| Russian Mathematical Surveys |
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This article is cited in 4 scientific papers (total in 4 papers) Brief Communications Asymptotic properties of Hermite–Padé polynomials and Katz points S. P. Suetin Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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     1. In a number of topical physical problems considered in the framework of the Rayley– Schrödinger perturbation theory, in calculations based on series of perturbation theory authors use Hermite–Padé polynomnials of the first type: see [1], [7], and [2]. In this case, in accordance with Katz’s results [4], authors assume that the multivalued analytic function corresponding to the series of perturbation theory takes real values on the real line and has only complex-conjugate square-root branch points. The original series of perturbation theory is a series in a small parameter $\varepsilon$, and the value of the function defined by this series must be found for $\varepsilon=1$. If this function has branch points in the unit disc, then one cannot calculate this value from the partial sums of the original series. Such branch points are called Katz points. When they are present, to calculate the value of the function at $\varepsilon=1$, in place of partial sums authors use Padé approximants, Shafer quadratic approximants, or other approximnts based on Hermite–Padé polynomials (see [10] and [5]). Thus determining Katz points numerically is a challenging problem. Note the following. Let $S$ be the Stahl compact set (with respect to $\varepsilon=0$) corresponding to our series of perturbation theory. By Stahl points we will mean the branch points of the function defined by the series of perturbation theory that occur on $S$. Stahl points in the unit disc are just Katz points. 2. It will be convenient to us below to deal with Laurent series at $z=\infty$. In accordance with Katz’s results, we consider the class of multivalued analytic functions $f$ all of whose branch points are of the second order. In addition, we assume that each $f$ has a holomorphic germ $f_\infty\in\mathscr H(\infty)$ at the point $z=\infty$ such that the corresponding Stahl compact set has no Chebotarev points. In this case we write $(f,f_\infty)\in\mathscr F$. For $(f,f_\infty)\in\mathscr F$ and an arbitrary $n\in\mathbb N$ let $Q_{n,j}$, $j=0,1,2$, denote the Hermite–Padé polynomials of the first type for the tuple $[1,f,f^2]$ and multi-index $(n,n,n)$:
$$
\begin{equation}
(Q_{n,0}+Q_{n,1}f+Q_{n,2}f^2)(z)=O(z^{-2n-2}),\qquad z\to\infty.
\end{equation}
\tag{1}
$$
The polynomial $D_n:=Q^2_{n,1}-4Q_{n,0}Q_{n,2}$ is the discriminant of the quadratic equation corresponding to (1) and arising in connection with Shafer approximations. ‘Generically’, $\deg{Q_{n,j}}=n$ and $\deg{D_n}=2n$.
An analysis of the behaviour of the zeros of $D_n$ for variable $n$ underlies the numerical search for Katz points; see [1] and [2]. It follows from a result stated in [6] that for meromorphic functions on a three-sheeted Riemann surface the zeros of the $Q_{n,j}$ and $D_n$ have the same limiting distribution. On the other hand numerical experiments performed with the use of specially designed software [3] show that the polynomials $D_n$ have the following property: a finite number of their zeros are attracted to Stahl points as $n\to\infty$. In the general case this should be regarded as a conjecture; to prove it one must find suitable formulae for the strong asymptotics of the polynomials $Q_{n,j}$. Conjecture. Let $(f,f_\infty)\in\mathscr F$, let $S=S(f_\infty)$ be the corresponding Stahl compact set, $\Sigma_S(f_\infty)=\{a_1,\dots,a_m\}\subset S$ be the set of Stahl points, and $D_n$, $n=0,1,\dots$, be the discriminants for the tuple $[1,f,f^2]$. Then the discriminant has $m$ zeros $\zeta_{n,j}$, $j=1,\dots,m$, such that $\zeta_{n,j}\to a_j$ as $n\to\infty$. 3. At the moment we can only prove this conjecture in the special class $\mathscr Z(\Delta)$ of functions $f$, $(f,f_\infty)\in\mathscr F$, defined in terms of the inverse Joukowsky function:
$$
\begin{equation*}
f(z)=\biggl[\biggl(A-\frac{1}{\varphi(z)}\biggr) \biggl(B-\frac{1}{\varphi(z)}\biggr)\biggr]^{-1/2},
\end{equation*}
\notag
$$
where $1<A<B$ and $\varphi(z)=z+(z^2-1)^{1/2}\sim 2z$ as $z\to\infty$. The function $f$ has four branch points, $\pm1$, $a$, and $b$, where $a=(A+1/A)/2$ and $b=(B+1/B)/2$. Under the above condition on $\varphi$ the interval $\Delta=[-1,1]$ is the Stahl compact set for $f_\infty$, and $\pm1$ are the Stahl points. The pair of functions $(f,f^2)$ forms a Nikishin system (see [9]).
Theorem 1. Let $f\in\mathscr Z(\Delta)$, and let $D_n$ be the discriminant corresponding to the Hermite–Padé polynomials for the tuple $[1,f,f^2]$. Then $D_n$ has zeros $\zeta_{n,1}$ and $\zeta_{n,2}$ such that $\zeta_{n,1}\to-1$ and $\zeta_{n,2}\to1$ as $n\to\infty$. The proof is based on the asymptotic representations for Hermite–Padé polynomials that were obtained in [8].
Citation in format AMSBIB
\Bibitem{Sue22} |
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