Shanta Laishram, T. N. Shorey
In this paper, we show that for each $n\geq 1$, the generalised Hermite-Laguerre Polynomials $G_{\frac{1}{4}}$ and $G_{\frac{3}{4}}$ are either irreducible or linear polynomial times an irreducible polynomial of degree $n-1$.
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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 polynomials which are of great interest. It was proved earlier that for $0 \leq s \leq 60,$ these polynomials are irreducible over $\mathbb{Q}.$ In this paper we improve this result upto $s \leq 88.$
For a rational $q=u+\fracα{d}$ with $u, α, d\in \ACOBZ$ with $u\ge 0, 1\le α<d$, $\gcd(α, d)=1$, the \emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \begin{align*} G_q(x)&=a_nx^n+a_{n-1}(α+(n-1+u)d)x^{n-1}+\cdots\\ &\quad+a_1\left(\prod^{n-1}_{i=1}(α+(i+u)d)\right)x+a_0 \left(\prod^{n-1}_{i=0}(α+(i+u)d)\right) \end{align*} where $a_0, a_1, \cdots, a_n$ are arbitrary integers. We prove some irreducibility results of $G_q(x)$ when $q\in \{\frac{1}{3}, \frac{2}{3}\}$ and extend some of the earlier irreducibility results when $q$ of the form $u+\frac{1}{2}$. We also prove a…
For integers $a_0,a_1,\ldots,a_n$ with $|a_0a_n|=1$ and either $α=u$ with $1\leq u \leq 50$ or $α=u+ \frac{1}{2}$ with $1 \leq u \leq 45$, we prove that $ψ_n^{(α)}(x;a_0,a_1,\cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},α=89/2$ are necessary. The above result with $0\leqα\leq 10$ is due to Filaseta, Finch and Leidy and with $α\in \{-1/2,1/2\}$ due to Schur.
For a positive integer $n$ and a real number $α$, the generalized Laguerre polynomials are defined by \begin{align*} L^{(α)}_n(x)=\sum^n_{j=0}\frac{(n+α)(n-1+α)\cdots (j+1+α)(-x)^j}{j!(n-j)!}. \end{align*} These orthogonal polynomials are solutions to Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties. He obtained irreducibility results of $L^{(\pm \frac{1}{2})}_n(x)$ and $L^{(\pm \frac{1}{2})}_n(x^2)$ and derived that the Hermite polynomials $H_{2n}(x)…
Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters
本文针对正整数参数n的广义拉盖尔多项式，通过推导潜在整数根的严格约束并运用牛顿多边形几何性质，将其在有理数域上不可约的参数范围从$0 \le a \le 50$大幅扩展至$a \le 1000$。
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 $r\geq 0, n\geq 1$. For different values of parameter $r$, this family provides polynomials which are of great interest. Hajir conjectured that for integers $r\geq 0$ and $n\geq 1$, $L_{n}^{(-1-n-r)}(x)$ is an irreducible polynomial whose Galois group contains $A_n$, the alternating group on $n$ symbols. Extending earlier results of Schur, Hajir, Sell, Nair and Shorey, we co…
We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions (n, m) with m <= 2, and (n, n) when n + 1 is a square. Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials. We also explain how irreducibili…
We prove that the generalised Laguerre polynomials $L_{n}^{(α)}(x)$ 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 statement.
The Krawtchouck polynomials arise naturally in both coding theory and probability theory and have been studied extensively from these points of view. However, very little is known about their irreducibility and Galois properties. Just like many classical families of orthogonal polynomials (e.g. the Legendre and Laguerre), the Krawtchouck polynomials can be viewed as special cases of Jacobi polynomials. In this paper we determine the Newton Polygons of certain Krawtchouck polynomials and show that they are very similar to those of the Legendre polynomials (and exhibit new cases of irreducibilit…
We obtain a series transformation formula involving the classical Hermite polynomials. We then provide a number of applications using appropriate binomial transformations. Several of the new series involve Hermite polynomials and harmonic numbers, Lucas sequences, exponential and geometric numbers. We also obtain a series involving both Hermite and Laguerre polynomials, and a series with Hermite polynomials and Stirling numbers of the second kind.
We consider the irreducibility of polynomial $L_n^{(α)} (x) $ where $α$ is a negative integer. We observe that the constant term of $L_n^{(α)} (x) $ vanishes if and only if $n \geq |α| = -α$. Therefore we assume that $α= -n-s-1$ where $s$ is a non-negative integer. Let $$ g(x) = (-1)^n L_n^{(-n-s-1)}(x) = \sum\limits_{j=0}^{n} a_j \frac{x^j}{j!} $$ and more general polynomial, let $$ G(x) = \sum\limits_{j=0}^{n} a_j b_j \frac{x^j}{j!} $$ where $b_j$ with $0 \leq j \leq n$ are integers such that $|b_0| = |b_n| = 1$. Schur was the first to prove the irreducibility of $g(x)$ for $s=0$. It has bee…
On the irreducibility of extended Laguerre Polynomials
本文研究扩展拉盖尔多项式$L_{m,\alpha}^\phi(x)$的有理不可约性，利用$\phi$-牛顿多边形与解析数论结果，证明其除有限个$m$外不可约，并给出$\alpha$取0到4时的例外情形。
On Schur's irreducibility results and generalised $ϕ$-Hermite polynomials
本文推广 Schur 关于偶次 Hermite 型多项式的不可约性定理，证明在模小素数不可约首一多项式 $ϕ$ 及容度条件下，广义 $ϕ$-Hermite 多项式在 $\mathbb{Q}$ 上不可约。
We use expansions with functions related to some special functions such as Hermite or Laguerre to get some conjectural expansions of the Riemann Zeta function in the critical strip involving a set of polynomials which have their zeros on the line Re(s)=1/2..
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