Related papers: On The Indefinite Sum In Fractional Calculus
Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left(…
The decompositions of an element of a finite von Neumann algebra into the sum of a normal operator plus an s.o.t.-quasinilpotent operator, obtained using the Haagerup--Schultz hyperinvariant projections, behave well with respect to…
We study Birkhoff sums over rotations (series of the form $\sum_{r=1}^{N}\phi(r\alpha)$), in which the summed function $\phi$ may be unbounded at the origin. Estimates of these sums have been of significant interest and application in pure…
In our last work, we formulate a Fourier transformation on the infinite-dimensional space of functionals. Here we first calculate the Fourier transformation of infinite-dimensional Gaussian distribution $\exp(-\pi…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
We use the displacement operator to derive an infinite series of integer order derivatives for the Gr\"{u}nwald-Letnikov fractional derivative and show its correspondence to the Riemann-Liouville and Caputo fractional derivatives. We…
In calculus, an indefinite integral of a function $f$ is a differentiable function $F$ whose derivative is equal to $f$. In present paper, we generalize this notion of the indefinite integral from the ring of real functions to any ring. The…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus…
For a family of infinite-dimensional diffusions with degenerate noise, we develop a modified $\Gamma$ calculus on finite-dimensional projections of the equation in order to produce explicit functional inequalities that can be scaled to…
Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial…
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
Let $f$ be a holomorphic function on the strip $\{z\in C: -\alpha<Im z<\alpha\}, \alpha > 0$, belonging to the class $H(\alpha,-\alpha;\epsilon)$ defined below. It is shown that there exist holomorphic functions $w_1$ on $\{z\in C: 0<Im z…
In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the…
A partial fraction decomposition of the Fermi function resulting in a finite sum over simple poles is proposed. This allows for efficient calculations involving the Fermi function in various contexts of electronic structure or electron…
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.
Many classical identities arise from nothing more mysterious than looking at the same object in two different ways. A number, a function, or a combinatorial object may admit several natural decompositions, and by disassembling it in one way…
We study the convolution of functions of the form \[ f_\alpha (z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^\alpha - 1}{2 \alpha}, \] which map the open unit disk of the complex plane onto polygons of 2 edges when $\alpha\in(0,1)$. We…
The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both…