A simple proof of the Rogers–Selberg identities

A short and simple proof of the Rogers–Selberg identities is obtained from the work of Andrew Sills on identities of the Rogers–Ramanujan type.     

A multilateral Bailey lemma and multiple Andrews–Gordon identities

A multilateral Bailey lemma is proved, and multiple analogues of the Rogers–Ramanujan identities and Euler’s pentagonal theorem are constructed as applications. The extreme cases of the Andrews–Gordon identities are also generalized using the multilateral Bailey lemma where their final form are written in terms of determinants of theta functions.     

On the Andrews–Schur proof of the Rogers–Ramanujan identities

In this article, we study one of Andrews’ proofs of the Rogers–Ramanujan identities published in 1970. His proof inspires connections to some famous formulas discovered by Ramanujan. During the course of study, we discovered identities such as $$\sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{\sqrt{5}}\Biggl(\beta \prod_{n=1}^{\infty}\frac{1}{1+\alpha q^{n/5}+q^{2n/5}}-\alpha \prod_{n=1}^{\infty}\frac{1}{1+\beta q^{n/5}+q^{2n/5}}\Biggr),$$ where β=?1/α is the Golden Ratio.     

Euler’s partition theorem for all moduli and new companions to Rogers–Ramanujan–Andrews–Gordon identities

The Ramanujan Journal - We generalise Euler’s partition theorem involving odd parts and distinct parts for all moduli and provide new companions to...     

A motivated proof of the Göllnitz–Gordon–Andrews identities

We present what we call a “motivated proof” of the Göllnitz–Gordon–Andrews identities. A similar motivated proof of the Rogers–Ramanujan identities was previously given by G. E. Andrews and R. J. Baxter, and was subsequently generalized to Gordon’s identities by J. Lepowsky and M. Zhu. We anticipate that the present proof of the Göllnitz–Gordon–Andrews identities will illuminate certain twisted vertex-algebraic constructions.     

Arc spaces and the Rogers–Ramanujan identities

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations, we obtain a new approach to the classical Rogers–Ramanujan Identities. The linking object is the Hilbert–Poincaré series of the arc space over a point of the base variety. In the case of the double point, this is precisely the generating series for the integer partitions without equal or consecutive parts.     

Shift-plethystic trees and Rogers–Ramanujan identities

The Ramanujan Journal - By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the...     

Topological strings,quiver varieties,and Rogers–Ramanujan identities

The Ramanujan Journal - 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...     

Bressoud polynomials,Rogers–Ramanujan type identities,and applications

In his paper providing an easy proof of the Rogers–Ramanujan identities, D. Bressoud extended his work to multiple series identities. Intrinsic in his works are polynomials with diverse applications to several aspects of \(q\)-series. This paper provides an initial exploration of these polynomials.     

Polygonal numbers and Rogers–Ramanujan–Gordon theorem

The Ramanujan Journal - Let $$B_{k,r}(n)$$ be the number of partitions of the form $$n=b_1+b_2+cdots +b_s$$ , where $$b_i-b_{i+k-1}hbox {,char 062,}2$$ and at most $$r-1$$ of the $$b_i$$ are...     

Harmonic number identities via the Newton–Andrews method

By means of the Bell polynomials, we establish explicit expressions of the higher-order derivatives of the binomial coefficient \(\binom{x+n}{m}\) and its reciprocal \(\binom{x+n}{m}^{-1}\) , and extend the application field of the Newton–Andrews method. As examples, we apply the results to the Chu–Vandermonde–Gauss formula and the Dougall–Dixon theorem and obtain a series of harmonic number identities. This paper generalizes some works presented before and provides a way to establish infinite harmonic number identities.     

A variation of the Andrews–Stanley partition function and two interesting q-series identities

The Ramanujan Journal - In this paper we present a new formula for the number of unrestricted partitions of n. We do this by introducing a correspondence between the number of unrestrited...     

The Central value of the Rankin–Selberg L-Functions

Let f be a Maass form for SL which is fixed and u _j be an orthonormal basis of even Maass forms for SL , we prove an asymptotic formula for the average of the product of the Rankin–Selberg L-function of f and u _j and the L-function of u _j at the central value 1/2. This implies simultaneous nonvanishing results of these L-functions at 1/2. Received: November 2006, Revision: March 2007, Accepted: March 2007     

Proofs of some conjectures on the reciprocals of the Ramanujan–Gordon identities

The Ramanujan Journal - Recently, Lin and Wang introduced two special partition functions $$RG_1(n)$$ and $$RG_2(n)$$ , the generating functions of which are the reciprocals of two identities due...     

Nonlinearity inH ∞-control theory,causality in the commutant lifting theorem,and extension of intertwining operators

The problems studied in this note have been motivated by our work in generalizing linearH control theory to nonlinear systems. These ideas have led to a design procedure applicable to analytic nonlinear plants. Our technique is a generalization of the linearH theory. In contrast to previous work on this topic ([9], [10]), we now are able to explicitly incorporate a causality constraint into the theory. In fact, we show that it is possible to reduce a causal optimal design problem (for nonlinear systems) to a classical interpolation problem solvable by the commutant lifting theorem [8]. Here we present the complete operator theoretical background of our research together with a short control theoretical motivation.This work was supported in part by grants from the Research Fund of Indiana University, the National Science Foundation DMS-8811084 and ECS-9122106, by the Air Force Office of Scientific Research F49620-94-1-0098DEF, and by the Army Research Office DAAL03-91-G-0019 and DAAH04-93-G-0332     

Modular equations for cubes of the Rogers–Ramanujan and Ramanujan–Göllnitz–Gordon functions and their associated continued fractions

In this paper, we prove modular identities involving cubes of the Rogers–Ramanujan functions. Applications are given to proving relations for the Rogers–Ramanujan continued fraction. Some of our identities are new. We establish analogous results for the Ramanujan–Göllnitz–Gordon functions and the Ramanujan–Göllnitz–Gordon continued fraction. Finally, we offer applications to the theory of partitions.     

The Eigenstructure of Operators Linking the Bernstein and the Genuine Bernstein–Durrmeyer operators

We study the eigenstructure of a one-parameter class of operators ${U_{n}^{\varrho}}$ of Bernstein–Durrmeyer type that preserve linear functions and constitute a link between the so-called genuine Bernstein–Durrmeyer operators U _n and the classical Bernstein operators B _n . In particular, for ${\varrho\rightarrow\infty}$ (respectively, ${\varrho=1}$ ) we recapture results well-known in the literature, concerning the eigenstructure of B _n (respectively, U _n ). The last section is devoted to applications involving the iterates of ${U_{n}^{\varrho}}$ .