Related papers: Exploring Integration by Differentiation
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
We aim to introduce a new extension of Mittag-Leffler function via q-analogue and obtained their significant properties including integral representation, q-differentiation, q-Laplace transform, image formula under q-derivative operators.…
The Ramanujan $_1\psi_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Bailey's very-well-poised…
A calculation formula of volume of revolution with integration by parts of definite integral is derived based on monotone function, and extended to a general case that curved trapezoids is determined by continuous, piecewise strictly…
The strengthening of linear relaxations and bounds of mixed integer linear programs has been an active research topic for decades. Enumeration-based methods for integer programming like linear programming-based branch-and-bound exploit…
In this article we extend a theorem of Andrews, Crippa, and Simon on the asymptotic behavior of polynomials defined by a general class of recursive equations. Here the polynomials are in the variable $q$, and the recursive definition at…
Path integral-based simulation methodologies play a crucial role for the investigation of nuclear quantum effects by means of computer simulations. However, these techniques are significantly more demanding than corresponding classical…
Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. 'Integration by…
Ramanujan (1916) expressed quotients of certain q-series as polynomials of Eisenstein series of degree 2, 4, 6 and derived the famous Ramanujan's differential equations. We continue this research with the variants of Eisenstein-type series…
In this paper, we are concerned with the construction and analysis of a new class of methods obtained as double jump compositions with complex coefficients and projection on the real axis. It is shown in particular that the new integrators…
We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses…
In this paper, we investigate applications of the ordinary derivative operator, instead of the $q$-derivative operator, to the theory of $q$-series. As main results, many new summation and transformation formulas are established which are…
Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…
Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
We study certain $q$-difference raising operators for Macdonald polynomials (of type $A_{n-1}$) which are originated from the $q$-difference-reflection operators introduced in our previous paper. These operators can be regarded as a…
We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and…
In this paper we present experimental ways of evaluating Ramanujan`s quantities which as someone can see are related with algebraic numbers. The good thing with algebraic numbers is that can be found in a closed form, from there…
In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so we reveal a wealth of divisibility theorems and…
Integrals involving derivatives of Legendre polynomials frequently arise in applications ranging from multipole expansions for processes involving electromagnetic probes to spectral methods in numerical physics. Despite their practical…
Using a lemma of Davis on Gram matrices applied to the classical Orthogonal Polynomials to generate reproducing kernel interpolation over the classical domains for polynomials. These kernels have terms which are exact over the rational…