The expansion of functions in ultraspherical polynomials

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 T n ( x ) of the first kind, corresponding to λ = 0; the Legendre polynomials P n ( x ) for which λ = ½; and the Chebyshev polynomials U n ( x ) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since .