The expansion of functions in ultraspherical polynomials

Abstract

The ultraspherical polynomial (x) of degree n and order λ is defined by for n = 0, 1, 2, …. Of these polynomials, the most commonly used are the Chebyshev polynomials Tn(x) of the first kind, corresponding to λ = 0; the Legendre polynomials Pn(x) for which λ = ½; and the Chebyshev polynomials Un(x) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since.