Related papers: Rational decompositions of complex meromorphic fun…

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k…

It was earlier conjectured by the second and the third authors that any rational curve $g:{\mathbb C}P^1\to {\mathbb C}P^n$ such that the inverse images of all its flattening points lie on the real line ${\mathbb R}P^1\subset {\mathbb…

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D \geq 3$ and standard module $V$. Recently Ito and Terwilliger introduced four direct sum decompositions of $V$; we call these the $(\mu,\nu)$--{\it split…

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers.…

The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most…

Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$…

The main goal of this paper is to establish the higher-dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions ( holomorphic maps) in several variables, such as the proximity function,…

We treat shared value problems for rational functions $R(z)$ and their derivative $R'(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and $\partial_{z} R = \lambda w \cdot R'(w)$ on ${\mathbb C}…

This paper addresses a theory of R(p,q)-deformed combinatorics in discrete probability. It mainly focuses on R(p,q)-deformed factorials, binomial coefficients, Vandermonde's formula, Cauchy's formula, binomial and negative binomial…

In this paper, we continue to study the sharing value problems for higher order derivatives of meromorphic functions with its linear difference and $q$-difference operators. Some of our results generalize and improve the results of…

We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type $(C^\vee_n,C_n)$. A certain representation in terms of difference-reflection operators naturally leads to the definition…

In this paper we study the derived sets for the rational deformations of multiple zeta-star values. By using the theory of bounded variation functions, we will give function decompositions which describe the metric structure of the derived…

In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in…

Let $P$ and $Q$ be two polynomials in two variables with coefficients in an algebraic closed field of characteristic zero. We consider the rational function $f=P/Q$. For an indeterminacy point $\text{x}$ of $f$ and a value $c$, we compute…

We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that…

Decoupling multivariate polynomials is useful for obtaining an insight into the workings of a nonlinear mapping, performing parameter reduction, or approximating nonlinear functions. Several different tensor-based approaches have been…

In this paper, we consstruct a new extended q-Bernoulli numbers and poly nomials. From these numbers, we derive the multiple zeta functions and give some relations between multiple Bernoulli numbers and multiple zeta functions.

Using non-archimedean q-integrals on Zp defined in [15, 16], we define a new Changhee q-Euler polynomials and numbers which are different from those of Kim [7] and Carlitz [2]. We define generating functions of multiple q-Euler numbers and…

Let k be an algebraically closed field of characteristic zero. An element F from k(x_1,...,x_n) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x_1,...,x_n). We prove that a rational function…