Pade approximants of special functions

摘要

Let f_γ(x) = ∑~∞_(k=0) T_k(x) /(γ)_k, where (γ)_k =γ(γ+1) … (γ+k-1) and T_k (x) = cos (k arccos x) are Pade-Chebyshev polynomials. For such functions and their Pade-Chebyshev approximations π_(n,m)~(ch)(x; f_γ), we find the asymptotics of decreasing the difference f_γ(x) - π~(ch)_(n,m)(x; f_γ) in the case where 0 ≤ m ≤ m(n), m(n) = o (n), as n → ∞ for all x ∈[-1, 1]. Particularly, we determine that, under the same assumption, the Pade-Chebyshev approximations converge to f_γ uniformly on the segment [-1, 1] with the asymptotically best rate.