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manuscript we derive and discuss the following partial dynamic equation on time scales. +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 $x,b\in \Lambda^2$ \[ - (t^{2m+1})^{\Delta} = - \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} + - \frac{\partial P(m,b,x)}{\Delta b}\bigg |_{x = t, \; b = t} + \frac{\Delta t^{2m+1}}{\Delta t} = +% \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} + + \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] such that $\sigma(t) > t$ is forward jump operator. In addition, we discuss various derivative operators in context of partial cases of above equation, diff --git a/src/sections/definitions-notations-and-conventions.tex b/src/sections/definitions-notations-and-conventions.tex index c47ea37..581be58 100644 --- a/src/sections/definitions-notations-and-conventions.tex +++ b/src/sections/definitions-notations-and-conventions.tex @@ -1,73 +1,61 @@ -We now set the following notation, which remains fixed for the remainder of this paper: +We now set the following notation such that remains fixed for the remainder of this manuscript \begin{itemize} - \setlength\itemsep{1em} + \setlength\itemsep{1.6em} \item Let be a function $f\colon \mathbb{T} \to \mathbb{R}$ and $t\in\mathbb{T}^{\kappa}$ then $f^{\Delta}(t)$ - is delta time-scale derivative~\cite{Bohner2001DynamicEO} of $f$ - \[ - f^{\Delta} (t) = \frac{f(\sigma(t)) - f(t)}{\mu(t)} = \frac{f(\sigma(t)) - f(t)}{\sigma(t) - t}, - \] - where $\mu(t) = \sigma(t) - t, \; \mu(t) \neq 0$ and $\sigma(t) > t$ is forward jump operator. - - \item $\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}, \; f^{\Delta_i}_{t_i}(t)$ is delta partial derivative - of $f\colon \Lambda^n \to \mathbb{R}$ on $n-$dimensional time scale - $\Lambda^n$~\cite{bohner2004partial, ahlbrandt2002partial,JACKSON2006391}, - defined as a limit - \[ - f^{\Delta_i}_{t_i}(t) = \lim \limits_{\substack{s_i \to t_i \\ s_i \neq \sigma_i(t_i)}} + is delta timescale derivative~\cite{Bohner2001DynamicEO} + \begin{align*} + f^{\Delta} (t) = \frac{f(\sigma(t)) - f(t)}{\sigma(t) - t} + \end{align*} + where $\sigma(t) - t \neq 0$ and $\sigma(t) > t$ is forward jump operator. + + \item $\dfrac{\partial f(t_1,\ldots,t_n)}{\Delta_i t_i}$ is the delta partial derivative + of $f\colon \Lambda^n \to \mathbb{R}$ on $n$-dimensional timescale + $\Lambda^n$ defined via the limit~\cite{bohner2004partial, ahlbrandt2002partial,JACKSON2006391} + \begin{align*} + f^{\Delta_i}_{t_i}(t) = \lim \limits_{\substack{s_i \to t_i}} \frac{ f(t_1, \ldots, t_{i-1}, \sigma_i(t_i), t_{t+1}, \ldots, t_n) - f(t_1, \ldots, t_{i-1}, s_i, t_{t+1}, \ldots, t_n) - }{\sigma_i(t_i) - s_i}, - \] + }{\sigma_i(t_i) - s_i} + \end{align*} where $\sigma_i(t_i) > t_i$ and $\sigma_i(t_i) - s_i \neq 0$. \item $\qderivative{f(x)}$ is $q-$derivative~\cite{jackson_1909,ernst2000history,ernst2008different,kac2001quantum} - \[ - \qderivative{f(x)} = \frac{f(qx)-f(x)}{qx-x}, - \] + \begin{align*} + \qderivative{f(x)} = \frac{f(qx)-f(x)}{qx-x} + \end{align*} where $x\neq 0, \; x\in\mathbb{R}, \; q\in\mathbb{R}$. \item $\nqderivative{f(t)}$ is $q-$power derivative~\cite{aldwoah2011power} - \[ - \nqderivative{f(t)} = \frac{f(qt^n) - f(t)}{qt^n - t}, - \] + \begin{align*} + \nqderivative{f(t)} = \frac{f(qt^n) - f(t)}{qt^n - t} + \end{align*} where $qt^n - t \neq 0$ and $n$ is odd positive integer and $0 < q < 1$. \item $\qpowerDerivative{f(x)}$ is $q-$power derivative - \[ - \qpowerDerivative{f(x)} = \frac{f(x^q)-f(x)}{x^q-x}, - \] + \begin{align*} + \qpowerDerivative{f(x)} = \frac{f(x^q)-f(x)}{x^q-x} + \end{align*} where $x^q \neq x, \; x\in\mathbb{R}, \; q\in\mathbb{R}$. - \item $\polynomialP{m}{b}{x}, \; x,b\in\mathbb{R}, \; m\in\mathbb{N}$ is $2m+1-$degree integer-valued polynomial~\cite{kolosov2016link} + \item $\polynomialP{m}{b}{x}, \; x,b\in\mathbb{R}, \; m\in\mathbb{N}$ is $2m+1$-degree + polynomial in $x,b$ \begin{equation} - \polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r, + \polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \label{eq:polynomial_p} \end{equation} - where $\coeffA{m}{r}, \; m\in\mathbb{N}$ is a real coefficient defined recursively - \[ - \coeffA{m}{r} = - \begin{cases} - (2r+1) - \binom{2r}{r}, & \text{if } r=m,\\ - (2r+1) \binom{2r}{r} \sum_{d=2r+1}^{m} \coeffA{m}{d} \binom{d}{2r+1} \frac{(-1)^{d-1}}{d-r} - \bernoulli{2d-2r}, & \text{if } 0 \leq rm, - \end{cases} - \] - where $\bernoulli{t}$ are Bernoulli numbers~\cite{WeissteinBernoulli}. - It is assumed that $\bernoulli{1}=\frac{1}{2}$. + where $\coeffA{m}{r}$ is a real coefficient defined recursively, see~\cite{kolosov2016link}. - \item $\mathbb{Z}$ is an integer time scale such that $\sigma(t) = t+1$ and $\mu(t) = 1$. + \item $\mathbb{Z}$ is an integer timescale such that $\sigma(t) = t+1$ and $\mu(t) = 1$. - \item $\mathbb{R}$ is a real time scale such that $\sigma(t) = t+\Delta t$ and $\mu(t) = \Delta t, \; \Delta t \to 0$. + \item $\mathbb{R}$ is a real timescale such that $\sigma(t) = t+\Delta t$ and $\mu(t) = \Delta t, \; \Delta t \to 0$. - \item $q^\mathbb{R}$ is a quantum time scale such that $\sigma(t) = qt$ and $\mu(t) = qt - t$, + \item $q^\mathbb{R}$ is a quantum timescale such that $\sigma(t) = qt$ and $\mu(t) = qt - t$, [page 18~\cite{Bohner2001DynamicEO}]. - \item $\mathbb{R}^q$ is a quantum power time scale such that $\sigma(t) = t^q$ and $\mu(t) = t^q - t$. + \item $\mathbb{R}^q$ is a quantum power timescale such that $\sigma(t) = t^q$ and $\mu(t) = t^q - t$. - \item $q^{\mathbb{R}^n}$ is a pure quantum power time scale + \item $q^{\mathbb{R}^n}$ is a pure quantum power timescale such that $\sigma(t) = qt^n > t, \; 0