorphic analogue of the almost holomorphic second Eisenstein series coincides with the physical propagator. In addition, the generalized Ramanujan identities match precisely the physical consistency conditions that need to be imposed on the propagator.