On nth order Euler polynomials of degree n that are Eisenstein

被引：1

作者：

Filaseta, Michael ^{[1
]}

Luckner, Thomas ^{[2
]}

机构：

[1] Univ South Carolina, Dept Math, Columbia, SC 29208 USA

[2] Flagler Coll, Dept Math, St Augustine, FL 32084 USA

来源：

INDAGATIONES MATHEMATICAE-NEW SERIES | 2024年 / 35卷 / 01期

关键词：

Bernoulli number; Eisenstein polynomial; Euler polynomial; Genoocchi number; Irreducible polynomial; GENOCCHI NUMBERS; CONGRUENCES;

D O I：

10.1016/j.indag.2023.09.001

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

For m an even positive integer and p an odd prime, we show that the generalized Euler polynomial mp (x) is in Eisenstein form with respect to p if and only if p does not divide m(2m - 1)Bm. As a consequence, we deduce that at least 1/3 of the generalized Euler polynomials En(n)(x) are in Eisenstein form with respect to a prime p dividing n and, hence, irreducible over Q. (c) 2023 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

引用

页码：76 / 86

页数：11

共 12 条

[1]

Abramowitz M., 1964, Handbook of mathematical functions with formulas, graphs, and mathematical tables, V55

[2]

Adams J.C., 1878, J. Reine Angew. Math, V85, P269

[3] Congruences of p-adic integer order Bernoulli numbers [J].

Adelberg, A .

JOURNAL OF NUMBER THEORY, 1996, 59 (02) :374-388

[4]

Adelberg A., 2002, Colloq. Math, V93, P21

[5]

Borevich ZI, 1966, NUMBER THEORY

[6]

ERDOS P, 1999, CRM P LECT NOTES, V19, P87

[7] Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture [J].

Hu, Su ;

Kim, Min-Soo ;

Moree, Pieter ;

Sha, Min .

JOURNAL OF NUMBER THEORY, 2019, 205 :59-80

[8]

Kummer EE, 1852, J REINE ANGEW MATH, V44, P93

[9] Applications of an explicit formula for the generalized Euler numbers [J].

Liu, Guo Dong ;

Zhang, Wen Peng .

ACTA MATHEMATICA SINICA-ENGLISH SERIES, 2008, 24 (02) :343-352

[10] Prime divisors of l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l [J].

Moree, Pieter ;

Sgobba, Pietro .

JOURNAL OF NUMBER THEORY, 2023, 251 :147-184

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