Related papers: An Asymptotic Formula for the Sequence ||exp(i n h…

The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied for integer n>=-1 and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of…

Motivated by studies in accelerator physics this paper computes the asymptotic behavior of the series $\displaystyle \sum_{k=1}^N \varphi(k) I_N\left(\frac{1}{k}\right)$, where $\varphi(k)$ is Euler's Totient function and $\displaystyle…

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation…

Asymptotic expansions are derived for associated Legendre functions of degree $\nu$ and order $\mu$, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument…

In this paper, we study the asymptotics of the Hahn polynomials Q_n(x; {\alpha}, {\beta}, N) as the degree n grows to infinity, when the parameters {\alpha} and {\beta} are fixed and the ratio of n/N = c is a constant in the interval (0,…

We will study the asymptotic behavior of summation functions of a natural argument, including the asymptotic behavior of summation functions of a prime argument in the paper. A general formula is obtained for determining the asymptotic…

We introduce a new approach to the the asymptotic iteration method (AIM) by means of which we establish the standard AIM connection with the continued fractions technique and we develop a novel termination condition in terms of the…

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ where…

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in [11]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential…

This article is focused on the asymptotic expansions, as time tends to infinity, of solutions of a system of ordinary differential equations with non-smooth nonlinear terms. The forcing function decays to zero in a very complicated but…

We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order $n^{\delta }$, where $0<\delta <1,$ and $f\left( z\right) $ is a power series satisfying…

In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main…

We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for some…

The paper compares the asymptotic of the expressions $\frac {1} {x} \sum\limits_{n \leq x} {f(n)}$ and $\sum\limits_{n \leq x} {\frac {f(n)} {n}}$, $\frac {1} {x} \sum\limits_{p \leq x} {f(p)}$ and $\sum\limits_{p \leq x} {\frac {f(p)}…

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[F(a+\epsilon\lambda,m;c+\lambda;x),\qquad \lambda\to+\infty\] for $x<1$ and positive integer $m$ when the parameter $\epsilon>1$ and the constants $a$ and…

In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm SL}_2…

We consider the asymptotic expansion of the sum \[S_p(a;w)=\sum_{n=1}^\infty n^{-w}\e^{-an^p}\] as $a\rightarrow 0$ in $|\arg\,a|<\pi/2$ for arbitrary finite $p>$ and $w>0$. Our attention is concentrated mainly on the case when $p$ and $w$…

Using the inner product formula of the canonical Hilbert space of fractional Brownian motion on an interval $[0,T]$ with Hurst parameter $H\in (0,1)$ given by Alazemi et al., we show the asymptotic expansion of the norm of…

We use a Mellin-Barnes integral representation for the Lerch transcendent $\Phi(z,s,a)$ to obtain large $z$ asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that $s$ is an integer. For…

We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z\}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$…