Related papers: On "mixed" modular equations of degree 21

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…

It was recently shown that $q\omega(q)$, where $\omega(q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less…

We study and classify faithfully balanced modules for the algebra of lower triangular $n$ by $n$ matrices. The theory extends known results about tilting modules, which are classified by binary trees, and counted with the Catalan numbers.…

In this paper, we derive eight basic identities of symmetry in three variables related to $q$-Euler polynomials and the $q$-analogue of alternating power sums. These and most of their corollaries are new, since there have been results only…

We give a comprehensive treatment on how $F$-signatures, splitting primes, splitting ratios, and test modules behave under finite covers. To this end, we expand on the notion of transposability along a section section of the relative…

We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a…

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…

In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular,…

We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.

We introduce and study twelve multivariable theta functions defined by pfaffians with elliptic function entries. We show that, when the crossing parameter is a cubic root of unity, the domain wall partition function for the…

In this article, we study the multiple zeta functions (MZF) and some of its variants at identical arguments. Using the harmonic product, these functions can be expressed as polynomials in the Riemann zeta function. Firstly, we note that an…

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes…

We give an algebraic analog of the functional equation of Riemann's theta function. More precisely, we define a `theta multiplier' line bundle over the moduli stack of principally polarized abelian schemes with theta characteristic and…

In recent years, the generalized sum-of-divisor functions of MacMahon have been unified into the algebraic framework of $q$-multiple zeta values. In particular, these results link partition theory, quasimodular forms, $q$-multiple zeta…

The addition relation for the Riemann theta functions and for its limits, which lead to the appearance of exponential functions in soliton type equations is discussed. The presented form of addition property resolves itself to the…

This thesis studies modular forms from a classical and adelic viewpoint. We use this interplay to obtain results about the arithmetic of the Fourier coefficients of modular forms and their generalisations. In Chapter 2, we compute lower…

We deduce $q$-continued fractions $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$ of order fourteen, and continued fractions $V_{1}(q)$, $V_{2}(q)$ and $V_{3}(q)$ of order twenty-eight from a general continued fraction identity of Ramanujan. We…

It is known, that under the assumption of the generalized Riemannian hypothesis, the function $\pi(x, q, 1) - \pi(x, q, a)$ has infinitely many sign changes. In this article we give an upper bound for the least such sign change. Similarly,…

We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for…

In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we…