Related papers: On zeros of polynomials and allied functions satis…
We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed by A. Seidenberg but no complete solution was…
We present new bounds for the numerical radius of bounded linear operators and $2\times 2$ operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new…
We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In…
Laguerre's theorem regarding the number of non-real zeros of a polynomial and its image under certain linear operators is generalized. This generalization is then used to (1) exhibit a number of previously undiscovered complex zero…
We study the effect of finite difference operators of finite order on the distribution of zeros of polynomials and entire functions.
Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated to $\mathrm{PSL}_2(\mathbb{Z})$, and Bannai, Kojima, and Miezaki's partial proof for the case of $\Gamma_0^*(2)$, we show that the zeros of certain modular…
The paper presents some heuristic results about the distribution of zeros of Hermite-Pade polynomials of first kind for the case of three functions $1,f,f^2$, where $f$ has the form $f(z): = \prod\limits_ {j = 1 } ^3 (z-a_j) ^ {\alpha_j} $,…
Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.
We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrodinger…
In this paper, we are concerned with the problem of locating the zeros of polynomials of a quaternionic variable with quaternionic coefficients. We derive some new Cauchy bounds for the zeros of a polynomial by virtue of maximum modulus…
In this paper we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on $ [-1,1] $ and behaves like $ h(x)|x - x_0|^\lambda dx $ for some $ x_0 \in…
The purpose of this note is twofold: firstly, it intends to bring to light an apparently unknown property of the product of the extreme zeros of Laguerre polynomials, which in a very particular case leads to a twenty-year-old conjecture for…
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-P\'olya 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…
We study the zeros of exceptional Hermite polynomials associated with an even partition $\lambda$. We prove several conjectures regarding the asymptotic behavior of both the regular (real) and the exceptional (complex) zeros. The real zeros…
We obtain bounds for the numerical radius of $2 \times 2$ operator matrices which improve on the existing bounds. We also show that the inequalities obtained here generalize the existing ones. As an application of the results obtained here…
We analyze the polynomial solutions of the linear differential equation $p_2(x)y''+p_1(x)y'+p_0(x)y=0$ where $p_j(x)$ is a $j^{\rm th}$-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…
We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials…
In this paper, we give results that partially prove a conjecture which was discussed in our previous work (arXiv:1307.4991). More precisely, we prove that as $n\to \infty,$ the zeros of the polynomial$${}_{2}\text{F}_{1}\left[…
Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected…