What are...Chebyshev polynomials?

[@michaelchristinarichardson9660]

Your enthusiasm is adorable!! Thank you for helping me again! ❤

[@VisualMath]

Thank you so much, you are very welcome. I am glad that the video was helpful ☺

[@youw6876]

It was an amazing presentation!! Thank you! Liebe Grüsse aus Zürich! :)

[@VisualMath]

Gruss zurück nach Zürich ;-)

[@robnicolaides3070]

So beautiful! I bumped into these when working with Coxeter Groups once but didn’t realise how ubiquitous they were. Definitely going to keep my on them. I guess you might expect them to be interesting in Fourier Analysis?

[@VisualMath]

Fun: I also found them via Coxeter theory. I am not an expert on Fourier analysis. But yes, I would expect that the first kind (the T not the U) play a role. At one point I read about it, the cos-roots show up course, but I can not remember where that actually was. However, I found www.geophysik.uni-muenchen.de/~igel/Lectures/NMG/05_orthogonalfunctions.pdf which should be a good starting point.

[@shashvatshukla]

You seem to have a video on so many bits of math I need for quantum algorithms

[@VisualMath]

Hmm, that is very interesting. I heard something very similar from someone else, and they also explicitly mentioned quantum algorithm and quantum computing. Now I just need to figure out why ;-) I guess its time for me to learn more about these subjects! Anyway, I am glad that the video was helpful!

[@shashvatshukla]

@@VisualMath you sure it wasn't just me ? :P

[@VisualMath]

@@shashvatshukla Well, not 100%. But it was in real life so I give it a 99% sure ;-)

[@thomasolson7447]

All polynomials generated in this fashion have a scalar of Chebyshev Polynomials in the solutions. So a solution would be (x-c*(d_1))(x-c*(d_2)..(x-c*(d_(n-1))(x-c*(d_(n)), where c is the scalar and d is what's left over in the solution. I had a function for that, but I would have to look it up. It's surprisingly simple. Now that I think about it, it's probably not perfected because of multiple solutions. But do you see how it's weird that there is a scalar inside the solutions? Anyway, it looks like I have a lot of work to do if I want to write this paper. I still yet have to learn how to do the notation correctly. Sorry.

[@VisualMath]

Sorry, I am not sure what you mean. Can you elaborate?

[@thomasolson7447]

@@VisualMath err, sorry. I'm referring to polynomial sequences generated by Binet Formula. a*x^2+b*x+c=0 (Finbonacci Polynomials would be a=1, b=-X, c=-1). (r[1]^n-r[2]^n)/(r[1]-r[2]) Lucas Polynomials r[1]^n+r[2]^n Chebyshev First Kind (a=1, b=-2X, c=1) r[1]^n+r[2]^n Chebyshev Second Kind (a=1, b=-2X, c=1) (r[1]^n-r[2]^n)/(r[1]-r[2]) The scalar would be S=-2*x*sqrt(a*c)/b So every polynomial generated in this fashion would be (x-S*r[1])*(x-S*r[2])..(x-S*r[n]) where the roots are the roots of Chebyshev, First or Second kind. Depending on how you generated them. I suppose I could show you a paper I'm trying to write, but you're going to give me hell for bad form. Also, I lost track of what I was trying to say. I'm not a mathematician (G12 pre-calc). This paper isn't ever going to get finished. It's way too complicated for me.

[@VisualMath]

@@thomasolson7447 That is a nice way of summarizing these polynomials, thanks for sharing! And never give up!

[@thomasolson7447]

@@VisualMath I just found something interesting. x^2-2*x*X+1 X±sqrt(X^2-1) Do you see that? Try the 3 4 5 in the unit circle 3/5±i*4/5 cos(θ)+i*sin(θ) Chebyshev First Kind (a=1, b=-2X, c=1) r[1]^n+r[2]^n 2*T_n=(cos(θ)+i*sin(θ))^n+(cos(θ)-i*sin(θ))^n Is this cool? The first part (cos(θ)+i*sin(θ))^n is a typical circle function. We see this in a lot of places. The second part is the complex conjugate. We are subtracting the imaginary part, leaving behind the real part. The above polynomial has the trig identity within it. As a result, these Chebyshev polynomials pop out. Chebyshev Second Kind (a=1, b=-2X, c=1) (r[1]^n-r[2]^n)/(r[1]-r[2]) U_n=((cos(theta)+i*sin(theta))^n-(cos(theta)+i*sin(theta))^n)/(2*i*sin(theta)) What is a cool property of Lucas Numbers or Fibonacci Numbers that you like? I'll try and do it with Chebyshev Polynomials.

[@VisualMath]

@@thomasolson7447 Not sure what you mean. What and I supposed to see from x^2-2*x*X+1 and its roots X±sqrt(X^2-1)? Hmm, in any case, I like all properties of Lucas and Fibonacci numbers ;-) Just kidding, what about the various nice matrix descriptions: en.wikipedia.org/wiki/Fibonacci_number#Matrix_form