Asymptotics of type I Hermite-Padé polynomials for semiclassical functions
Type I Hermite-Pad\'e polynomials for set of functions $f_0, f_1,..., f_s$ at infinity, $(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s)(z)=O(\frac{1}{z^{sn+s}}), z\rightarrow \infty$ with the degree of all $Q_{n,k}<=n$. We describe an approach for finding the asymptotic zero distribution...
| Main Authors: | , , |
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
| Published: |
2017
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| Online Access: | http://hdl.handle.net/10835/4880 |
| _version_ | 1789406582183821312 |
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| author | Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Suetin, Sergey P. |
| author_facet | Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Suetin, Sergey P. |
| author_sort | Martínez-Finkelshtein, Andrei |
| collection | DSpace |
| description | Type I Hermite-Pad\'e polynomials for set of functions $f_0, f_1,..., f_s$ at infinity, $(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s)(z)=O(\frac{1}{z^{sn+s}}), z\rightarrow \infty$ with the degree of all $Q_{n,k}<=n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\rightarrow \infty$ under the assumption that all $f'_js$ are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients.
We discuss in more detail the case when $f'_k$s are powers of the same function $f (f_k=f^k)$; for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed.
From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Pad\'e approximants (when $s=1$). |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-4880 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2017 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-48802023-04-12T19:38:33Z Asymptotics of type I Hermite-Padé polynomials for semiclassical functions Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Suetin, Sergey P. Type I Hermite-Pad\'e polynomials for set of functions $f_0, f_1,..., f_s$ at infinity, $(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s)(z)=O(\frac{1}{z^{sn+s}}), z\rightarrow \infty$ with the degree of all $Q_{n,k}<=n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\rightarrow \infty$ under the assumption that all $f'_js$ are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when $f'_k$s are powers of the same function $f (f_k=f^k)$; for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Pad\'e approximants (when $s=1$). 2017-06-21T10:04:11Z 2017-06-21T10:04:11Z 2016 info:eu-repo/semantics/article http://hdl.handle.net/10835/4880 en Attribution-NonCommercial-NoDerivatives 4.0 Internacional http://creativecommons.org/licenses/by-nc-nd/4.0/ info:eu-repo/semantics/openAccess First published in Contemp. Math. in 661 (2016), 199-228, published by the American Mathematical Society |
| spellingShingle | Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Suetin, Sergey P. Asymptotics of type I Hermite-Padé polynomials for semiclassical functions |
| title | Asymptotics of type I Hermite-Padé polynomials for semiclassical functions |
| title_full | Asymptotics of type I Hermite-Padé polynomials for semiclassical functions |
| title_fullStr | Asymptotics of type I Hermite-Padé polynomials for semiclassical functions |
| title_full_unstemmed | Asymptotics of type I Hermite-Padé polynomials for semiclassical functions |
| title_short | Asymptotics of type I Hermite-Padé polynomials for semiclassical functions |
| title_sort | asymptotics of type i hermite-padé polynomials for semiclassical functions |
| url | http://hdl.handle.net/10835/4880 |
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