Asymptotics of type I Hermite-Padé polynomials for semiclassical functions. (English) Zbl 1355.30038

Hardin, Douglas P. (ed.) et al., Modern trends in constructive function theory. Constructive functions 2014 conference in honor of Ed Saff’s 70th birthday, Vanderbilt University, Nashville, TN, USA, May 26–30, 2014. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2534-0/pbk; 978-1-4704-2934-8/ebook). Contemporary Mathematics 661, 199-228 (2016).
Let \(\mathbf{f}=(f_0,f_1,\dots, f_s)\), \(s\in\mathbb{N}\), be a vector of analytic functions. Denote by \(\mathbb{P}_n\) the space of algebraic polynomials with complex coefficients and degree \(\leq n\). Let \(Q_{n,0},Q_{n,1},\dots, Q_{n,s}\), \(n\in\mathbb{N}\), \(Q_{n,k}\in\mathbb{P}_n\), \(k=0,1,\dots,s\), be the associated vector of type I Hermite-Padé polynomials corresponding to \(\mathbf{f}\), and let \[ R_n(z):=\big(Q_{n,0}f_0+Q_{n,1}f_1+\cdots+Q_{n,s}f_s\big)(z)=O\left(\frac1{z^{sn+s}}\right),\quad z\to\infty\,, \] be the remainder. The authors describe an approach for finding the asymptotic zero distribution of these polynomials as \(n\to\infty\) under the assumption that all \(f_k\)’s are semiclassical, i.e., their logarithmic derivatives are rational functions. In particular, let \(f(z,\alpha)\) be a holomorphic branch of the function
\[ \Big(\frac{z-1}{z+1}\Big)^\alpha, \quad \alpha\in\Big(\frac12,-\frac12\Big)\setminus\{0\}. \] Then all zeros of the polynomials \(Q_{n,k}\) belong to \(\mathbb{R}\setminus[-1,1]\), while \[ Q_{n,1}(x)+2\cos(\alpha\pi)Q_{n,2}(x)(f^++f^-)(x),\quad x\in(-1,1)\,, \] has at least \(2n+1\) zeros on \((-1,1)\). Here \(f^+\) (resp. \(f^-\)) is the boundary value of the function \(f\) on \((-1,1)\) from the upper (resp. lower) half-plane. The authors consider also some differential equations and asymptotics for the Hermite-Padé polynomials.
For the entire collection see [Zbl 1343.00038].


30E10 Approximation in the complex plane
41A20 Approximation by rational functions
41A21 Padé approximation
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