Chebyshev Polynomial of the First Kind Tₙ(x) Calculator

Enter the degree n and x to evaluate the first-kind Chebyshev polynomial Tₙ(x) with the three-term recurrence. Shows neighboring degrees, the derivative, a coefficient table, and a graph.

Input

Evaluate the first-kind Chebyshev polynomial Tₙ(x) with the three-term recurrence. Enter the degree n and x.

Degree n

Integer from 0 to 200

Value of x

Any real number (cosine form holds for −1 to 1)

Result

Value of Tₙ(x) (n = 5)

0.5

Value at x = 0.5

Degree n

T(n−1)(x) (n = 5)

-0.5

T(n+1)(x) (n = 5)

Derivative Tₙ'(x) (n = 5)

-5

Graph of Tₙ(x) (n = 5)

The horizontal axis is x (−1 to 1) and the vertical axis is Tₙ(x). The orange dot marks the entered x.

Coefficient table of Tₙ(x) (n = 5)

Coefficients for each power of x, ordered from the highest power.

Power of x	Coefficient
5	16
3	-20
1	5

How it works

The first-kind Chebyshev polynomial is defined by the three-term recurrence T0(x)=1, T1(x)=x, and T(n+1)(x)=2x·Tn(x)−T(n-1)(x). This tool advances the recurrence from degree 0 and returns Tn(x) together with the neighbors T(n-1)(x) and T(n+1)(x).
On the interval −1 ≤ x ≤ 1 the identity Tn(x)=cos(n·arccos x) holds, so the value always stays between −1 and 1. The graph is drawn over this interval.
The derivative uses the relation with the second-kind Chebyshev polynomial Un, namely the derivative of Tn equals n·U(n-1)(x), where U also follows the same three-term recurrence.
The coefficient table lists the coefficient of each power of x in the expansion of Tn(x), ordered from the highest power. For example T5(x)=16·x^(5)−20·x^(3)+5·x.
Enter the degree n as an integer from 0 to 200. Values outside this range or non-integers produce an error.

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