Gegenbauer Polynomial Calculator

Enter degree n, parameter lambda, and x to evaluate the ultraspherical polynomial C_n^lambda(x) via recurrence, with neighbor values and a curve.

Input

Enter the degree n, parameter lambda, and x to evaluate the Gegenbauer (ultraspherical) polynomial C_n^lambda(x) by recurrence.

Degree n

Integer from 0 to 200

Parameter lambda

lambda greater than −1/2 and not 0 (0.5 gives Legendre, 1 gives Chebyshev 2nd kind)

Variable x

Any real number. The orthogonality interval is −1 to 1

C_n^lambda(x) at n=3, lambda=1, x=0.5

-1

Degree n

Parameter lambda

Neighbor C_(n−1)^lambda(x)

Neighbor C_(n+1)^lambda(x)

-1

Curve of C_n^lambda(x) on −1 to 1

Computes the Gegenbauer (ultraspherical) polynomial C_n^lambda(x) with the three-term recurrence: C_0^lambda(x)=1, C_1^lambda(x)=2 lambda x, and n C_n^lambda(x)=2(n+lambda−1)x C_(n−1)^lambda(x)−(n+2 lambda−2)C_(n−2)^lambda(x).
The parameter lambda corresponds to the weight (1−x^2)^(lambda−1/2). This tool accepts lambda greater than −1/2 and not equal to 0.
Special case: when lambda=1/2 the polynomial reduces to the Legendre polynomial P_n(x).
When lambda=1 it equals the Chebyshev polynomial of the second kind U_n(x). The limit lambda toward 0 (with normalization) relates to the Chebyshev polynomial of the first kind.
They are orthogonal on −1 to 1 with respect to the weight (1−x^2)^(lambda−1/2). Values can grow quickly with degree n, so the chart auto-scales its vertical axis.

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