Functiones et Approximatio Commentarii Mathematici

Irreducibility of generalized Hermite-Laguerre polynomials

Shanta Laishram and Tarlok N Shorey

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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

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

First available in Project Euclid: 25 September 2012

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Zentralblatt MATH identifier

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

irreducibility Hermite-Laguerre polynomials arithmetic progressions primes


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.

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  • M. Allen and M. Filaseta, A generalization of a third irreducibility theorem of I. Schur, Acta Arith. 114 (2004), 183–197.
  • Z. Cao, A note on the Diophantine equation $a^x+b^y=c^z$, Acta Arith. 91 (1999), 85–93.
  • Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Ph.D thesis, Université de Limoges, 1998.
  • P. Dusart, Inégalitiés explicites pour $\psi(X), \theta(X), \pi(X)$ et les nombres premiers, C. R. Math. Rep. Acad. Sci. Canada 21(1)(1999), 53–59, 55.
  • C. Finch and N. Saradha, On the irreducibility of a certain polynomials with coefficients that are products of terms in an arithmetic progression, Acta Arith. 143 (2010), 211–226.
  • S. Laishram and T.N. Shorey, Number of prime divisors in a product of terms of an arithmetic progression, Indag. Math. 15(4) (2004), 505–521.
  • S. Laishram and T.N. Shorey, The greatest prime divisor of a product of terms in an arithmetic progression, Indag. Math. 17(3) (2006), 425–436.
  • S. Laishram and T.N. Shorey, Grimm's Conjecture on consecutive integers, Int. Jour. Number Theory 2 (2006), 207–211.
  • D.H. Lehmer, On a problem of Störmer, Illinois J. of Math. 8 (1964), 57–79.
  • T. Nagell, Sur une classe d'équations exponentielles, Ark. Mat. 3 (1958), 569–582.
  • O. Ramaré and R. Rumely, Primes in Arithmetic Progression, Math. Comp. 65 (1996), 397–425.
  • H. Robbins, A remark on stirling's formula, Amer. Math. Monthly 62 (1955). 26–29.
  • T.N. Shorey and R. Tijdeman, Generalizations of some irreducibility results by Schur, Acta Arith. 145 (2010), 341–371.
  • I. Shur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, II, Sitzungsber. Preuss. Akad. Wiss. Berlin Phys.-Math. Kl. 14 (1929), 370–391.
  • I. Schur, Affektlose Gleichungen in der Theorie der Laguerreschen und Hermitschen Polynome, J. Reine Angew. Math. 165 (1931), 52–58.