Factors and roots of the van der Pol polynomials

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.