Multidimensional Padé approximation of binomial functions: Equalities

08/01/2021

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by   Michael A. Bennett, et al.

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Let ω_0,…,ω_M be complex numbers. If H_0,…,H_M are polynomials of degree at most ρ_0,…,ρ_M, and G(z)=∑_m=0 ^M H_m(z) (1-z)^ω_m has a zero at z=0 of maximal order (for the given ω_m,ρ_m), we say that H_0,…,H_M are a multidimensional Padé approximation of binomial functions, and call G the Padé remainder. We collect here with proof all of the known expressions for G and H_m, including a new one: the Taylor series of G. We also give a new criterion for systems of Padé approximations of binomial functions to be perfect (a specific sort of independence used in applications).