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Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic…

We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…

We produce examples of groups of type F_3 with 2-dimensional Dehn functions of the form exp^n(x) (a tower of exponentials of height n), where n is any natural number.

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…

Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…

Integrable two-dimensional models which possess an integral of motion cubic or quartic in velocities are governed by a single prepotential, which obeys a nonlinear partial differential equation. Taking into account the latter's invariance…

We establish exponential laws for certain spaces of differentiable functions over a valued field K. For example, we show that the topological vector spaces C^{r,s}(U x V,E) and C^r(U,C^s(V,E)) are isomorphic if U and V are open subsets of…

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…

In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…

In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series…

Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…

Polynomial sequences $p_n(x)$ of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express $p_n(x)$ as a \emph{path integral} in the ``phase space'' $\Space{N}{} \times {[-\pi,\pi]}$. The Hamiltonian…

Formulas to calculate multivector exponentials in a base-free representation and in a orthonormal basis are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on the analysis of roots of characteristic…

In this paper we give an integral representation of an $n$-convex function $f$ in general case without additional assumptions on function $f$. We prove that any $n$-convex function can be represented as a sum of two $(n+1)$-times monotone…

For a second-order linear differential equation with two irregular singular points of rank three, multiple Laplace-type contour integral solutions are considered. An explicit formula in terms of the Stokes multipliers is derived for the…

In this paper, we first introduce certain forms of extended incomplete Pochhammer symbols which are then used to define families of extended incomplete generalized hypergeometric functions. For these functions, we investigate various…

Based on the recent developments in the irregular Riemann-Hilbert correspondence for holonomic D-modules and the Fourier-Sato transforms for enhanced ind-sheaves, we study the Fourier transforms of some irregular holonomic D-modules. For…

We give a formula for matrix exponentials and partial fraction decompositions.