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

TL;DR

This paper establishes a new identity linking the delta derivative of odd-power polynomials on time scales with their partial derivatives, encompassing various derivative types like finite difference, classical, and q-derivatives.

Contribution

It introduces a novel identity connecting time scale derivatives of polynomials with their partial derivatives, unifying multiple derivative concepts within a single framework.

Findings

Derived a key identity relating delta derivatives and partial derivatives of polynomials.

Unified various derivative operators, including finite difference, classical, and q-derivatives.

Provided insights into the behavior of polynomial derivatives on time scales.

Abstract

Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. Let be a two-dimensional timescale $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$ 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 $(x,b) \in \Lambda^2$ \[ \frac{\Delta x^{2m+1}}{\Delta x}(t) = \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] such that $\sigma(t) > t$ 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.