Related papers: On Finite Mellin Transform via Ramanujan's Master …
We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term…
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…
At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $\alpha_n$. All those results were proved by Berndt et. al by employing Weber-Ramanujan's class invariants. In this paper, we initiate to derive the…
We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria…
We describe the complex of solutions of the algebraic Mellin transform of a $\mathcal{D}$-module $\mathcal{M}$ in terms of the solutions of $\mathcal{M}$. In order to do that, we define a Mellin functor on sheaves. We show the Mellin…
A method for the evaluation of the epsilon expansion of multi-loop massless Feynman integrals is introduced. This method is based on the Gegenbauer polynomial technique and the expansion of the Gamma function in terms of harmonic sums.…
Deriving a comprehensive set of reduction rules for Feynman integrals has been a longstanding challenge. In this paper, we present a proposed solution to this problem utilizing generating functions of Feynman integrals. By establishing and…
Several expansions of the solutions to the confluent Heun equation in terms of incomplete Beta functions are constructed. A new type of expansion involving certain combinations of the incomplete Beta functions as expansion functions is…
We prove a general result on representing the Riemann zeta function as a convergent infinite series in a complex vertical strip containing the critical line. We use this result to re-derive known expansions as well as to discover new series…
The modular forms are revisited from a geometric and an algebraic point of view leading to a geometric interpretation of the weak Maass forms connecting them to the Ramanujan Mock Theta functions and to the cusp forms generated from the…
The classical Ruckert-Lefschetz scheme of analysis of implicit functions (defined by finite systems of n analytical equations with n unknowns) is studied from the point of view of calculations with finite number coefficients in Taylor…
In this paper we obtain a closed form expression of the zeta function $Z(X_\Gamma, u)$ of a finite quotient $X_\Gamma = \Gamma \backslash PGL_3(F)/PGL_3(O_F)$ of the Bruhat-Tits building of $PGL_3$ over a nonarchimedean local field $F$.…
We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formul\ae. Here, the integer $h\ge 0$ is called, as usual, the {\it shift}…
We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's…
We have developed a finite-element micromagnetic simulation code based on the FEniCS package called magnum.fe. Here we describe the numerical methods that are applied as well as their implementation with FEniCS. We apply a transformation…
A new class of integrals involving the product of $\Xi$-functions associated with primitive Dirichlet characters is considered. These integrals give rise to transformation formulas of the type $F(z, \alpha,\chi)=F(-z,…
In this paper we prove theorems related to the Ramanujan-type series for $1/\pi$ (type $_3F_2$) and to the Ramanujan-like series, discovered by the author, for $1/\pi^2$ (type $_5F_4$). Our developments for the cases $_3 F_2$ and $_5 F_4$…
The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma…
We review the results having the property of maximal transcendentality.
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with…