Proof: Generating Functions for Chebyshev Polynomials (1-x^2)/(1-2tx+t^2)=∑^∞_n=0(U_(n+1) (x)t^n)

In this video, we present a step-by-step proof of the generating function for Chebyshev polynomials of the second kind. The generating function is given by:

\frac{1 - x^2}{1 - 2xt + t^2} = \sum_{n=0}^{\infty} U_{n+1}(x) t^n

We carefully derive this result, explaining each step of the proof and highlighting the connection between generating functions and orthogonal polynomials. This video is useful for students and researchers in mathematics who are studying **special functions, polynomial theory, or mathematical analysis**.

👉 Topics Covered:

* Definition of Chebyshev polynomials of the second kind $U_n(x)$
* Generating function approach
* Step-by-step derivation of the formula

📚 Perfect for learners in mathematics, applied mathematics, and physics.
Keywords / Tags (SEO):
Chebyshev polynomials, generating functions, orthogonal polynomials, special functions, proof of generating function, Chebyshev polynomials of the second kind, mathematics lecture, math proof, polynomial theory, approximation theory, applied mathematics, higher mathematics, infinite series, functional analysis.

Видео Proof: Generating Functions for Chebyshev Polynomials (1-x^2)/(1-2tx+t^2)=∑^∞_n=0(U_(n+1) (x)t^n) канала Afghan Mathematics Academy