T (n+1)(x)-2xT n(x)+T (n-1)(x)=0

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Theorem

The following formula holds for $n=0,1,2,\ldots$: $$T_{n+1}(x)-2xT_n(x)+T_{n-1}(x)=0,$$ where $T_n$ denotes Chebyshev polynomials of the first kind.

Proof

References