These steps raise numerous questions.
What is the reasoning behind choosing $\delta _Nx\equiv [x-a]/N$ ? This seems almost arbitrary. I understand that $a$ and $x$ eventually becomes the limits for the Riemann-Louiville fractional integral/derivative, but how is this known in advance? I also see that this expression is identical to the expression denoting partition length for the Riemann-sum definition of the integral, which through a very intuitive process creates integral limits of $a$ and $x$ through using that partition length expression, but what specifically is the connection between the two derivations and the reasoning behind this choice of a substitution?
How is the upper bound of the summation changed from $n$ (which denotes the order of the derivative) into $N-1$?
At the end, the finite sum becomes an infinite sum. How is it possible for such a substitution to be justified? It seems absurd for a seemingly arbitrary substitution that changes the substance of the whole expression, namely changing the upper bound of the summation from finite to infinite, to be valid. Although I do see that some of the expression does stay equivalent as it converges to the same value due to the limit, such as $\lim_{\delta _Nx \to 0} f(x-j\delta _Nx) = \lim_{N \to \infty } f(x-j[\frac{x-a}{N}])$
Thank you very much for your time spent considering this.