Related papers: On Sum-Of-Tails Identities
We provide new proofs to five of Ramanujan's intriguing identities on false theta functions without using the Rogers-Fine identity and Bailey transforms.
In his deathbed letter to Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotics matches the one of a modular form, though a choice of the modular function depends on the root of unity.…
Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of $n$ with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is…
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)…
Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock…
We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Toth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose…
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…
We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k…
The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials…
The G\"ollnitz-Gordon identities were found by G\"ollnitz and Gordon independently. In 1967, Andrews obtained a combinatorial generalization of the G\"ollnitz-Gordon identities, called the Andrews-G\"ollnitz-Gordon theorem. In 1980,…
We extend the Reed Dawson identity for Knuth's old sum with a complex parameter, and we offer two separate hypergeometric series-based proofs of this generalization, and we apply this generalization to introduce binomial-harmonic sum…
In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $$\sum_{n\ge 0} \frac{(-1)^n q^{3\binom{n}{2}+4n}(q;q^3)_n}{(q^9;q^9)_n} =…
Around 2007, Warnaar proved four identities related to Nahm sums associated with twice the inverse of the Cartan matrix of type $D_k$. Three of these had been conjectured by Flohr, Grabow, and Koehn, while special cases of two of the…
Recently Gordon and McIntosh introduced the third order mock theta function $\xi(q)$ defined by $$ \xi(q)=1+2\sum_{n=1}^{\infty}\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. $$ Our goal in this paper is to study arithmetic properties of…
We present other proofs, generalizations and analogues of the identities concerning multiple Dirichlet series by Tahmi and Derbal (2022). As applications, we obtain asymptotic formulas with remainder terms for certain related sums.
We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoglu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions.…
By considering even functions (mod $n$) we generalize a Menon-type identity by Li and Kim involving additive characters of the group ${\Bbb Z}_n$. We use a different approach, based on certain convolutional identities. Some other…
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our…
We propose a method to construct a variety of partition identities at once. The main application is an all-moduli generalization of some of Andrews' results in [5]. The novelty is that the method constructs solutions to functional equations…
In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned…