Related papers: Upper bounds for Erd\'{e}lyi's multivariate Laguer…
We study the asymptotic zero distribution of the rescaled Laguerre polynomials, $\displaystyle L_n^{(\alpha_n)}(nz)$, with the parameter $\alpha_n$ varying in such a way that $\displaystyle \lim_{n\rightarrow \infty}\alpha_n/n=-1$. The…
In this paper, we study generating functions of Erd\'{e}lyi's multivariate Laguerre polynomials $L_{n_1,\cdots,n_k}^{(\alpha)}(x_1,\cdots,x_k)$ with a varying complex parameter. Our main result is a multiple generating function from which…
The notion of multipolynomials was recently introduced and explored by T. Velanga in [10] as an attempt to encompass the theories of polynomials and multi- linear operators. In the present paper we push this subject further, by proving…
We propose a simple uniform lower bound on the spacings between the successive zeros of the Laguerre polynomials $L_n^{(\alpha)}$ for all $\alpha>-1$. Our bound is sharp regarding the order of dependency on $n$ and $\alpha$ in various…
In this paper, we obtain a sharp upper bound for the sum of the first $k$-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of…
We present a new set of infinitely many shape invariant potentials and the corresponding exceptional (X_{\ell}) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly…
The analogous quaternionic polynomials of a class of bivariate orthogonal polynomials (arXiv: 1502.07256, 2014) introduced. The ladder operators for these quaternionic polynomials also studied. For the quaternionic case, the ladder…
Laguerre polynomials are orthogonal polynomials defined on positive half line with respect to weight $e^{-x}$. They have wide applications in scientific and engineering computations. However, the exponential growth of Laguerre polynomials…
In this paper, we extend our investigation into semiclassical multiple discrete orthogonal polynomials by considering an arbitrary number of weights. We derive multiple versions of the Toda equations and the Laguerre-Freud equations for the…
We obtain all spectral type differential equations satisfied by the Sobolev-type Laguerre polynomials. This generalizes the results found in 1990 by the first and second author in the case of the generalized Laguerre polynomials defined by…
This paper deals with both the higher order Tur\'an inequalities and the Laguerre inequalities for quasi-polynomial-like functions -- that are expressions of the form $f(n)=c_l(n)n^l+\cdots+c_d(n)n^d+o(n^d)$, where $d,l\in\mathbb{N}$ and…
We prove that the generalised Laguerre polynomials $L_{n}^{(\alpha)}(x)$ with $0\le \al\le 50$ are irreducible except for finitely many pairs $(n, \al)$ and that these exceptions are necessary. In fact it follows from a more general…
The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an…
We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…
Motivated by the work of Prajapati \emph{et al.} \cite{PAA}, here we study some explicit form of the generalized Laguerre polynomials $L_{\lfloor\frac{n}{q}\rfloor}^{(\alpha,\beta)}(z)$, when $q=1$.
We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different…
In recent years, the log-concavity or log-convexity of combinatorial sequences and their root sequences, higher order Tur{\'a}n inequalities, and Laguerre inequalities of order two have been widely studied. However, the research of the…
We provide here a modest improvement upon a large sieve inequality for quadratic polynomial amplitudes orginally due to Liangyi Zhao.
The notions of higher-order weighted multilinear Poincar\'e and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.
In 1959, Erd\H{o}s and Szekeres posed a series of problems concerning the size of polynomials of the form $$ P_n(z) = \prod_{j=1}^n (1 - z^{s_j}), $$ where $s_1, \dots, s_n$ are positive integers. Of particular interest is the quantity $$…