Related papers: Ramanujan's Harmonic Number Expansion

We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums…

The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…

In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.

We present numerical evidence that a simple variational improvement of the ordinary perturbation theory of the quantum anharmonic oscillator can give a convergent sequence of approximations even in the extreme strong coupling limit, the…

Using functional equations, we derive noncommutative extensions of Ramanujan's 1-psi-1 summation.

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of…

In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m…

We study two aspects of Hecke symmetry in this note: first, we conjecture a generalization of the Ramanujan identities to the case of automorphic forms of Hecke groups; second, we conjecture a generalization of an inversion formula from the…

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a…

A new method to compute the incoherent scattering function of harmonic lattices is introduced. It is based in a saddle point approximation for each term of the phonon expansion, and is simple enough to be used in practice. The method gives…

We study Ramanujan-Fourier series of certain arithmetic functions of two variables. We generalize Delange's theorem to the case of arithmetic functions of two variables and give sufficient conditions for pointwise convergence of…

The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the…

Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.

Perturbation expansions up to third order for the generalized spiked harmonic oscillator Hamiltonians H = -d^2/dx^2+ x^2 + A/x^2 + lambda/x^alpha, A >= 0, 2gamma > alpha, gamma=1+(1/2)sqrt(1+4A), and small values of the coupling lambda > 0,…

Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in…

We obtain rates of convergence of numerical approximations of abstract linear parabolic evolution equations in Banach spaces. Our estimates extend known results from the literature of finite element approximations of parabolic equations to…

We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…

In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that…

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential…

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma…