Hardy-Petrovitch-Hutchinson's problem and partial theta function

Abstract

In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminantal inequalities one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J.I.Hutchinson has shown that an entire function p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the property that any its finite segment a_ix^i+...+a_jx^j has all real roots if and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or equal to 4. In the present paper we give sharp lower bounds on the ratios a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular, we show that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.

References in corpus (2)

• The leading root of the partial theta function
• New Multiplier Sequences via Discriminant Amoebae

Cited by in corpus (13)

• The closest to $0$ spectral number of the partial theta function
• The Poisson binomial distribution -- Old & New
• Thermodynamic phases in first detected return times of quantum many-body systems
• A property of a partial theta function
• On the conditions for a special entire function relative to the partial theta-function and the Euler function to belong to the Laguerre-Pólya class
• A generalization of Schönemann's theorem via a graph theoretic method
• Asymptotic expansions of zeros of a partial theta function
• Partial theta function identities from Wang and Ma's conjecture
• Asymptotics of generalized partial theta functions with a Dirichlet character
• Uniform bounds on locations of zeros of partial theta function
• On the zero set of the partial theta function
• One helpful property of functions generating Pólya frequency sequences
• Addendum: A separation in modulus property of the zeros of a partial theta function