How to prove Chepyshev's polynomials generating function identity?

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Homework Help Overview

The discussion revolves around proving the generating function identity for Chebyshev's polynomials, specifically the equation (1-xt)/(1-2xt+t^2)=sum(Tn(x)t^n.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the definition of Chebyshev's polynomials and consider the ordinary differential equation (ODE) that these polynomials satisfy. There is a request for hints or suggestions on how to approach the proof.

Discussion Status

The discussion is ongoing, with participants raising questions about definitions and potential methods for proving the identity. Some guidance has been offered regarding the use of the ODE related to Chebyshev's polynomials, but no consensus has been reached on a specific approach.

Contextual Notes

There is a mention of a typographical error in the spelling of "Chebyshev," which does not appear to affect the mathematical content of the discussion.

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(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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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:
Tn(x)=y
(1-x^2)y" - xy' + (n^2)y=0
Am i have to solve this equation?
 
i know how to write Chebyshev but it is just a mistake cause of fast typing.
 

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