{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-study-on-partial-dynamic-equation-on-time","title":"A study on partial dynamic equation on time scales involving derivatives of polynomials","arxiv_id":"1608.00801","date":"2016-08-02","proceeding":null,"authors":["Petro Kolosov"],"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.","url_abs":"https://arxiv.org/abs/1608.00801v8","url_pdf":"https://arxiv.org/pdf/1608.00801v8.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-study-on-partial-dynamic-equation-on-time","repo_url":"https://github.com/kolosovpetro/astudyondynamicequations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}