Related papers: Dicrete Analogues of the Laguerre Inequality

Expressions for the summation of the series involving the Laguerre polynomials \[S_m(\pm\nu, \pm p)\equiv e^{-x}\sum_{n=0}^\infty \frac{x^n\,L_n^{(\nu)}(x)}{(1\pm \nu\pm p)_n}\frac{(f+m)_n}{(f)_n}\] for any non-negative integers $m$ and $p$…

We shall give bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class and satisfying a second order differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the…

A conjecture of I. Krasikov is proved. Several discrete analogues of classical polynomial inequalities are derived, along with results which allow extensions to a class of transcendental entire functions in the Laguerre-P\'olya class.

For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards…

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a…

Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 <A, x_k >B,$ which are uniform in all the parameters involved. Together with…

We consider the irreducibility of Generalised Laguerre Polynomials for negative integral values given by $L_n^{(-1-n-s)}(x)=\displaystyle\sum_{j=0}^{n}\binom{n-j+s}{n-j}\frac{x^j}{j!}.$ For different values of $s,$ this family gives…

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde L^\alpha_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\cdots ,K$, are real…

This contribution aims to obtain several connection formulae for the polynomial sequence, which is orthogonal with respect to the discrete Sobolev inner product \[ \langle f, g\rangle_n=\langle {\bf u}, fg\rangle+ \sum_{j=1}^M \mu_{j}…

Let $f(z)=e^{-bz^2}f_1(z)$ where $b \geq 0$ and $f_1(z)$ is a real entire function of genus 0 or 1. We give a necessary and sufficient condition in terms of a sequence of inequalities for all of the zeros of $f(z)$ to be real. These…

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well…

We investigate interlacing properties of zeros of Laguerre polynomials $ L_{n}^{(\alpha)}(x)$ and $ L_{n+1}^{(\alpha +k)}(x),$ $ \alpha > -1, $ where $ n \in \mathbb{N}$ and $ k \in {\{ 1,2 }\}$. We prove that, in general, the zeros of…

We study the algebraic properties of Generalized Laguerre polynomials for negative integral values of a given parameter which is $L_{n}^{(-1-n-r)}(x)= \sum\limits_{j=0}^{n} \binom{n-j+r}{n-j} \frac{x^{j}}{j!}$ for integers $r\geq 0, n\geq…

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials…

This paper introduces new mixed formulations and discretizations for $m$th-Laplace equations of the form $(-1)^m\Delta^m u=f$ for arbitrary $m=1,2,3,\dots$ based on novel Helmholtz-type decompositions for tensor-valued functions. The new…

We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the…

We make progress on a conjecture of Cilleruelo on the growth of the least common multiple of consecutive values of an irreducible polynomial $f$ on the additional hypothesis that the polynomial be even. This strengthens earlier work of…

Let $s_0,s_1,\dots,s_{m-1}$ be complex numbers and $r_0,\dots,r_{m-1}$ rational integers in the range $0\le r_j\le m-1$. Our first goal is to prove that if an entire function $f$ of sufficiently small exponential type satisfies…

We consider the irreducibility of polynomial $L_n^{(\alpha)} (x) $ where $\alpha$ is a negative integer. We observe that the constant term of $L_n^{(\alpha)} (x) $ vanishes if and only if $n \geq |\alpha| = -\alpha$. Therefore we assume…

Let $m\geq 1$ and $a_m$ be integers. Let $\alpha$ be a rational number which is not a negative integer such that $\alpha = \frac{u}{v}$ with $\gcd(u,v) = 1, v>0$. Let $\phi(x)$ belonging to $\Z[x]$ be a monic polynomial which is irreducible…