Stable regions of Tur\'an expressions

TL;DR

This paper investigates the stability of Turán expressions for polynomial sequences satisfying specific differential recurrences, proving weak Hurwitz stability in certain cases and addressing a problem related to Bell polynomials.

Contribution

It establishes conditions under which Turán expressions are weakly Hurwitz stable for sequences satisfying particular differential recurrences, including solutions to a problem posed by S. Fisk.

Findings

Turán expressions for certain polynomial sequences are weakly Hurwitz stable.

Confirmed Fisk's conjecture for Bell polynomials.

Extended stability results to Chebyshev, Hermite, Laguerre, Bessel, and Jensen polynomials.

Abstract

Consider polynomial sequences that satisfy a first-order differential recurrence. We prove that if the recurrence is of a special form, then the Tur\'an expressions for the sequence are weakly Hurwitz stable (non-zero in the open right half-plane). A special case of our theorem settles a problem proposed by S. Fisk that the Tur\'an expressions for the univariate Bell polynomials are weakly Hurwitz stable. We obtain related results for Chebyshev and Hermite polynomials, and propose several extensions involving Laguerre polynomials, Bessel polynomials, and Jensen polynomials associated to a class of real entire functions.