Absolutely monotonic functions related to the asymptotic formula for the complete elliptic integral of the first kind

Let $\mathcal{K}\left(x\right)$ be the complete elliptic integral of the first kind and \begin{equation*} $\mathcal{G}$_{p}\left( x\right) =e^{$\mathcal{K}$\left( \sqrt{x} \right) }-\frac{p}{\sqrt{1-x}} \end{equation*} for $p\in \mathbb{R}$ and $x\in (0,1)$. In this paper we find the necessary and sufficient conditions for the functions $\pm {\mathcal{G}}_p^{\left(k\right)}\left(x\right)$ ($k\in \mathbb{N}\cup \left\{0\right\}$) to be absolutely monotonic on $(0,1)$, which extend previous known results and yield several new functional inequalities involving the complete elliptic integral of the first kind. More importantly, we provide a new method to deal with those absolute monotonicity problem by proving the monotonicity of a sequence generated by the coefficients of the power series of ${\mathcal{G}}_p\left(x\right)$.