Chebyshev series expansion of inverse polynomials

The Chebyshev series expansion ∑ ′ n = 0 ∞ a n T n ( x ) of the inverse of a polynomial ∑ j = 0 k b j T j ( x ) is well defined if the polynomial has no roots in [ - 1 , 1 ] . If the inverse polynomial is decomposed into partial fractions, the a n are linear combinations of simple functions of the polynomial roots. Also, if the first k of the coefficients a n are known, the others become linear combinations of these derived recursively from the b j ʹs. On a closely related theme, finding a polynomial with minimum relative error towards a given f ( x ) is approximately equivalent to finding the b j in f ( x ) / ∑ 0 k b j T j ( x ) = 1 + ∑ k + 1 ∞ a n T n ( x ) ; a Newton algorithm produces these if the Chebyshev expansion of f ( x ) is known.