• ### Minimal Completion of the Lie Algebra arising from Fractional Calculus

Abstract: We showed the group $G_\mathbb{R}$ and Lie algebra $\mathfrak{g}_\mathbb{R}$ generated by $\{\frac{d^a}{dx^a}|a\in\mathbb{R}\}$ and $\{x^a|a\in\mathbb{R}\}$ and their generating operators $\log(\frac{d}{dx})$ and $\log x$ have unique maximal normal subgroup $\mathsf{A}_\mathbb{R}$ and maximal ideal $\mathfrak{a}$. It was shown taking suitable completion $\bar{\mathfrak{a}}$, $\vartheta$; $\vartheta g=g^{-1}\frac{dg}{dx}$ gives a monomorphism from $\mathsf{A}_\mathbb{R}$ into $\bar{\mathfrak{a}}$([6].). But to search “good” completion is remained as a problem.
In this paper, we propose the good completion (in some sense, minimal) is $\mathfrak{a}_{\exp}$; $\mathfrak{a}_{\exp}=\{\sum_n c_n\Psi^{(n)}(1+s)|\sum_n c_nx^n\in\mathrm{Exp}(\mathbb{C})\}.$
Here $\mathrm{Exp}(\mathbb{C})$ is the space of finite exponential type functions. Corresponding extension of $\mathsf{A}_\mathbb{R}$ is also proposed.