Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Factors and roots of the van der Pol polynomials


Author: F. T. Howard
Journal: Proc. Amer. Math. Soc. 53 (1975), 1-8
MSC: Primary 10A40
MathSciNet review: 0379347
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The van der Pol polynomials $ {V_n}(a)$ are defined by means of

$\displaystyle {x^3}{e^{xa}}{[6x({e^x} + 1) - 12({e^x} - 1)]^{ - 1}} = \sum\limits_{n = 0}^\infty {{V_n}} (a){x^n}/n!.$

In this paper new properties of these polynomials are derived. It is shown that neither $ {V_{2n}}(a)$ nor $ {V_{2n + 1}}(a)/(a - 1/2)$ has rational roots, and that if $ n = 2 \bullet {3^m},m \geqslant 0$, or $ n = {3^m} + {3^t},m > t > 0$, or $ n = m(p - 3),p$ a prime number, $ 3m < p$, then $ {V_n}(a)$ and $ {V_{n + 1}}(a)/(a - 1/2)$ are both irreducible over the rational field. It is also shown that if $ n = {2^k}$, then $ {V_n}(a)$ is irreducible over the rational field. Finally, possible factors of the van der Pol polynomials are discussed.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10A40

Retrieve articles in all journals with MSC: 10A40


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0379347-9
PII: S 0002-9939(1975)0379347-9
Keywords: van der Pol numbers and polynomials, Rayleigh function, Bernoulli and Euler polynomials, irreducible over the rational field, Eisenstein's irreducibility criterion, Eisenstein polynomial
Article copyright: © Copyright 1975 American Mathematical Society