Partition identities arising from theta function identities

The authors show that certain theta function identities of Schroeter and Ramanujan imply elegant partition identities.     

On three identities of Ramanujan

The Ramanujan Journal - In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to...     

Four identities related to third-order mock theta functions

The Ramanujan Journal - 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...     

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...     

Some theta function identities related to the Rogers-Ramanujan continued fraction

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction . In order to prove some of these claims, G. N. Watson established two important theorems about that he found in Ramanujan's notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

     

Two families of constant term identities

The Ramanujan Journal - In 1985, Bressoud and Goulden derived the formula for the constant term in $$\prod _{(i,j)\in T} \frac{x_j}{x_i} \prod _{0\le i     

Partition identities and Ramanujan's modular equations

We show that certain modular equations and theta function identities of Ramanujan imply elegant partition identities. Several of the identities are for t-cores.     

The dual form of Beck type identities

The Ramanujan Journal - Inspired by Andrews and Merca’s recent work on the number of even parts over all partitions into distinct parts, we introduce a new kind of Beck type identities, which...     

Farkas’ identities with quartic characters

The Ramanujan Journal - Farkas in (On an Arithmetical Function II. Complex Analysis and Dynamical Systems II. Contemporary Mathematics, American Mathematical Society, Providence, 2005) introduced...     

The space of convolution identities on divisor functions

The Ramanujan Journal - We consider convolutions of divisor functions in arbitrary length with modular congruence restrictions, and introduce the notion of a space of convolution identities over...     

Ramanujan type identities and congruences for partition pairs

Using elementary methods, we establish several new Ramanujan type identities and congruences for certain pairs of partition functions.     

Some identities involving Inverse Gamma integrals

The Ramanujan Journal - In this paper we will give some identities related with the Fransén–Robinson constant and the Inverse Gamma function. The main result is to use Riemann...     

Rogers–Ramanujan type identities via Abel’s lemma on summation by parts

The Ramanujan Journal - The Abel’s lemma on summation by parts is employed to review identities of Rogers–Ramanujan type. Twenty examples are illustrated including several new RR...     

Ramanujan-type partial theta identities and conjugate Bailey pairs

Residual identities of Ramanujan-type partial theta identities are tailor-made for producing conjugate Bailey pairs. This is carried out for partial theta identities in Ramanujan??s lost notebook, a number of the partial theta identities of Warnaar, and for some new ones as well.     

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.     

New theta constant identities

The residue theorem is employed to obtain new identities amongpthe powers of theta constants with rational characteristics. The technique is then used to derive some known identities of Ramanujan. Dedicated to John Thompson for his many original contributions to mathematics Research by HMF partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Some identities for Carlitz type q-Daehee polynomials

The Ramanujan Journal - Motivated by Carlitz’s q-Bernoulli polynomials, properties and identities are given for a new type of polynomial, Carlitz type q-Daehee polynomials. Higher order...     

Hypergeometric identities for 10 extended Ramanujan-type series

We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter a when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.      

Vanishing coefficients and identities concerning Ramanujan’s parameters

The Ramanujan Journal - In this paper we investigate several infinite products with vanishing Taylor coefficients in arithmetic progressions. These infinite products are closely related to...     

m-Dissections of some infinite products and related identities

The Ramanujan Journal - For integers t and m with $$m\ge 5$$ relatively prime to 6 such that $$1 \le t n_0$$ , the signs of the $$c_n$$ are periodic with period m. In addition, a formula is given...