Padé approximants of special functions
Let $ {f_{\gamma }}(x) = \sum\nolimits_{{k = 0}}^{\infty } {{{{T_k (x)}} \left/ {{{{\left( \gamma \right)}_k}}} \right.}} $
 , 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 $ \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $
 , we find the asymptotics of decreasing the difference $ {f_{\gamma }}(x) - \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $
 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.