Related papers: An explicit formula for the discrete power functio…
In this paper, we consider the discrete power function associated with the sixth Painlev\'e equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this…
The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from…
Using methods developed in multivariate splines, we present an explicit formula for discrete truncated powers, which are defined as the number of non-negative integer solutions of linear Diophantine equations. We further use the formula to…
A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $\tau$-function for certain Painlev\'e transcendents. The coefficients in this expansion are given in terms…
We consider the Sobolev space over $\mathbb{R}^d$ of square integrable functions whose gradient is also square integrable with respect to some positive weight. Tt is well known that smooth functions are dense in the weighted Sobolev space…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the…
We introduce discrete analogues of the exponential, sine, and cosine functions. Then using a discrete trigonometric version of a non-polynomial divided difference, we define discrete analogues of the trigonometric B-splines. We derive a…
To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann…
Taylor's formula holds significant importance in function representation, such as solving differential difference equations, ordinary differential equations, partial differential equations, and further promotes applications in visual…
By means of a new technique, we develop further a discrete subordination approach to the functional calculus of power bounded and Ritt operators initiated by N. Dungey in [19]. This allows us to show, in particular, that (infinite) convex…
It is known, that among the formal solutions of the sixth Painlev\'e equation there met series with integer power exponents of the independent variable $x$ with coefficients in form of formal Laurent series (with finite main parts) in…
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference…
We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…
This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two…
Two distinct systems of commutative complex numbers in 6 dimensions of the polar and planar types of the form u=x_0+h_1x_1+h_2x_2+h_3x_3+h_4x_4+h_5x_5 are described in this work, where the variables x_0, x_1, x_2, x_3, x_4, x_5 are real…
By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the…
In the present work we establish a simple relation between the Dirac equation with a scalar and an electromagnetic potentials in a two-dimensional case and a pair of decoupled Vekua equations. In general these Vekua equations are bicomplex.…
A discrete analog of the holomorphic map $z^{\gamma}$ is studied. It is given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are embedded and described by special…
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…