Related papers: Crank equidistribution and $(k,j)$-overlined parti…

One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple…

The periodic points of the algebraic function defined by the equation $g(x,y) = x^3(4y^2+2y+1)-y(y^2-y+1)$ are shown to be expressible in terms of Ramanujan's cubic continued fraction $c(\tau)$ with arguments in an imaginary quadratic field…

In 2021, Andrews mentioned that George Beck introduced partition statistics $M_w(r,m,n)$, which denote the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$. Recently, a number of congruences and identities…

We prove that the generating function of partitions into $k$-th powers is strongly Gaussian in the sense of B\'aez-Duarte. Within the probabilistic framework of Khinchin families, the Hardy--Ramanujan asymptotic formula for the…

We discuss the appearance of Jacobi automorphic forms in the theory of superconformal vertex algebras, explaining it by way of supercurves and formal geometry. We touch on related topics such as Ramanujan's differential equations for…

By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8\phi_7$-series. Their duplicate forms under the partition…

We develop an algorithm to compute Fourier expansions of vector valued modular for Weil representations. As an application, we compute explicit linear equivalences of special divisors on modular varieties of orthogonal type. We define three…

Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for…

We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in…

Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(\pi)$, appears $k$ times and every overlined part is bigger than $s(\pi)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote…

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian…

In this paper, we apply high level versions of Jacobi's derivative formula to number theory such as quarternary quadratic forms and convolution sums of some arithmetical functions.

Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space…

Let $p$ be an odd prime. The bar partitions with sign and $p$-bar-core partitions with sign respectively label the spin characters and $p$-defect zero spin characters of the double cover of the symmetric group, and by restriction, those of…

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on the number of $k$-compositions of $n$ satisfying some coprimality conditions. We substantially refine the error term concerning the number of $k$-compositions of…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type…

Amdeberhan and Merca recently studied arithmetic properties of the sequence $a(n)$, the reciprocal of the crank parity function, which counts the number of integer partitions of weight $n$ whose even parts are monochromatic and whose odd…

Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$…