The closed span of some Exponential system $E_{\mathrm{\Lambda }}$ in the spaces $L^p(\gamma ,\beta )$, properties of a Biorthogonal family to $E_{\mathrm{\Lambda }}$ in $L^2(\gamma ,\beta )$, Moment problems, and a differential equation of Carleson

A set of complex numbers $\mathrm{\Lambda }=\{{\lambda }_n,{\mu }_n{\}}_{n=1}^{\mathrm{\infty }}$ with multiple terms $$\{{\lambda }_n,{\mu }_n{\}}_{n=1}^{\mathrm{\infty }}:=\{\underset{{\mu }_1-times}{\underbrace{{\lambda }_1,{\lambda }_1,\dots ,{\lambda }_1}},\underset{{\mu }_2-times}{\underbrace{{\lambda }_2,{\lambda }_2,\dots ,{\lambda }_2}},\dots ,\underset{{\mu }_k-times}{\underbrace{{\lambda }_k,{\lambda }_k,\dots ,{\lambda }_k}},\dots \}$$ is said to belong to the $\mathbf{A}\mathbf{B}\mathbf{C}$ class if it satisfies three conditions: $(\mathbf{A})$ ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n\mathrm{/}\mathrm{\mid }{\lambda }_n\mathrm{\mid }<\mathrm{\infty }$, $(\mathbf{B})$ ${\sup }_{n\in \mathbb{N}}\mathrm{\mid }\arg {\lambda }_n\mathrm{\mid }<\pi \mathrm{/}2$, $(\mathbf{C})$ $\mathrm{\Lambda }$ is an interpolating variety for the space of entire functions of exponential type zero. Assuming that $\mathrm{\Lambda }\in \mathbf{A}\mathbf{B}\mathbf{C}$, we characterize in the spirit of the Müntz-Szász theorem, the closed span of its associated exponential system $$E_{\mathrm{\Lambda }}:=\{x^k{e}^{{\lambda }_{n}x}:n\in \mathbb{N},k=0,1,2,\dots ,{\mu }_n-1\}$$ in the Banach spaces $L^p(\gamma ,\beta )$, where $-\mathrm{\infty }<\gamma <\beta <\mathrm{\infty }$ and $p\ge 1$. Related to $E_{\mathrm{\Lambda }}$, we explore the properties of its unique biorthogonal sequence $$r_{\mathrm{\Lambda }}=\{r_{n,k}:n\in \mathbb{N},k=0,1,\dots ,{\mu }_n-1\}\subset \overline{\text{span}}(E_{\mathrm{\Lambda }})$$ in $L^2(\gamma ,\beta )$. As a result, we find a solution to the Moment problem $${\int }_{\gamma }^{\beta }f(t)\cdot t^k{e}^{\overline{{\lambda }_n}t}dt=d_{n,k},\mathrm{\forall }n\in \mathbb{N}\text{and}k=0,1,\dots ,{\mu }_n-1,d_{n,k}=O(e^{a\mathrm{\Re }{\lambda }_n})fora<\beta \mathrm{.}$$ Finally, we characterize the solution space of a differential equation of infinite order, studied by L. Carleson.