Rogers-Ramanujan Identities Research Articles

New Proofs of the Septic Rogers-Ramanujan Identities

New Proofs of the Septic Rogers-Ramanujan Identities

A New (But Very Nearly Old) Proof of the Rogers-Ramanujan Identities

We present a new proof of the Rogers-Ramanujan identities. Surprisingly, all its ingredients are available already in Rogers seminal paper from 1894, where he gave a considerably more complicated proof.

Legendre Theorems for a Class of Partitions with Initial Repetitions

AbstractPartitions with initial repetitions were introduced by George Andrews. We consider a subclass of these partitions and find Legendre theorems associated with their respective partition functions. The results in turn provide partition-theoretic interpretations of some Rogers–Ramanujan identities due to Lucy J. Slater.

Some 𝑞-identities derived by the ordinary derivative operator

In this paper, we investigate applications of the ordinary derivative operator, instead of the q q -derivative operator, to the theory of q q -series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the q q -binomial theorem, Ramanujan’s 1 ψ 1 {}_1\psi _1 formula, the quintuple product identity, Gasper’s q q -Clausen product formula, and Rogers’ 6 ϕ 5 {}_6\phi _5 formula, etc. Among these results is a finite form of the Rogers-Ramanujan identity and a short way to Eisenstein’s theorem on Lambert series.

Some separable integer partition classes

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus [Formula: see text]. We also extend separable integer partition classes with modulus [Formula: see text] to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.

Generalized Alder-Type Partition Inequalities

In 2020, Kang and Park conjectured a "level $2$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $3$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $n$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities.

A proof of the second Rogers-Ramanujan identity via Kleshchev multipartitions

A proof of the second Rogers-Ramanujan identity via Kleshchev multipartitions

Neighborly partitions and the numerators of Rogers–Ramanujan identities

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), monomial ideals and some infinite graphs.

A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities

A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities

An Invitation to Formal Power Series

This is an account on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton’s binomial theorem, Jacobi’s triple product, the Rogers–Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan’s partition congruences, generating functions of Stirling numbers and Jacobi’s four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahon’s master theorem.

Three-way Combinatorial Interpretations of Rogers–Ramanujan Identities

Combinatorial interpretations of the Rogers–Ramanujan identities are provided in terms of n–color partitions. Further interpretations in terms of ordinary partitions are obtained by using bijective maps. These results lead to the interpretations of two fifth order mock theta functions by attaching weights.

Overpartition analogues for the generalized Rogers–Ramanujan identities of Bressoud

In the past two decades, there have been a lot of research centred around overpartitions, some of which concern overpartition analogues of Rogers–Ramanujan type identities. In this paper, we present Rogers–Ramanujan type overpartition identities by considering Bressoud’s even moduli generalization of the Rogers–Ramanujan identity and its overpartition analogue of Chen, Sang and Shi given in 2015. We first introduce another overpartition function C¯k,a(n) and show that C¯k,a(n) equals the overpartition function B¯k,a,0(n) of Chen, Sang and Shi. Next, we study parity constrains on parts of overpartitions. Recently, Sang, Shi and Yee obtained Rogers–Ramanujan type identities for overpartitions by adding some parity constraints to even or odd parts of overpartitions Chen, Sang and Shi introduced in 2013. We make some modifications and add constraints to even or odd parts of overpartitions counted by C¯k,a(n) and B¯k,a,0(n) obtaining further Rogers–Ramanujan type overpartition identities.

Revisiting a Classic Identity That Implies the Rogers–Ramanujan Identities II

We give a new proof of an identity due to Ramanujan. From this identity, he deduced the famous Rogers–Ramanujan identities. We prove this identity by establishing a simple recursion Jk=qkJk−1, where |q|<1. This is a sequel to our recent work.

Even–odd partition identities of Rogers–Ramanujan type

We prove a theorem which adds a new companion to the Rogers–Ramanujan identities. This new companion counts partitions with different type of constraints on even and odd parts. Generalizing this theorem, we obtain two families of partition identities.

On double sum generating functions in connection with some classical partition theorems

We focus on writing double sum representations of the generating functions for the number of partitions satisfying some gap conditions. Some example sets of partitions to be considered are partitions into distinct parts and partitions that satisfy the gap conditions of the Rogers–Ramanujan, Göllnitz–Gordon, and little Göllnitz theorems. We refine our representations by imposing a bound on the largest part and find finite analogues of these new representations. These refinements lead to many q-series and polynomial identities. Additionally, we present a different construction and a double sum representation for the products similar to the ones that appear in the Rogers–Ramanujan identities.

Applications of Generalized q-Difference Equations for General q-Polynomials

Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(∞,−1,1,q) is the generation of the fifth-order mock theta functions. In the present investigation, several interesting types of generating functions for this q-polynomial using q-difference equations is deduced. Besides that, a generalization of Andrew’s result in form of a multilinear generating function for q-polynomials is also given. Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities. We observe that most of the parameters involved in our results are symmetric to each other. Our results are shown to be connected with several earlier works related to the field of our present investigation.

Bounds for d-distinct partitions

Euler's identity and the Rogers-Ramanujan identities are perhaps the most famous results in the theory of partitions. According to them, 1-distinct and 2-distinct partitions of n are equinumerous with partitions of n into parts congruent to ±1 modulo 4 and partitions of n into parts congruent to ±1 modulo 5, respectively. Furthermore, their generating functions are modular functions up to multiplication by rational powers of q. For d ≥ 3, however, there is neither the same type of partition identity nor modularity for d-distinct partitions. Instead, there are partition inequalities and mock modularity related with d-distinct partitions. For example, the Alder-Andrews Theorem states that the number of d-distinct partitions of n is greater than or equal to the number of partitions of n into parts which are congruent to ±1 (mod d+3). In this note, we present the recent developments of generalizations and analogs of the Alder-Andrews Theorem and establish asymptotic lower and upper bounds for the d-distinct partitions. Using the asymptotic relations and data obtained from computation, we propose a conjecture on a partition inequality that gives an upper bound for d-distinct partitions. Specifically, for d ≥ 4, the number of d-distinct partitions of n is less than or equal to the number of partitions of n into parts congruent to ±1 (mod m), where m ≤ 2dπ^2 / [3 log^2 (d)+6 log d] .

Looking for a New Version of Gordon’s Identities

We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in "Gordon's identities", which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals we conjecture a family of partition identities which extend Gordon's identities. This family is indexed by r >=2. We prove the conjecture for r=2 and r=3.

Cylindric partitions and some new $A_2$ Rogers–Ramanujan identities

We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo $8$ with quadruple sums, related with work of Andrews, Schilling, and Warnaar.

Bounded Littlewood identities

We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R,S) in terms Macdonald polynomials of type A, are q,t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups, important in the theory of plane partitions. As applications of our results we obtain combinatorial formulas for characters of affine Lie algebras, Rogers-Ramanujan identities for such algebras complementing recent results of Griffin et al., and transformation formulas for Kaneko-Macdonald-type hypergeometric series.