Related papers: Discrete Generating Series and Linear Difference E…
Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on…
A Chen generating series, along a path and with respect to $m$ differential forms,is a noncommutative series on $m$ letters and with coefficients which are holomorphic functionsover a simply connected manifold in other words a series with…
This paper presents a study of power series distributions (PSD) with prescribed covariance characteristics. Such distributions constitute a fundamental class in probability theory and mathematical statistics, as they generalize a wide range…
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…
A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side…
We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and $p$-adic integral (the Volkenborn integral). By using these…
This is the first of two articles devoted to a comprehensive exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
Despite recent advances, goal-directed generation of structured discrete data remains challenging. For problems such as program synthesis (generating source code) and materials design (generating molecules), finding examples which satisfy…
Discrete orthogonal matrices have several applications in information technology, such as in coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize…
Consider a set of single-input, single-output nonlinear systems whose input-output maps are described only in terms of convergent Chen-Fliess series without any assumption that finite dimensional state space models are available. It is…
We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The particular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the…
A rational function $f(x)$ is rationally summable if there exists a rational function $g(x)$ such that $f(x)=g(x+1)-g(x)$. Detecting whether a given rational function is summable is an important and basic computational subproblem that…
D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…
We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field…
Diffusion generative models have recently been applied to domains where the available data can be seen as a discretization of an underlying function, such as audio signals or time series. However, these models operate directly on the…
Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the…
We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of $PSL(2,{\bold C})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in…