Delta gk

Let $\mathbb{T}$ be a time scale and let $t,s \in \mathbb{T}$. The $g_k$ monomials are defined by the recurrence $$\left\{ \begin{array}{ll} g_0(t,s)=1 \\ g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau. \end{array} \right.$$
 $\mathbb{T}=$ $g_k(t,t_0)=$ $\mathbb{R}$ $g_k(t,t_0)=\dfrac{(t-t_0)^k}{k!}$ $\mathbb{Z}$ $g_k(t,t_0)=$ $h\mathbb{Z}$ $g_k(t,t_0)=$ $\mathbb{Z}^2$ $g_k(t,t_0)=$ $\overline{q^{\mathbb{Z}}}, q > 1$ $g_k(t,t_0)=$ $\overline{q^{\mathbb{Z}}}, q < 1$ $g_k(t,t_0)=$ $\mathbb{H}$ $g_k(t,t_0)=$
 $\Delta$-special functions on time scales $\cos_p$ $\cosh_p$ $e_p$ $g_k$ $h_k$ $\sin_p$ $\sinh_p$