This is for a recreational math project which I hope to turn into a video.
If you start with a sequence $f_n$, finding the difference of that sequence $f_n-f_{n-1}$ is analogous to taking the derivative of a continuous function $f(t)$. Similarly, finding the sum of that sequence $\sum_{k=0}^n{f_k}$ is analogous to taking an integral of a continuous function $f(t)$.
If you start with a continuous function $f(t)$ and create a series of sequences sampling it at progressively smaller intervals $dt$, and modify the sum and difference formulae to multiply and divide by dt respectively, $\lim_{dt\to 0}$ will be the integral/derivative of $f(t)$.
The sum formula is equivalent to a discrete convolution by the sequence $1, 1, 1...$ and the difference formula is equivalent to a discrete convolution by the sequence $1, -1, 0, 0, 0...$. These convolutions cancel each other out much the same way integrals and derivatives do.
My thought was, can this be used to define the fractional-order derivative and integral operators as limits? I took the sum convolution sequence $1, 1, 1, 1...$, translated it to the power series $1+x+x^2+x^3...={1\over{1-x}}$, and took the root $\sqrt{1\over {1-x}}$. Plugging $\sqrt{1\over {1-x}}$ into Wolfram produced the power series $1+{x\over 2}+{3x^2\over 128}+{5x^3\over 16}+{35x^4\over 128}+{63x^5\over 256}+{231x^6\over 1024}+{429x^7\over 2048}+{6435x^8\over 32768}...$ which I then translated back to the convolution sequence ${1\over 1},{1\over 2},{3\over 8},{5\over 16},{35\over 128},{63\over 256},{231\over 1024},{429\over 2048},{6435\over 32768}...$. I found the full numerator and denominator sequences on OEIS. Surprisingly, convoluting by this new sequence interpolated between sums, with $2n$ new sequence convolutions equivalent to $n$ sum convolutions. The same thing worked for the root of the difference polynomial $\sqrt{(1-x)}$.
Now, the big question is, can you define an operation on a continuous function by sampling sequences at intervals $dt$, adding a factor to the half-sum sequence to compensate for $dt$, and taking the limit of the convolutions as $dt$ approaches zero? And, would that function be equivalent to half an integral?
I've been working above my knowledge so far and relying heavily on resources like Wolfram and OEIS without really understanding how they're reaching their answers. My next step would be to precisely define the continuous limiting convolution I'm thinking of, and try and derive some rules for it (like integration/differentiation rules) if possible, but I've reached a point where I've no idea how to continue on paper, or even how to ask a good calculator, so I'm stuck.
Does anyone have advice on what techniques to look into learning so I can approach this problem myself? If not, are there computer resources that could do the work for me?