Related papers: On summatory arithmetic functions and a Volterra I…
We analyze the conditions on the Taylor coefficients of an analytic function to admit global analytic continuation, complementing a recent paper of Breuer and Simon on general conditions for natural boundaries to form. A new summation…
In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation…
It is shown how to calculate asymptotics of integrals over the positive semi-axis of two functions related to the Degenerate Third Painlev\'e Equation (dP3). As an example, the corresponding results for the meromorphic solution of the dP3…
Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.
In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an…
We use partial zeta functions to analyse the asymptotic behaviour of certain smooth arithmetical sums over smooth k-free integers.
In the asymptotic analysis of regular sequences as defined by Allouche and Shallit, it is usually advisable to study their summatory function because the original sequence has a too fluctuating behaviour. It might be that the process of…
We calculate some finite and infinite sums containing the digamma function in closed-form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative…
We propose a formula for finding the horizontal, oblique or curvilinear asymptote of any rational polynomial function of any positive degree, as a sum of matrix determinants formed directly from the coefficients of the terms in the given…
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian…
In the paper, the authors establish some asymptotic formulas and double inequalities for the factorial $n!$ and the gamma function $\Gamma$ in terms of the tri-gamma function $\psi'$.
In this paper the incomplete gamma function $\gamma(\alpha,x)$ and its derivative is considered for negative values of $\alpha $ and the incomplete gamma type function $\gamma_*(\alpha,x_-)$ is introduced. Further the polygamma functions…
We improve the upper bounds of the following inequalities proved in [H. Alzer and N. Batir, Monotonicity properties of the gamma function, Appl. Math. Letters, 20(2007), 778-781]. \begin{equation*}…
We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some…
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such…
Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice $\Lambda \subseteq \mathbb{R}^d$ inside the $d$-sphere of radius $R$. In contrast to…
We exploit the properties of a sequence of functions that approximate the divisor functions and combine them with an analytical formula of a delta-like sequence to give a new proof of a theorem of Gronwall on the asymptotic of the divisor…
We study some mapping properties of Volterra type integral operators and composition operators on model spaces. We also discuss and give out a couple of interesting open problems in model spaces where any possible solution of the problems…