In my spare time, I have been studying and analysing continued fractions.
I was having a conversation with someone on Discord in a Mathematics server and he was telling me that continued fractions can be related to quantum physics. He didn't go into it too much and the concept he was describing seemed a little vague to me. I am aware that Pincherle's theorem[1] asserts there is an intimate relationship with three-term recurrence relations of the form $x_{n+1}=b_nx_n+a_nx_{n-1}$ and continued fractions, more accurately with their partial convergents, given that this recurrence relation has a minimum if a related cfrac converges. But I am not quite the physicist myself to compare physics with the properties of cfracs.
Although I can go on Google and do some research on this, I figured it might serve useful to have a post on this in this beta community, but I do apologise if it is too broad or open-ended and thus upsets any regulations here.
Any thoughts on this?
References
[1] Pincherle, S. (1894). Delle funzioni ipergeometriche e di varie questioni ad esse attinenti. Giorn. Mat. Battaglini. 32:209–29
[2] Parusnikov, V. I. A Generalization of Pincherle's Theorem to k-Term Recursion Relations. Math Notes 2005, 78 (5-6), 827–840. DOI: 10.1007/s11006-005-0188-7.