Related papers: New Indefinite Summation Formulas and Some Applica…

We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function,…

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…

Classical and nonclassical symmetries of the nonlinear heat equation $$u_t=u_{xx}+f(u),\eqno(1)$$ are considered. The method of differential Gr\"obner bases is used both to find the conditions on $f(u)$ under which symmetries other than the…

The indefinite integral $$ \int x^\alpha e^{\eta x^\beta}\,_pF_q (a_1, a_2, \cdot\cdot\cdot a_p; b_1, b_2, \cdot\cdot\cdot, b_q; \lambda x^{\gamma})dx, $$ where $\alpha, \eta, \beta, \lambda, \gamma\ne0$ are real or complex constants and…

A new approximation of the discriminant of a second order periodic differential equation is presented as a recursive summation of the evaluation of its excitation function at different values of time. The new approximation is obtained, at…

Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for…

For $\nu\in[0,1]$ and a complex parameter $\sigma,$ $Re\, \sigma>0,$ we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane $z\in\mathbb{C}$: \[…

This paper offers a solution method that allows one to find exact values for a large class of convergent series of rational terms. Sums of this form arise often in problems dealing with Quantum Field Theory.

We prove new exact formulas for the generalized sum-of-divisors functions, $\sigma_{\alpha}(x) := \sum_{d|x} d^{\alpha}$. The formulas for $\sigma_{\alpha}(x)$ when $\alpha \in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over…

We establish a new $W^{1,2\frac{n-1}{n-2}}$ estimate for the extremal solution of $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$, which is convex, for arbitrary positive and increasing nonlinearities $f\in…

Given a function f with an absolutely convergent Fourier series, we define the norm of f as ||f||_A = the sum of absolute values of the Fourier coefficients of f. We study the behavior of ||f^n||_A as n goes to infinity, for f of the form…

We study higher order linear differential equation $y^{(k)}+A_1(z)y=0$ with $k\geq2$, where $A_1=A+h$, $A$ is a transcendental entire function of finite order with $\frac{1}{2}\leq \mu(A)<1$ and $h\neq0$ is an entire function with…

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.

We consider nonnegative solutions to $-\Delta u=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under…

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive…

In the underlying study it is shown how the linear method of the Yosida-approximation of the derivative applies to solve possibly nonlinear and multivalued functional differential equations like: \begin{eqnarray*} u^\prime(t) &\in&…

Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2), where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked…

From the algebraic solution of $x^{n}-x+t=0$ for $n=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these…

The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…

Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity \[ S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where $[t]$ is…