Chebyshev Polynomial of the Second Kind Uₙ(x) Calculator

Enter degree n and x to evaluate the Chebyshev polynomial of the second kind Uₙ(x) by recurrence. Shows neighboring degrees, derivative, coefficient table, and a plot.

Input

Evaluate the Chebyshev polynomial of the second kind Uₙ(x) using the three-term recurrence U(n+1)=2x·Un−U(n-1). Enter the degree n and x.

Degree n

Integer of 0 or more (max 200)

Value of x

Any real number

Result

Value of U_5(x) for n = 5

at x = 0.5

Degree n

Previous U(n-1)(x)

-1

Next U(n+1)(x)

Derivative Un'(x)

-8

Plot of U_5(x) over -1 to 1

Behavior of Un(x) over the interval -1 to 1. The orange dot marks the entered value of x.

Coefficients of U_5(x) by power of x

Each row shows a power of x and the coefficient of that term. Terms with a zero coefficient are omitted.

Power of x	Coefficient
5	32
3	-32
1	6

How it works

The Chebyshev polynomial of the second kind starts from U0(x)=1 and U1(x)=2x, then advances with the three-term recurrence U(n+1)(x)=2x·Un(x)−U(n-1)(x).
On the interval -1 to 1, where x equals cosθ, the identity Un(cosθ)=sin((n+1)θ)/sinθ holds and the curve oscillates. On this interval the absolute value of Un(x) never exceeds n+1.
The derivative uses (x²−1)Un'(x)=n·x·Un(x)−(n+1)·U(n-1)(x), with the endpoints x=±1 filled in by closed forms such as Un'(1)=n(n+1)(n+2)/3.
The coefficient table lists the coefficients of each power of x in the expansion of Un(x), built as integers through the recurrence applied to the coefficients themselves.
When degree n is large and the absolute value of x exceeds 1, Un(x) grows rapidly and floating point rounding error becomes significant.

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