On process, and on the derivatives of the powers of sine

Nov 2019
2
0
Rozebeke, Flanders
Some background:
A few months ago, I was on holiday in Prague.
And I saw a nice clock there.
Which led me to think about the equation of time.
Which led me to think about \(\displaystyle y = x + e\,\sin y\)
Which led me to think about \(\displaystyle \frac{e^k}{k!} \frac{\partial^{k-1}}{\partial x^{k-1}} [\sin^k x]\)
Which led me to think about the derivatives of the powers of \(\displaystyle \sin\).
I had some time there in Prague, waiting for a guided tour to start, so I took out a notebook and played a bit with these notions.


I found some nice expressions for the derivatives of the powers of \(\displaystyle \sin\).
Later I realised that analogue expressions hold for \(\displaystyle \cos\), \(\displaystyle \sinh\), and \(\displaystyle \cosh\).

So the past few weekends I took some time to write it all out.
And then I did some typesetting in LaTeX.
A couple of days ago, I submitted my work to ArXiV and last night it got announced:
On the derivatives of the powers of trigonometric and hyperbolic sine and cosine
https://arxiv.org/pdf/1911.01386.pdf

I know this is not earth-shattering
but as far as I can tell, this is original (except for the bit that was already published by Qi (2015), which I properly cited)
and I personally find it kind of neat and quite beautiful. Do you agree? Or not? Why?

My real question is: Where do I go from here?
How do I bring this to the attention of the kind of people who are interested in this kind of thing?
Where and how do I invite feedback, both on substance and on presentation?

Is anybody helped if I seek to get this published in a peer-reviewed journal?
Which one? (How and based on what criteria do I choose a journal?) (Is getting published expensive?)