Functiones et Approximatio Commentarii Mathematici

Irreducibility of generalized Hermite-Laguerre polynomials

Shanta Laishram and Tarlok N Shorey

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Abstract

For a rational $q=u+\frac{\alpha}{d}$ with $u, \alpha, d\in \ACOBZ$ with $u\ge 0, 1\le \alpha<d$, $\gcd(\alpha, d)=1$, the \emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \begin{align*} G_q(x)&=a_nx^n+a_{n-1}(\alpha +(n-1+u)d)x^{n-1}+\cdots\\ &\quad+a_1\left(\prod^{n-1}_{i=1}(\alpha +(i+u)d)\right)x+a_0 \left(\prod^{n-1}_{i=0}(\alpha +(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 new improved lower bound for greatest prime factor of product of consecutive terms of an arithmetic progression whose common difference is $2$ and $3$.

Article information

Source
Funct. Approx. Comment. Math., Volume 47, Number 1 (2012), 51-64.

Dates
First available in Project Euclid: 25 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1348578276

Digital Object Identifier
doi:10.7169/facm/2012.47.1.4

Mathematical Reviews number (MathSciNet)
MR2987110

Zentralblatt MATH identifier
1196.33009

Subjects
Primary: 11A41: Primes
Secondary: 11B25: Arithmetic progressions [See also 11N13] 11N05: Distribution of primes 11N13: Primes in progressions [See also 11B25] 11C08: Polynomials [See also 13F20] 11Z05: Miscellaneous applications of number theory

Keywords
irreducibility Hermite-Laguerre polynomials arithmetic progressions primes

Citation

Laishram, Shanta; Shorey, Tarlok N. Irreducibility of generalized Hermite-Laguerre polynomials. Funct. Approx. Comment. Math. 47 (2012), no. 1, 51--64. doi:10.7169/facm/2012.47.1.4. https://projecteuclid.org/euclid.facm/1348578276


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