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Distribution of the zeros of Hermite-Padé polynomials. (English. Russian original) Zbl 1346.42034
Russ. Math. Surv. 70, No. 6, 1179-1181 (2015); translation from Usp. Mat. Nauk 70, No. 6, 211-212 (2015).
The authors study a multivalued analytic function defined by \[ f(z)=\prod_{j=1}^q \left({z-e_{2j-2}\over z-e_{2j}}\right)^{\alpha}, \] with \(2\alpha\in\mathbf C\setminus\mathbf Z\) and \(-1=e_1<\cdots <e_{2q}=1\); let \(E=\cup_{j=1}^q\,[e_{2j-1},e_{2j}]\). The main result is:
Theorem 1. Let \(f\) be a function as defined above and let \(Q_{n,j}\,(j=0,1,2)\) be the corresponding Hermite-Padé polynomials, each of degree \(n\), for the system \(\{1,f,f^2\}\). Then the zeros of these polynomials have a limiting distribution which coincides with the equilibrium measure \(\eta_F\) : \[ {1\over n}\chi(Q_{n,j})\overset{*}\rightarrow \eta_F,\;n\rightarrow\infty,\;j=0,1,2, \] (this equilibrium measure follows from the logarithmic potential being constant on \(E\)).
For the diagonal Padé polynomials for the functions \(\{1,f\}\) this follows from H. Stahl’s general theory [Constr. Approx. 2, 225–240 (1986; Zbl 0592.42016); J. Approx. Theory 91, No. 2, 139–204, Art. No. AT973141 (1997; Zbl 0896.41009)].

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
41A21 Padé approximation
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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