Functiones et Approximatio Commentarii Mathematici

Irreducibility of extensions of Laguerre polynomials

Shanta Laishram, Saranya G. Nair, and Tarlok N. Shorey

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For integers $a_0,a_1,\ldots,a_n$ with $|a_0a_n|=1$ and either $\alpha =u$ with $1\leq u \leq 50$ or $\alpha=u+ \frac{1}{2}$ with $1 \leq u \leq 45$, we prove that $\psi_n^{(\alpha)}(x;a_0,a_1,\cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore, all exceptions other than $n=2^{12},\alpha=89/2$ are necessary. The above result with $0\leq\alpha \leq 10$ is due to Filaseta, Finch and Leidy and with $\alpha \in \{-1/2,1/2\}$ due to Schur.

Article information

Funct. Approx. Comment. Math., Volume 62, Number 2 (2020), 143-164.

First available in Project Euclid: 9 November 2019

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Mathematical Reviews number (MathSciNet)

Primary: 11A41: Primes 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 Laguerre polynomials primes Newton polygons


Laishram, Shanta; Nair, Saranya G.; Shorey, Tarlok N. Irreducibility of extensions of Laguerre polynomials. Funct. Approx. Comment. Math. 62 (2020), no. 2, 143--164. doi:10.7169/facm/1748.

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