Related papers: Permutable entire functions satisfying algebraic d…
This note concerns exponential sheaves and the "universal" Fourier transform on them. Fourier invertibility and the subsequent Fourier miracle is demonstrated. Further, t-structures and realizations are constructed and shown to have…
We introduce a new factorial function which agrees with the usual Euler gamma function at both the positive integers and at all half-integers, but which is also entire. We describe the basic features of this function.
We give a new proof of the result that if f and g are transcendental entire functions, then the composite function f(g) has infinitely many fixed points. The method yields a number of generalization of this result. In particular, it extends…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
This paper aims to study the periodicity of a transcendental entire function of hyper-order less than one. For a transcendental entire function of hyper order less than one and a non-zero complex constant $c$, $\mathfrak{f} (z) \equiv…
Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure $\mathbf{A}$ is called…
In the present paper, we give sufficient conditions on the elements of the continued fractions $A$ and $B$ that will assure us that the continued fraction $A^B$ is a transcendental number. With the same condition, we establish a…
Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for…
Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can, in…
Permutation polynomials over finite fields have taken an important role in vast areas in mathematics as well as engineering. Recently, Tu et al. gave some classes of complete permutation polynomials over finite fields of even…
Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the…
We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
For a second order linear differential equation $f''+A(z)f'+B(z)f=0$, with $ A(z)$ and $B(z)$ being transcendental entire functions under some restriction, we have established that all non-trivial solutions are of infinite order. In…
We prove the existence of one or more solutions to a singularly perturbed elliptic problema with two potential functions.
We show that the class of representable substitution algebras is characterized by a set of universal first order sentences. In addition, it is shown that a necessary and sufficient condition for a substitution algebra to be representable is…
The existence of entire solutions to quadratic trinomial Fermat type differential-difference equations and \(q\)-difference differential equations involving second-order derivatives is studied by using Nevanlinna theory, and the exact form…
We study flat deformations of quotients of a polynomial algebra in a class of graded commutative associative algebras. Functional equations and their solutions in terms of theta functions play important role in these studies. An analog of…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that…