A note on the zeros of Jensen polynomials
A recent result of Griffin, Ono, Rolen and Zagier on Jensen polynomials related with the Riemann zeta function is improved.
math.CV↗
arXiv subjects
Young-One Kim
Publications and source records attributed to Young-One Kim.
The convergence of a sequence of polynomials and the distribution of their zeros
Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$ lie in the closed upper half plane $\rm{Im}\ z\geq 0$, or if $f_1, f_2, \dots$ are real polynomials and the numbers of their non-real zeros are uniformly bounded, then the sequence converges uniformly on compact sets in the complex plane. The results imply a theorem of Benz and a conjecture of Pólya.
math.CV↗
On the Pólya-Wiman properties of Differential Operators
Let $ϕ(x)=\sum α_n x^n$ be a formal power series with real coefficients, and let $D$ denote differentiation. It is shown that "for every real polynomial $f$ there is a positive integer $m_0$ such that $ϕ(D)^mf$ has only real zeros whenever $m\geq m_0$" if and only if "$α_0=0$ or $2α_0α_2 - α_1^2 <0$", and that if $ϕ$ does not represent a Laguerre-Pólya function, then there is a Laguerre-Pólya function $f$ of genus $0$ such that for every positive integer $m$, $ϕ(D)^mf$ represents a real entire function having infnitely many nonreal zeros.
math.CV↗
On the flips for a synchronized system
It is shown that if an infinite synchronized system has a flip, then it has infinitely many non-conjugate flips, and that the result cannot be extended to the class of coded systems.
math.DS↗
On the number of fixed points of sofic flip systems
In the case when $X$ is a sofic shift and $ϕ: X \to X$ is a homeomorphism such that $ϕ^2 = \text{id}_X$ and $ϕσ_X = σ_X^{-1} ϕ$, the number of points in $X$ that are fixed by $σ_X^m$ and $σ_X^n ϕ$, $m=1,2,...$, $n\in\Bbb Z$, is expressed in terms of a finite number of square matrices: The matrices are obtained from Krieger's joint state chain of a sofic shift which is conjugate to $X$.
math.DS↗