×

Zero-free regions for Bernoulli polynomials. (English) Zbl 0532.30005

The author continues his study of the complex zeros of the Bernoulli polynomials [begun in ibid. 5, 189-194 (1983; Zbl 0526.30007)]. He shows that the region \(x^ 2<c| y|,\quad c=2/33,\) contains no zero of \(B_ n(z+1/2)\) for \(n\geq 129.\) He also shows that the strip \(- .1577<x<1.1577\) contains no non-real zero of \(B_ n(z)\) for \(n\geq 1\), improving a result of R. Spira [Proc. Am. Math. Soc. 17, 1466-1467 (1966; Zbl 0154.319)]. In addition, he shows that the sections of the Maclaurin series for the sine and cosine have no zeros in the region \(x^ 2<c| y|\) with \(c=1/2\).
Reviewer: D.W.Boyd

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
33E99 Other special functions
30C10 Polynomials and rational functions of one complex variable
PDFBibTeX XMLCite