Padé approximants of the Mittag-Leffler functions

The asymptotic form of the Hermite-Padé approximations for a system of mittag-leffler functions

“…In this connection there is of interest the problem of proof of analogs of paper [37] for the system of exponents {e λ j z } k j=1 and the system of Mittag-Leffler functions {F j γ (z)} k j=1 , where λ j are various real or imaginary values. Note that for n + m → +∞ the asymptotic behavior of Pad´e approximations π n, m (z, F 1 γ ) for Mittag-Leffler functions F 1 γ (z) (case k = 1) was investigated in [39].…”

Discrete-space time-fractional processes

“…The authors begin with an integral representation of the function that can be used to calculate values to arbitrary accuracy in a finite element setting, then using these exact values of the function for a set of points x in [0.1, 15] the coefficients of the approximant are solved for. These results are superior to those of [48] because it is the exact values of the function at intermediate values being approximated with a rational function, compared to just the Taylor series. Nonetheless, we find it interesting to derive rational approximations in a different manner.…”

“…Padé approximations for the Mittag-Leffler function are discussed in Starovoitov and Starovoitova [48], where the authors show that the approximants serve uniformly on the compact set {|z| ≤ 1}. While the Padé approximation is better than a truncated Taylor series, it is not necessarily compatible with the asymptotic behaviour for large arguments.…”

Padé approximants of special functions

“…In [28], it was established (see [28,Theorem 1]) that, for nonnegative integers n and m, the Padé-Baker approximations π n, m (x; Φ γ ) exist and belong to the class R n, m , and, uniformly over all |z| 1 and m, m n, as n → ∞,…”

“…A method used in the proof of Theorem 1.2 is different from Bernstein's one and is based on the connection between the Padé-Chebyshev approximations of a function f and the trigonometric Padé-Baker approximations of the induced function f (cos x) and on the results of work [28]. Therefore, we consider the approximational properties of the trigonometric Padé-Baker operator in § 2.…”