Factors and roots of the van der Pol polynomials
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- by F. T. Howard PDF
- Proc. Amer. Math. Soc. 53 (1975), 1-8 Request permission
Abstract:
The van der Pol polynomials ${V_n}(a)$ are defined by means of \[ {x^3}{e^{xa}}{[6x({e^x} + 1) - 12({e^x} - 1)]^{ - 1}} = \sum \limits _{n = 0}^\infty {{V_n}} (a){x^n}/n!.\] 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
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Additional Information
- © Copyright 1975 American Mathematical Society
- 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