Gamma function

We define $$p_f(t,s)=e_{\frac{f}{\mathrm{id}}}(t,s),$$ where $\mathrm{id}$ denotes the identity map $\mathrm{id} \colon \mathbb{T} \rightarrow \mathbb{T}$ and $e_{\cdot}$ denotes the time scale exponential. Define the operations $$f \boxplus_{\mu} g := f+g+\dfrac{1}{\mathrm{id}}fg\mu$$ and $$f \boxminus_{\mu} g := \dfrac{(f-g)\mathrm{id}}{\mathrm{id} + g \mu}.$$ With these definitions, we have the gamma operator $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$