Related papers: On the Poincar\'{e} functional equation
We investigate a formulation of Poincar\'e invariant quantum mechanics where the dynamical input is Euclidean invariant Green functions or their generating functional. We argue that within this framework it is possible to calculate…
We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the…
We define a new algebraic extension of the Poincar\'e symmetry; this algebra is used to implement a field theoretical model. Free Lagrangians are explicitly constructed; several discussions regarding degrees of freedom, compatibility with…
In this article I present a fast and direct method for solving several types of linear finite difference equations (FDE) with constant coefficients. The method is based on a polynomial form of the translation operator and its inverse, and…
This thesis contains some new results on the uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sums
The main objective of this paper is to introduce an algorithm for solving fractional and classical differential equations based on a new generalized fractional power series. The algorithm relies on expanding the solution of an FDE or an ODE…
The purpose of this paper is to give the explicit formulae of p-adic l-functions and sums of powers which are related to Euler numbers.
The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions and equivariant forms.
By introducing a kind of special functions namely exponent-like function, cosine-like function and sine-like function, we obtain explicitly the basic structures of solutions of initial value problem at the original point for this kind of…
A systematic study of non-trivial cubic extensions of the four-dimensional Poincar\'e algebra is undertaken. Explicit examples are given with various techniques (Young tableau, characters etc).
New methods for obtaining functional equations for Feynman integrals are presented. Application of these methods for finding functional equations for various one- and two- loop integrals described in detail. It is shown that with the aid of…
In view of the usefulness and importance of the kinetic equation in certain physical problems, the authors derive the explicit solution of a fractional kinetic equation of general character, that unifies and extends earlier results.…
Necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials on lattices are stated. Moreover, the functional Rodrigues formula and a closed…
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of…
In the present paper, we deal with Fourier-transformation of Frobenius-Euler polynomials. We shall give its applications by using infinite series. Our applications possess interesting properties which we state in this paper.
For any positive integer n, a new family of periodic functions in power series form and of period n is used to solve in closed form a class of polynomial equation of order n. The n roots are the values of the appropriate function from that…
Several years ago the second author playing with different "recognizers of real constants", e.g., the LLL algorithm, the Plouffe inverter, etc. found empirically the following formula. Let $p_n/q_n$ denote the $n$th convergent of the…
We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors…
We give presentation of composition inverse of formal power serie in a logarithmic form.