Related papers: The asymptotic estimation for two classes of gener…

In a paper in the American Mathematical Monthly, the corresponding author asks for an asymptotic of a gcd-sum function \begin{align}\sum_{ab\leq N}\tau(\gcd(a,b))\label{eqn:taugcdsum}\end{align} We extensively study generalizations of the…

The paper aims at reconsidering the famous Le Cam LAN theory. The main features of the approach which make it different from the classical one are as follows: (1) the study is nonasymptotic, that is, the sample size is fixed and does not…

In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable…

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary…

An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a…

In this paper, we consider a generalization of Horadam sequence fwng which is defined by the recurrence relation wn = x(n)wn-1+ cwn-2; where x(n) = a if n is even, x(n) = b if n is odd with arbitrary initial conditions w0;w1 and nonzero…

We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n) in {(7 + 5k,5), (8 + 5k,5)} where k is greater than or equal to 0. This together with previously known results yields a complete classification of the finite F(r,n),…

Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to…

This paper concerns the asymptotic behavior of a random variable $W_\lambda$ resulting from the summation of the functionals of a Gibbsian spatial point process over windows $Q_\lambda \uparrow R^d$. We establish conditions ensuring that…

Let $F_n$ be the $n$th Fibonacci number. Let $m, n$ be positive integers. Define a sequence $(G(k,n,m))_{k\geq 1}$ by $G(1,n,m) = F^m_n$, and $G(k+1,n,m) = F_{nG(k,n,m)}$ for all $k\geq 1$. We show that $F_n^{k+m-1}\mid G(k,n,m)$ for all…

Methods were initiated by Mark Kac and Richard Feynman to evaluate random functionals of the form $\int^t_0V(X_s)ds$ for a nonnegative $V$ and a Markov process $X_t$. Their results evolved into the well known Feynman Kac formula.…

The Fibonacci polynomials are defined recursively as $f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x)$, where $f_0(x) = 0$ and $f_1(x)= 1$. We generalize these polynomials to an arbitrary number of variables with the $r$-Fibonacci polynomial. We extend…

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is…

An asymptotic formula for the sum $\sum L(1,\chi)$ is established for a family of hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$ as $g\rightarrow\infty$ making use of the analogue of the approximate functional…

It is shown how some of the recent results of de Souza et al. [1] can be generalized to describe Hamiltonians whose eigenvalues are given as k-generalized Fibonacci numbers. Here k is an arbitrary integer and the cases considered by de…

Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large…

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of…

Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given…

This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze…

This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that…