Related papers: On representations of integers by indefinite terna…

For an integer $q\ge2$, a $q$-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of~$q$. In this article, $q$-recursive sequences are studied and the asymptotic behavior of their summatory…

In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.

We develop a new infinite dimensional gluing method for fractional elliptic equations. As a model problem, we construct solutions of the fractional Allen--Cahn equation vanishing on a rotationally symmetric surface which resembles a…

We revisit the derivation of a formula for the $q$-generalised multinomial coefficient rooted in the $q$-deformed algebra, a foundational framework in the study of nonextensive statistics. Previous approximate expressions in the literature…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

Let $F_Q$ be the Farey sequence of order $Q$ and let $F_{Q,o}$ and $F_{Q,e}$ be the set of those Farey fractions of order $Q$ with odd, respectively even denominators. A fundamental property of $F_Q$ says that the sum of denominators of any…

Let $q$ be a quadratic form over a field $F$ and let $L$ be a field extension of $F$ of odd degree. It is a classical result that if $q_L$ is isotropic (resp. hyperbolic) then $q$ is isotropic (resp. hyperbolic). In turn, given two…

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an in- definite weight on the nonlinearity f (u, r). In particular we are interested in the case…

We have for positive integers $n$, $k$ and finite field $\mathbb{F}_q$, $c(n,k,q)$, as the number of simultaneous similarity classes of $k$-tuples of commuting $n\times n$ matrices over the $\mathbb{F}_q$. In this paper, it has been shown…

We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic…

Let $$ \zeta_E(s,q)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+q)^{s}} $$ be the alternating Hurwitz (or Hurwitz-type Euler) zeta function. In this paper, we obtain the following asymptotic expansion of $\zeta_{E}(s,q)$ $$ \zeta_E(s,q)\sim\frac12…

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form $A=T(a)+E$ where $T(a)$ is the Toeplitz matrix with entries $(T(a))_{i,j}=a_{j-i}$, for $a_{j-i}\in\mathbb C$, $i,j\ge 1$, while $E$ is a matrix representing a compact…

This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-\Delta)_{p}^{s}u+\varepsilon^{qs}(-\Delta)_q^su+…

We study functions $f$ on $\mathbb Q$ which statisfy a ``quantum modularity'' relation of the shape $$ f(x+1)=f(x), \qquad f(x) - |x|^{-k} f(-1/x) = h(x) $$ where $h:\mathbb R_{\neq 0} \to \mathbb C$ is a function satisfying various…

We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;\lambda)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-\lambda)}, \qquad |q|>1, \quad \lambda\notin q^{\mathbb Z_{>0}}, $$ that includes as special cases…

In this paper we show that certain sets are dense in $\mathbb{R}$. We give some applications. For example, we show an analytical proof that $q^{\frac{1}{n}}$, $q$ is a prime number and $e$; are irrational numbers. As another application we…

In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form $Q$ that is not extremely well approxiable by rational forms, and for…

Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to…

This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows…

We initiate a systematic development of $F_N(a, b; t)$, a finite analogue of Fine's function $F(a, b; t)$. Our results are transformations between $F_N(a, b; t)$ and $F_N(aq^{\ell}, bq^{m}; tq^{n})$, where $\ell,m$ and $n$ take the values…