Chebyshevin

Chebyshevin is a family of orthogonal polynomials in one variable, introduced as a two-parameter generalization of Chebyshev polynomials of the first kind. The polynomials are denoted C_n^{(α,β)}(x) and depend on two shape parameters α and β, with α > -1 and β > -1, governing their weighting on the interval [-1,1].

They are orthogonal on [-1,1] with respect to the weight w_{α,β}(x) = (1 - x)^α (1 + x)^β. The family is defined by a three-term recurrence: C_0(x) = 1, C_1(x) = x, and for n ≥ 1, C_{n+1}(x) = (2x) C_n(x) - λ_n(α,β) C_{n-1}(x), where λ_n(α,β) = (n - 1 + α)(n - 1 + β) / ((n + α)(n + β)). This normalization ensures that, for α = β = -1/2, the polynomials reduce to the classical Chebyshev polynomials of the first kind up to a constant factor.

Chebyshevin polynomials are closely related to Jacobi polynomials P_n^{(α,β)}(x) and inherit many of their structural properties, including a real, simple zero set contained in (-1,1) and interlacing behavior as degree increases. They can be expressed in terms of Jacobi polynomials and share comparable asymptotic and orthogonality features.

Applications of the Chebyshevin family appear in approximation theory and numerical analysis. They are used in spectral methods for differential equations, polynomial approximation with endpoint weighting, and data smoothing where a tunable weight at the endpoints helps tailor convergence and conditioning.