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Anna Vishnyakova

Publications and source records attributed to Anna Vishnyakova.

At least 19 recordsLinked to original sources
arXiv2026

Polynomially Deformed Normalized Pochhammer Sequences Having Generating Functions With Only Real Non-positive Zeros

For a given real number $a>0$ and a given real polynomial $P_n\in \mathbb{R}[x]$ of degree $n=0, 1, 2, \ldots$ it is easy to see that $ \sum_{k=0}^\infty \frac{(a)_k}{k!} P_n(k) z^k =\frac{S_{n, a}(z)}{(1-z)^{a+ n}}, \ |z|<1, $ where $S_{n, a}$ is a real polynomial of degree not greater than $n.$ Here $(a)_k =a(a+1)\cdot \ldots \cdot (a+k-1),\ (a)_0 = 1,$ is the rising factorial, or the Pochhammer symbol. We consider the following open problem: to describe the set of real polynomials $P_n\in \mathbb{R}[x]$ of degree $n=0, 1, 2, \ldots,$ such that the corresponding polynomial $S_{n, a}$ has all real non-positive zeros. In the case $a=1$ this problem has been studied in \cite{vish}. We establish several new necessary conditions and several sufficient conditions, present a number of important examples, and formulate several open problems.

math.CV↗
arXiv2026

Polynomials interpolated totally positive sequences

A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $ \left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (where $a_k=0$ for $k<0$). In this paper, we investigate the following question: for which real polynomials $P$ the sequence $(P(k))_{k=0}^\infty$ is totally positive? We establish a few new necessary conditions, sufficient conditions, present a number of important examples and formulate several open problems.

math.CV↗
arXiv2025

Convolution operators preserving the set of totally positive sequences

A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $ \left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $a_k=0$ for $k<0$). In this paper, which continues our earlier work \cite{kv}, we investigate the set of real sequences $(b_k)_{k=0}^\infty$ with the property that for every totally positive sequence $(a_k)_{k=0}^\infty,$ the sequense of termwise products $(a_k b_k)_{k=0}^\infty$ is also totally positive. In particular, we show that for every totally positive sequence $(a_k)_{k=0}^\infty$ the sequence $\left(a_k a^{-k (k-1)}\right)_{k=0}^\infty$ is totally positive whenever $a^2\geq 3{.}503.$ We also propose several open problems concerning convolution operators that preserve total positivity.

math.CV↗
arXiv2025

Generalized $f$-Eulerian polynomials: zeros and hypergeometric representations with applications

In this paper, we explore (slightly generalized) $f$-Eulerian polynomials introduced by Stanley and frequently appearing in combinatorics. Notable special cases include the classical Eulerian polynomials, the generating polynomials of order polynomials for certain labeled posets, and the $d$-Narayana polynomials. We establish simple sufficient conditions for the reality (and sign) of their zeros and present implications for total positivity of sequences generated by values of polynomials at integers. We further relate these polynomials to generalized Euler's transformations for the generalized hypergeometric functions with integral parameter differences. Exploiting this and other hypergeometric connections, we provide purely hypergeometric proofs for various known and some new properties of $d$-Narayana polynomials. Another family encompassed by our definition of the generalized $f$-Eulerian polynomials is that of Jacobi-Piñeiro type II multiple orthogonal polynomials. Their zero location can thus be analyzed, for both canonical and non-canonical parameter values, without invoking orthogonality. Finally, we present several connection formulas relating $d$-Narayana polynomials to particular Jacobi-Piñeiro polynomials.

math.CO↗
arXiv2025

Unimodality preservation by ratios of functional series and integral transforms

An elementary, but very useful lemma due to Biernacki and Krzyż (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their respective coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios as well as for unimodality in place of monotonicity. This paper continues this line of research: we consider ratios of general functional series and integral transforms and furnish natural sufficiency conditions for preservation of unimodality by such ratios. Numerous series and integral transforms appearing in applications satisfy our sufficiency conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. Finally, we illustrate our general results by exhibiting certain statements on monotonicity patterns for ratios of some special functions. The key role in our considerations is played by the notion of sign regularity.

math.CA↗
arXiv2024

On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture

For a given real polynomial $p$ we study the possible number of real roots of a differential polynomial $H_{\varkappa}[p](x) = \varkappa\left(p'(x)\right)^2-p(x)p''(x), \varkappa \in \mathbb{R}.$ In the special case when all real zeros of the polynomial $p$ are simple, and all roots of its derivative $p'$ are real and simple, the distribution of zeros of $H_{\varkappa}[p]$ is completely described for each real $\varkappa.$ We also provide counterexamples to two Boris Shapiro's conjectures about the number of zeros of the function $H_{\frac{n-1}{n}}[p].$

math.CV↗
arXiv2024

In search of Newton-type inequalities

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide counterexamples to two earlier conjectures refining Descartes rule of signs.

math.CV↗
arXiv2024

An analog of multiplier sequences for the set of totally positive sequences

A real sequence $(b_k)_{k=0}^\infty$ is called totally positive if all minors of the infinite matrix $ \left\| b_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $b_k=0$ for $k<0$). In this paper, we investigate the problem of description of the set of sequences $(a_k)_{k=0}^\infty$ such that for every totally positive sequence $(b_k)_{k=0}^\infty$ the sequence $(a_k b_k)_{k=0}^\infty$ is also totally positive. We obtain the description of such sequences $(a_k)_{k=0}^\infty$ in two cases: when the generating function of the sequence $\sum_{k=0}^\infty a_k z^k$ has at least one pole, and when the sequence $(a_k)_{k=0}^\infty$ has not more than $4$ nonzero terms.

math.CV↗
arXiv2024

On the growth of resolvent of Toeplitz operators

We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on the Hardy space, in terms of the distance to its spectrum $σ(T_b)$. We are primarily interested in the case when the symbol $b$ is a Laurent polynomial (\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.

math.SP↗
arXiv2022

Hutchinson's intervals and entire functions from the Laguerre-Pólya class

We find the intervals $[α, β(α)]$ such that if a univariate real polynomial or entire function $f(z) = a_0 + a_1 z + a_2 z^2 + \cdots $ with positive coefficients satisfy the conditions $ \frac{a_{k-1}^2}{a_{k-2}a_{k}} \in [α, β(α)]$ for all $k \geq 2,$ then $f$ belongs to the Laguerre--Pólya class. For instance, from J.I.~Hutchinson's theorem, one can observe that $f$ belongs to the Laguerre--Pólya class (has only real zeros) when $q_k(f) \in [4, + \infty).$ We are interested in finding those intervals which are not subsets of $[4, + \infty).$

math.CV↗
arXiv2022

A sufficient condition for a complex polynomial to have only simple zeros and an analog of Hutchinson's theorem for real polynomials

We find the constant $b_{\infty}$ ($b_{\infty} \approx 4.81058280$) such that if a complex polynomial or entire function $f(z) = \sum_{k=0}^ ωa_k z^k, $ $ω\in \{2, 3, 4, \ldots \} \cup \{\infty\},$ with nonzero coefficients satisfy the conditions $\left|\frac{a_k^2}{a_{k-1} a_{k+1}}\right| >b_{\infty} $ for all $k =1, 2, \ldots, ω-1,$ then all the zeros of $f$ are simple. We show that the constant $b_{\infty}$ in the statement above is the smallest possible. We also obtain an analog of Hutchinson's theorem for polynomials or entire functions with real nonzero coefficients.

math.CV↗
arXiv2021

On the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k,$ $a_k >0,$ we define the sequence of the second quotients of Taylor coefficients $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$. We find new necessary conditions for a function with a non-decreasing sequence $Q$ to belong to the Laguerre--Pólya class of type I. We also estimate the possible number of nonreal zeros for a function with a non-decreasing sequence $Q.$

math.CV↗
arXiv2021

On the entire functions from the Laguerre-Pólya I class with non-monotonic second quotients of Taylor coefficients

We study the entire functions $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ with non-monotonic second quotients of Taylor coefficients, namely, such that $\frac{a_{2m-1}^2}{a_{2m-2}a_{2m}} = a>1$ and $\frac{a_{2m}^2}{a_{2m-1}a_{2m+1}} = b>1$ for all $m \in \mathbb{N}.$ We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-Pólya I class.

math.CV↗
arXiv2020

On the entire functions from the Laguerre-Pólya I class having the increasing second quotients of Taylor coefficients

We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq 2\sqrt[3]{2},$ then all but a finite number of zeros of $f$ are real and simple. We also present a criterion in terms of the closest to zero roots for such a function to have only real zeros (in other words, for belonging to the Laguerre--Pólya class of type I) under additional assumption on the sequence $Q.$

math.CV↗
arXiv2020

On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-Pólya I class

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that if $f$ belongs to the Laguerre-Pólya class, and the quotients $q_k := \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots $ satisfy the condition $q_2 \leq q_3,$ then $f$ has at least one zero in the segment $[-\frac{a_1}{a_2},0].$ We also give necessary conditions and sufficient conditions of the existence of such a zero in terms of the quotients $q_k$ for $k=2,3, 4.$

math.CV↗
arXiv2019

On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-Pólya class

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that $f$ does not belong to the Laguerre-Pólya class if the quotients $\frac{a_{n-1}^2}{a_{n-2}a_n}$ are increasing in $n$, and $c:= \lim\limits_{n\to \infty} \frac{a_{n-1}^2}{a_{n-2}a_n}$ is smaller than an absolute constant $q_\infty$ $(q_\infty\approx 3{.}2336) .$

math.CV↗
arXiv2019

Hermite-Poulain theorems for linear finite difference operators

We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form $$ Δ_{θ, h}(f)(z)=e^{iθ}f(z+ih)-e^{-iθ}f(z-ih), \quadθ\in[0,π),\ \ h\in\mathbb{C}\setminus\{0\}, $$ where $f$ is a polynomial or an entire function of a certain kind, and prove that the roots of $Δ_{θ, h}(f)$ are simple under some conditions. Moreover, we prove that the operator $Δ_{θ, h}$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of $Δ_{θ, h}(p)$ as $|h|\to\infty$ is found for any complex polynomial $p$. Some other interesting roots preserving properties of the operator $Δ_{θ, h}$ are also studied, and a few examples are presented.

math.CA↗