Related papers: Deformation Expression for Elements of Algebras (V…
This paper presents a preliminary version of the deformation theory of expressions of elements of algebras. The notion of *-functions is given. Several important problems appear in simplified forms, and these give an intuitive bird's-eye of…
In the recent development in a various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In this paper, we consider the deformations which have two purposes: to have them like…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
In this note, we mainly consider the extended Weyl algebra of two generators (u,v), that is, the algebra generated by u,v with the fundamental commutation relation. Weyl algebra is realized on the space of polynomials of u and v by defining…
We give a formula for matrix exponentials and partial fraction decompositions.
In this paper, we study some extended hypergeometric functions from matrix point of view. We have given the integral representations of these matrix functions. Finally, we obtain some generating function relations using fractional…
In a noncommutative algebra there is no canonical way to express elements in univalent way, which is often called "ordering problem". In this note we give product formula of the Weyl algebra in generic ordered expression. In particular, the…
Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are…
Special functions have always played a central role in physics and in mathematics, arising as solutions of particular differential equations, or integrals, during the study of particular important physical models and theories in Quantum…
Generating functions and functional equations of Dickson polynomials of the first and second kind are derived and continued analytically. These formulae are expressed in terms of the incomplete gamma function over complex variables of the…
We introduce a set of special functions called multiple polyexponential integrals, defined as iterated integrals of the exponential integral $\text{Ei}(z)$. These functions arise in certain perturbative expansions of the local solutions of…
This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we…
Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…
We present a deformed algebra related to the q-exponential and the q-logarithm functions that emerge from nonextensive statistical mechanics. We also develop a q-derivative (and consistently a q-integral) for which the q-exponential is an…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $e^x$ from classical differential calculus. A differential exponential map is an endomorphism which…
Ext-int.\ one affine functions are functions affine in the direction of one-divisible exterior forms, with respect to exterior product in one variable and with respect to interior product in the other. The purpose of this article is to…
For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for…
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…