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In this paper, we propose fully discrete analogues of a generalizedsine-Gordon (gsG) equation $u_{t x}=\left(1+\nu \partial_x^2\right) \sin u$.The bilinear equations of the discrete KP hierarchy and the proper definitionof discrete hodograph transformations are the keys to the construction. Then wederive semi-discrete analogues of the gsG equation from the fully discrete gsGequation by taking the temporal parameter $b\rightarrow0$. Especially, onefull-discrete gsG equation is reduced to a semi-discrete gsG equation in thecase of $\nu=-1$ (Feng {\it et al. Numer. Algorithms} 2023). Furthermore,$N$-soliton solutions to the semi- and fully discrete analogues of the gsGequation in the determinant form are constructed. Dynamics of one- andtwo-soliton solutions for the discrete gsG equations are discussed with plots.We also investigate the reductions to the sine-Gordon (sG) equation and theshort pulse (SP) equation. By introducing an important parameter $c$, wedemonstrate that the gsG equation reduces to the sG equation and the SPequation, and the discrete gsG equation reduces to the discrete sG equation andthe discrete SP equation, respectively, in the appropriate scaling limit. Thelimiting forms of the $N$-soliton solutions to the gsG equation also correspondto those of the sG equation and the SP equation.

Integrable discretizations for a generalized sine-Gordon equation and the reductions to the sine-Gordon equation and the short pulse equation