Properties of the van der Pol numbers and polynomials.

doi:10.1515/crll.1973.260.35

Cited by 4 publications

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“…In [1] (p. 235), see also Section 1 in [2], the sequence of rational numbers V k for k ≥ 0 was defined by means of…”

Section: Motivationsmentioning

confidence: 99%

“…where V n is the nth van der Pol number. See paper [2] for the properties of these numbers, including recurrence relations. In particular, the generating function is Equation (d) in Section 1.…”

Section: Motivationsmentioning

confidence: 99%

“…1−z cot z to be 3 at z = 0. According to the discussion in [2] (Section 6), the Maclaurin power series expansion (3) has a finite positive radius of convergence, and this radius of convergence, denoted by R, is located between 4π 3 and 3π 2 . In fact, the number R is the first positive zero of the equation tan x − x = 0.…”

Section: Motivationsmentioning

confidence: 99%

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Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent

Sun

Guo

2023

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In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients.

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“…In [1] (p. 235), see also Section 1 in [2], the sequence of rational numbers V k for k ≥ 0 was defined by means of…”

Section: Motivationsmentioning

confidence: 99%

Section: Motivationsmentioning

confidence: 99%