相关论文: On the infinite Borwein product raised to a positi…

In this paper we give a geometric condition which ensures that $(q,p)$-Poincar\'e-Sobolev inequalities are implied from generalized $(1,1)$-Poincar\'e inequalities related to $L^1$ norms in the context of product spaces. The concept of…

We obtain bounds on the number of triples that determine a given pair of dot products arising in a vector space over a finite field or a module over the set of integers modulo a power of a prime. More precisely, given $E\subset \mathbb…

We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it…

The paper is concerned with the Kontsevich-Zagier formal power series $$ f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n) $$ and its analytic properties. To begin with, we give an explicit formula for the Borel transform of the associated formal…

In this paper, we first show that the irreducible characters of a quotient table algebra modulo a normal closed subset can be viewed as the irreducible characters of the table algebra itself. Furthermore, we define the character products…

In the study of theta series and partition functions, Andrews and Merca, Guo and Zeng independently conjectured that a truncated Jacobi triple product series has nonnegative coefficients. This conjecture was proved analytically by Mao and…

Let $g \geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to %the product $E_1 \times \ldots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\ldots$, $E_g/\mathbb{Q}$, without complex…

We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of…

Borcherds lift for an even lattice of signature (p,q) is a lifting from weakly holomorphic modular forms of weight (p-q)/2 for the Weil representation. We introduce a new product operation on the space of such modular forms and develop a…

Recently it has been shown that the $\alpha$-Sun density $h(x)$ [{\it J. Math. Anal. Appl.}, {\bf 527} (2023), p. 127371] which interpolates between the Fr{\'e}chet density and that of the positive, stable distributions whose density is…

We study infinite series expansions for the Riemann xi function $\Xi(t)$ in three specific families of orthogonal polynomials: (1) the Hermite polynomials; (2) the symmetric Meixner-Pollaczek polynomials $P_n^{(3/4)}(x;\pi/2)$; and (3) the…

It is well known that the Bernoulli polynomials $\mathbf{B}_n(x)$ have nonintegral coefficients for $n \geq 1$. However, ten cases are known so far in which the derivative $\mathbf{B}'_n(x)$ has only integral coefficients. One may assume…

Several quantities related to the Zernike circle polynomials admit an expression as an infinite integral involving the product of two or three Bessel functions. In this paper these integrals are identified and evaluated explicitly for the…

Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and…

We prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j,…

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit…

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian…

We find an explicit form for the coproduct formula for PBW generators of quantum groups of infinite series $U_q(\frak{sp}_{2n})$ and $U_q(\frak{so}_{2n}).$ Similar formulas for $U_q(\frak{sl}_{n+1})$ and $U_q(\frak{so}_{2n+1})$ are already…

This paper shows Hurwitz integrality of the coefficients of expansion at the origin of the sigma function \(\sigma(u)\) associated to a certain plane curve which should be called a plane telescopic curve. For the prime \(2\), the expansion…

For every modulus $q\ge3$, we define a family of subsets $\mathcal{A}$ of the multiplicative group $(\mathbb{Z}/{q}\mathbb{Z})^\times$ for which the Euler product $\prod_{p\text{mod}q\in\mathcal{A}}(1-p^{-s})$ can be computed in double…