Related papers: A Generalization of the arctg function
This work introduces a new functional series for expanding an analytic function in terms of an arbitrary analytic function. It is generally applicable and straightforward to use. It is also suitable for approximating the behavior of a…
We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example…
Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide…
We define a modular function which is a generalization of the elliptic modular lambda function. We show this function and the modular invariant function generate the modular function field with respect to the principal congruence subgroup.…
A characterization of multiplicative (and additive) arithmetical functions is given. Using this characterization, we show that the group of multiplicative arithmetical functions is isomorphic to the group of additive arithmetical functions.
In this paper, we introduce a way to generalize the Euler's gamma function as well as some related special functions. With a given polynomial in one variable $f(t)\ge 0$, we can associate a function, so-called "gamma function associated…
This paper introduces a new method for redefining the Roman factorial using universally applicable functions that are not expressed in closed form. We present a set of foundational functions, similar to Boolean operations, to simplify the…
A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. H. Barendregt has shown that if we can represent, for a numeral system, the functions : Successor,…
In this paper we consider a class of conjugate equations, which generalizes de Rham's functional equations. We give sufficient conditions for existence and uniqueness of solutions under two different series of assumptions. We consider…
This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications…
The article discusses criteria for univalence of analytic functions in the unit disc. Various families of analytic functions depending on real parameters are considered. A unified method for creating new sets of conditions ensuring…
We prove that all groups of exponential growth support non-constant positive harmonic functions. In fact, out results hold in the more general case of strongly connected, finitely supported Markov chains invariant under some transitive…
In this paper, we give one possible definition for functions of several variables applied to endomorphisms of finite dimensional C-vector spaces. This definition is consistent with the usual notion of a function of a square matrix. Some…
For any given finite abelian group, we give factorizations of the group determinant in the group algebra of any subgroup. The factorizations are an extension of Dedekind's theorem. The extension leads to a generalization of Dedekind's…
Given a rational function of degree at least two defined over a number field k, we study the cardinality of the set of rational iterated preimages. We prove bounds for the cardinality of this set as the rational function varies in certain…
In this paper, we develop new identities for the inverse tangent integral by connecting it to the dilogarithmic (polylogarithmic) structure and to a carefully designed auxiliary arctangent integral $Ti_2(a)$ with a tunable endpoint. The…
We give a generalization of the random matrix ensembles, including all lassical ensembles. Then we derive the joint density function of the generalized ensemble by one simple formula, which give a direct and unified way to compute the…
We construct and study the class of continuous on $[0, 1]$ functions with continuum set of peculiarities (singular, nowhere monotonic, and non-differentiable functions are among them). The representative of this class is the function…
In this paper, we introduce a method of converting implicit equations to the usual forms of functions locally without differentiability. For a system of implicit equations which are equipped with continuous functions, if there are unique…
Real analytic generalized functions are investigated as well as the analytic singular support and analytic wave front of a generalized function in $\mathcal{G}(\Omega)$ are introduced and described.