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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
DOI: https://doi.org/10.1090/S0002-9939-1975-0379347-9
MathSciNet review: 0379347
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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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0379347-9
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

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