# Chebyshev T

From specialfunctionswiki

Chebyshev polynomials of the first kind are orthogonal polynomials defined for $n=0,1,2,\ldots$ and $-1 \leq x \leq 1$ by $$T_n(x) = \cos(n \mathrm{arccos}(x)),$$ where $\cos$ denotes cosine and $\mathrm{arccos}$ denotes arccos.

# Properties

T_(n+1)(x)-2xT_n(x)+T_(n-1)(x)=0

Orthogonality of Chebyshev T on (-1,1)

Relationship between Chebyshev T and hypergeometric 2F1

Relationship between Chebyshev T and Gegenbauer C

T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n

T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)

# References

- 1968: W.W. Bell:
*Special Functions for Scientists and Engineers*... (previous) ... (next): $(7.1)$ - 1978: T.S. Chihara:
*An Introduction to Orthogonal Polynomials*... (previous) ... (next) $(1.4)$ (*note: calls them Tchebichef polynomials of the first kind*)

**Orthogonal polynomials**