chebyshev

chebyshev (n, x)

The chebyshev function returns the Chebyshev polynomial of the first kind and of degree n.

The Chebyshev polynomials satisfy the recurrence relation:

$$\displaystyle T_{n+1}(x)=2 x T_{n}(x)-T_{n-1}(x)$$

with:

$$\displaystyle \begin{array}{l} T_{0}(x)=1 \\ T_{1}(x)=x \\ T_{2}(x)=2 x^{2}-1 . \end{array}$$

The orthogonality of the polynomial is expressed by the integral:

$$\displaystyle \int_{-1}^{1} \frac{T_{n}(x) T_{m}(x)}{\sqrt{1-x^{2}}} d x=\left\{\begin{array}{ll} 0 & m \neq n \\ \\ \pi / 2 & m=n \neq 0 \\ \\ \pi & m=m=0 \end{array} .\right.$$

References

Abramowitz, M., and I.A. Stegun, "Handbook of Mathematical Functions", 446 pp., Dover, New York, 1972.

See Also

chebyshevU