arXiv subjects

Petro Kolosov

Publications and source records attributed to Petro Kolosov.

5 recordsLinked to original sources

arXiv2025

An efficient method of spline approximation for power function

Let $P(m, X, N)$ be an $m$-degree polynomial in $X\in\mathbb{R}$ having fixed non-negative integers $m$ and $N$. The polynomial $P(m, X, N)$ is derived from a rearrangement of Faulhaber's formula in the context of Knuth's work entitled "Johann Faulhaber and sums of powers". In this manuscript we discuss the approximation properties of polynomial $P(m,X,N)$. In particular, the polynomial $P(m,X,N)$ approximates the odd power function $X^{2m+1}$ in a certain neighborhood of a fixed non-negative integer $N$ with a percentage error under $1\%$. By increasing the value of $N$ the length of convergence interval with odd-power $X^{2m+1}$ also increases. Furthermore, this approximation technique is generalized for arbitrary non-negative exponent $j$ of the power function $X^j$ by using splines.

math.GM↗

arXiv2024

On the link between binomial theorem and discrete convolution

Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree polynomial in $x$ and $b \in \mathbb{R}$ \[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r \] where $\mathbf{A}_{m,r}$ are real coefficients. In this manuscript, we introduce the polynomial $\mathbf{P}^{m}_{b}(x)$ and study its properties, establishing a polynomial identity for odd-powers in terms of this polynomial. Based on mentioned polynomial identity for odd-powers, we explore the connection between the Binomial theorem and discrete convolution of odd-powers, further extending this relation to the multinomial case. All findings are verified using Mathematica programs.

math.NT↗

arXiv2024

A study on partial dynamic equation on time scales involving derivatives of polynomials

Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. Let be a two-dimensional timescale $Λ^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$ such that $\mathbb{T}_1 = \mathbb{T}_2$. In this manuscript we derive and discuss an identity that connects the timescale derivative of odd-power polynomial with partial derivatives of polynomial $P(m,b,x)$ evaluated in particular points. For every $t\in\mathbb{T}_1$ and $(x,b) \in Λ^2$ \[ \frac{Δx^{2m+1}}{Δx}(t) = \frac{\partial P(m,b,x)}{Δx} (m, σ(t), t) + \frac{\partial P(m,b,x)}{Δb} (m, t, t) \] such that $σ(t) > t$ is forward jump operator. In addition, we discuss various derivative operators in context of partial cases of above equation, we show finite difference, classical derivative, $q-$derivative, $q-$power derivative on behalf of it.

math.CA↗

arXiv2023

Another approach to get derivative of odd-power

In this manuscript, we provide and discuss another approach to get derivative of odd-power such that is based on an identity in partial derivatives in terms of polynomial function $f_y$ defined as \[ f_{y} (x, z) = \sum_{k=1}^{z} \sum_{r=0}^{y} \mathbf{A}_{y,r} k^r (x-k)^r \] where $x, z\in \mathbb{R}$, $y$ is fixed constant $y \in \mathbb{N}$ and $\mathbf{A}_{y,r}$ are real coefficients.

math.GM↗

arXiv2021

An unusual identity for odd-powers

In this manuscript we provide a new polynomial pattern. This pattern allows to find a polynomial expansion of the form \[x^{2m+1} = \sum_{k=1}^{x}\sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r,\] where $x,m\in\mathbb{N}$ and $\mathbf{A}_{m,r}$ is real coefficient.

math.GM↗