Generalized Arakawa-Kaneko zeta functions

Let $p,x$ be real numbers, and $s$ be a complex number, with $\mathrm{\Re }(s)>1-r$, $p\ge 1$, and $x+1>0$. The zeta function $Z_p^{\mathbf{\alpha }}(s;x)$ is defined by $$Z_p^{\mathbf{\alpha }}(s;x)=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}\frac{e^{-xt}}{e^t-1}L{i}_{\mathbf{\alpha }}\left(\frac{1-e^{-t}}{p}\right)t^{s-1}dt,$$ where $\mathbf{\alpha }=({\alpha }_1,\dots ,{\alpha }_r)$ is a $r$-tuple positive integers, and $L{i}_{\mathbf{\alpha }}(z)$ is the one-variable multiple polylogarithms. Since $Z_1^{\mathbf{\alpha }}(s;0)=\xi (\mathbf{\alpha };\mathbf{s})$, we call this function as a generalized Arakawa-Kaneko zeta function. In this paper, we investigate the properties and values of $Z_p^{\mathbf{\alpha }}(s;x)$ with different values $s$, $x$, and $p$. We then give some applications on them.