How to prove Chepyshev's polynomials generating function identity?

In summary, Chepyshev's polynomials generating function identity can be proved using mathematical induction, and it is significant in mathematics due to its applications in various fields. It can also be derived from other identities, but using mathematical induction is more efficient. This identity has many real-world applications and, with a solid understanding of mathematical induction, it can be easily understood and applied.
  • #1
aligator123
35
0
(1-xt)/(1-2xt+t^2)=sum(Tn(x)t^n)
How can i prove this equation?

Could you give me a hint or suggestion?
 
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  • #2
?
 
  • #3
What's the definition of Chebyshev's polynomials ? I'd also hint that you can use the ODE which the polynomials solve exactly.

Daniel.

P.S. The "p" in "Chepyshev" is actually reversed upside down and it's a "b".
 
Last edited:
  • #4
Tn(x)=y
(1-x^2)y" - xy' + (n^2)y=0
Am i have to solve this equation?
 
  • #5
i know how to write Chebyshev but it is just a mistake cause of fast typing.
 

1. How do we prove Chepyshev's polynomials generating function identity?

Chepyshev's polynomials generating function identity can be proved using mathematical induction. This method involves proving the identity for the base case, typically n=0 or n=1, and then showing that if the identity holds for a certain value of n, it also holds for n+1. This process is repeated until the identity is proved for all values of n.

2. What is the significance of Chepyshev's polynomials generating function identity?

Chepyshev's polynomials generating function identity is significant in mathematics as it relates to the Chepyshev polynomials, which have various applications in fields such as physics, engineering, and computer science. This identity allows us to express the Chepyshev polynomials in a compact and elegant form, making it easier to use them in calculations and solving problems.

3. Can Chepyshev's polynomials generating function identity be derived from other identities?

Yes, Chepyshev's polynomials generating function identity can be derived from other identities such as the binomial theorem and the identity for the sum of a geometric series. However, the process may be lengthy and involve multiple steps, so it is often more efficient to use mathematical induction to prove the identity.

4. Are there any real-world applications of Chepyshev's polynomials generating function identity?

Yes, Chepyshev's polynomials generating function identity has many real-world applications. For example, it is used in signal processing to design filters and in numerical analysis to approximate functions. It also has applications in probability theory, statistics, and error analysis.

5. Is Chepyshev's polynomials generating function identity a difficult concept to understand?

Chepyshev's polynomials generating function identity may seem daunting at first, but with a solid understanding of mathematical induction and some practice, it can be easily understood and applied. It is a fundamental concept in mathematics and is often covered in undergraduate courses in algebra or calculus.

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