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Hardy-Petrovitch-Hutchinson’s problem and partial theta function. (English) Zbl 1302.30008
A polynomial \(p(x)=a_0+a_1x+a_2x^2+\ldots +a_nx^n\) with positive coefficients is called section-hyperbolic if, for all \(i=1,2,3,\ldots ,n\), each of its section \(a_0+a_1x+a_2x^2+\ldots +a_i x^i\) has only real roots. The paper deals with the class \(\Delta\) of all section-hyperbolic polynomials. The authors give the conclusive answer to the following question:

Problem (Hardy-Petrovitch-Hutchinson-Ostrovskii). For a given positive integer \(i\), find or estimate
\[ m_i=\inf_{p\in \Delta} \frac{a_i^2}{a_{i-1}a_{i+1}}. \]

Using an inductive procedure, they construct an entire function \(p_{\infty }=1+x+\sum_{n=2}^{\infty } A_n x^n\) such that, for any positive integer \( i,\) one has \( m_i=\frac{A_i^2}{A_{i-1}A_{i+1}}\), that is \(p_{\infty }\) minimizes all \(m_i\) simultaneously. The authors prove that the limit \(m_{\infty}=\lim_{i \to \infty} m_i\) exists and coincides with \(1/\tilde {q}\), where \(\tilde{q} >0\) is the maximal positive number for which the partial theta function \(\Psi(q,u)=\sum_{j=0}^{\infty }q^{j(j+1)/2}u^j\) has only real roots as a function of \(u\). They state that this number \(\tilde{q} \) is the unique solution of the equation \(\frac{1}{x}-\frac{2}{1-x}=\sum_{j=1}^{\infty} \frac{x^j}{1-x^{j+1}}\) beloning to the interval \((0,1)\).

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D15 Special classes of entire functions of one complex variable and growth estimates
30D10 Representations of entire functions of one complex variable by series and integrals
30D20 Entire functions of one complex variable, general theory
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
26C05 Real polynomials: analytic properties, etc.
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