Related papers: An Explicit Cohen-Style Threefield Identity

It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal…

A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of…

We present q-series proofs of four identities involving sixth order mock theta functions from Ramanujan's lost notebook. We also show how Ramanujan's identities can be used to give a quick proof of four sixth order identities of Berndt and…

Quite recently, the first author investigated vanishing coefficients of the arithmetic progressions in several $q$-series expansions. In this paper, we further study the signs of coefficients in two $q$-series expansions and establish some…

Andrews, Dyson, and Hickerson showed that 2 $q$-hypergeometric series, going back to Ramanujan, are related to real quadratic fields, which explains interesting properties of their Fourier coefficients. There is also an interesting relation…

Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, with $d$ a cube-free positive integer. Let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. By the aid of genus theory, arithmetic proprieties of the pure cubic…

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 identities. Recently, Andrews et al. provided different proofs by…

Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we…

In 1988, Andrews, Dyson and Hickerson initiated the study of q-hypergeometric series whose coefficients are dictated by the arithmetic in real quadratic fields. In this paper, we provide a dozen q-hypergeometric double sums which are…

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible…

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are…

Let $C_n(q)$ be the number of ideals of codimension $n$ of $\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$, where $\mathbb{F}_q$ is the finite field with $q$ elements. Kassel and Reutenauer [KasselReutenauer2015A] proved that $C_n(q)$ is a…

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in…

The notion of $\theta$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $\theta$-congruent if it is the area of a rational triangle with an angle $\theta$ whose cosine is rational.…

Andrews, Dyson and Hickerson proved many interesting properties of coefficients for a Ramanujan's $q$-hypergeometric series by relating it to real quadratic field $\Q(\sqrt{6})$ and using the arithmetic of $\Q(\sqrt{6})$, hence solved a…

In 2018 Celikbas, Laxmi, Kra\'skiewicz, and Weyman exhibited an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. All previously known…

In 1986, Andrews studied the function $\sigma(q)$ from Ramanujan's ``Lost" Notebook, and made several conjectures on its Fourier coefficients $S(n)$, which count certain partition ranks. In 1988, Andrews-Dyson-Hickerson famously resolved…

Very recently, Issa and Darrag [Arch. Math. (Basel) 123 (2024), no. 4, 379-383] determined partial Dedekind zeta values for certain ideal classes in the real quadratic fields of the form $\mathbb{Q}(\sqrt{9m^2+2m})$, where $9m^2+2m$ is…

An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the…

We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the…