Factors and roots of the van der Pol polynomials

Howard, F. T.

doi:10.1090/s0002-9939-1975-0379347-9

Cited by 3 publications

(1 citation statement)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Rayleigh functions can be represented in terms of the zeros of the Bessel function; see [3][4][5]. Howard [6] showed that Euler and Bernoulli polynomials have identical properties to the Van der Pol polynomials.…”

Section: Introductionmentioning

confidence: 99%

Starlikeness Associated with the Van Der Pol Numbers

et al. 2023

View full text Add to dashboard Cite

In this paper, we define a subclass of starlike functions associated with the Van der Pol numbers. For this class, we derive structural formula, radius of starlikeness of order α, strong starlikeness, and some inclusion results. We also study radii problems for various classes of analytic functions. Furthermore, we investigate some coefficient-related problems which include the sharp initial coefficient bounds and sharp bounds on Hankel determinants of order two and three.

show abstract

Section: Introductionmentioning

confidence: 99%

Starlikeness Associated with the Van Der Pol Numbers

et al. 2023

View full text Add to dashboard Cite

show abstract

Numbers generated by the reciprocal of 𝑒^{𝑥}-𝑥-1

Howard¹

1977

Math. Comp.

View full text Add to dashboard Cite

In this paper we examine the polynomials An(z) and the rational numbers An = An(0) defined by means of oo exzx2(ex-x-I)'1 =22 An(z)x"/n\. n=0 We prove that the numbers An are related to the Stirling numbers and associated Stirling numbers of the second kind, and we show that this relationship appears to be a logical extension of a simUar relationship involving Bernoulli and Stirling numbers. Other similarities between An and the Bernoulli numbers are pointed out. We also reexamine and extend previous results concerning An and An(z). In particular, it has been conjectured that An has the same sign as-cos nd, where re is the zero of exx-1 with smallest absolute value. We verify this for 1 < n < 14329 and show that if the conjecture is not true for An, then Icos i7Ô| < 10-'n-''. We also show that An(z) has no integer roots, and in the interval [0, 1), An(z) has either two or three real roots.

show abstract

Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)

Murugusundaramoorthy,

Bulboacă

2023

Mathematica Slovaca

View full text Add to dashboard Cite

The purpose of this paper is to find coefficient estimates for the class of functions ℳ N ( γ , ϑ , λ ) consisting of analytic functions f normalized by f(0) = f′(0) – 1 = 0 in the open unit disk D subordinated to a function generated using the van der Pol numbers, and to derive certain coefficient estimates for a 2, a 3, and the Fekete-Szegő functional upper bound for f ∈ ℳ N ( γ , ϑ , λ ) . Similar results were obtained for the logarithmic coefficients of these functions. Further application of our results to certain functions defined by convolution products with a normalized analytic functions is given, and in particular, we obtain Fekete-Szegő inequalities for certain subclasses of functions defined through the Poisson distribution series.

show abstract

Factors and roots of the van der Pol polynomials

Abstract: ABSTRACT. The van der Pol polynomials V (a) are defined by means of oo xlexa\jox(ex + 1) -12(e* -l)]-1 = 1 V (a)xrt/n\. n=0 "In this paper new properties of these polynomials are derived. It is shown that neither V', (a) nor K, ,,(a)/(a ~ 1/2) has rational roots, and that if

Cited by 3 publications

References 15 publications

Starlikeness Associated with the Van Der Pol Numbers

Starlikeness Associated with the Van Der Pol Numbers

Numbers generated by the reciprocal of 𝑒^{𝑥}-𝑥-1

Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)

Contact Info

Product

Resources

About