They both are are equally "good". Unlike conventional calculus in discrete calculus there are two equally valid differentiation operators with little reason to prefere one over the other - forward difference $\Delta f(x)$ and backward difference $\nabla f(x)$. In discrete multiplicative calculus there are also two similar operators - discrete multiplicative forward difference $\frac{f(x+1)}{f(x)}$ and discrete multiplicative backward difference $\frac{f(x)}{f(x-1)}$. They both have their respective inverse operators - forward discrete multiplicative integral and backward discrete multiplicative integral. So the $\Gamma(x)$ is the forward discrete multiplicative integral of $f(x)=x$ and $\Gamma(x+1)=x!$ is the backward discrete multiplicative integral of the same function.

In the scientific applications there is a preference to using forward difference rather than backward difference I think because if is possible to find forward differences of arbitrary order of a function defined on only positive integers. Among other considerations, it allows to represent in the form of Newton series a function which is defined only on natural numbers (Newton series with backward difference would require the function to be defined on negative integers).

They both are are equally "good". Unlike conventional calculus in discrete calculus there are two equally valid differentiation operators with little reason to prefere one over the other - forward difference $\Delta f(x)$ and backward difference $\nabla f(x)$. In discrete multiplicative calculus there are also two similar operators - discrete multiplicative forward difference $\frac{f(x+1)}{f(x)}=\exp(\Delta \ln f(x))$ and discrete multiplicative backward difference $\frac{f(x)}{f(x-1)}=\exp(\nabla \ln f(x))$. They both have their respective inverse operators - forward discrete multiplicative integral and backward discrete multiplicative integral. So the $\Gamma(x)$ is the forward discrete multiplicative integral of $f(x)=x$ and $\Gamma(x+1)=x!$ is the backward discrete multiplicative integral of the same function.

In the scientific applications there is a preference to using forward difference rather than backward difference I think because if is possible to find forward differences of arbitrary order of a function defined on only positive integers. Among other considerations, it allows to represent in the form of Newton series a function which is defined only on natural numbers (Newton series with backward difference would require the function to be defined on negative integers).

They both are are equally "good". Unlike conventional calculus in discrete calculus there are two equally valid differentiation operators with little reason to prefere one over the other - forward difference $\Delta f(x)$ and backward difference $\nabla f(x)$. In discrete multiplicative calculus there are also two similar operators - discrete multiplicative forward difference $\frac{f(x+1)}{f(x)}$ and discrete multiplicative backward difference $\frac{f(x)}{f(x-1)}$. They both have their respective inverse operators - forward discrete multiplicative integral and backward discrete multiplicative integral. So the $\Gamma(x)$ is the forward discrete multiplicative integral of $f(x)=x$ and $\Gamma(x+1)=x!$ is the backward discrete multiplicative integral of the same function.

In the scientific applications there is a subtle preference to using forward difference rather than backward difference but this can vary from one author to another.

They both are are equally "good". Unlike conventional calculus in discrete calculus there are two equally valid differentiation operators with little reason to prefere one over the other - forward difference $\Delta f(x)$ and backward difference $\nabla f(x)$. In discrete multiplicative calculus there are also two similar operators - discrete multiplicative forward difference $\frac{f(x+1)}{f(x)}$ and discrete multiplicative backward difference $\frac{f(x)}{f(x-1)}$. They both have their respective inverse operators - forward discrete multiplicative integral and backward discrete multiplicative integral. So the $\Gamma(x)$ is the forward discrete multiplicative integral of $f(x)=x$ and $\Gamma(x+1)=x!$ is the backward discrete multiplicative integral of the same function.

In the scientific applications there is a preference to using forward difference rather than backward difference I think because if is possible to find forward differences of arbitrary order of a function defined on only positive integers. Among other considerations, it allows to represent in the form of Newton series a function which is defined only on natural numbers (Newton series with backward difference would require the function to be defined on negative integers).