Solutions and existence results for difference equations of fermat typeSolutions and existence results for fifference equations of fermat typeN. Sarkar, P. Das

Our paper focuses on exploring the existence and characteristics of solutions for various complex difference equations of Fermat-type such as f(z)2+α(z)2(eP(z))2f(z+c)2=Q(z)e2β(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(z)^2 + \alpha (z)^2 (e^{P(z)})^2 f(z+c)^2 = Q(z) e^{2\beta (z)} $$\end{document} and f(z)2+α(z)2(eP(z))2(Δcf(z))2=Q(z)e2β(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(z)^2 + \alpha (z)^2 (e^{P(z)})^2 (\Delta _c f(z))^2 = Q(z) e^{2\beta (z)} $$\end{document}, where α(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha (z) $$\end{document}, β(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta (z) $$\end{document}, P(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P(z) $$\end{document}, and Q(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Q(z) $$\end{document} are non-zero polynomials in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {C}} $$\end{document} and c∈C\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ c \in {\mathbb {C}} {\setminus } \{0\} $$\end{document}. Our findings represent significant advancements over previous theorems established by Long Jian-ren and Qin Da-zhuan (Appl Math J Chin Univ 39:69-88, 2024), particularly in terms of both the existence and explicit forms of solutions. Moreover, some examples are provided to strengthen our results.