Functiones et Approximatio Commentarii Mathematici

Variations of the Ramanujan polynomials and remarks on $\zeta(2j+1)/\pi^{2j+1}$

Matilde N. Lalín and Mathew D. Rogers

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Abstract

We observe that five polynomial families have all of their roots on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang [MSW]. Our proofs rely upon theorems of Schinzel [S], and Lakatos and Losonczi [LL] and some generalizations.

Article information

Source
Funct. Approx. Comment. Math., Volume 48, Number 1 (2013), 91-111.

Dates
First available in Project Euclid: 25 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.facm/1364222831

Digital Object Identifier
doi:10.7169/facm/2013.48.1.8

Mathematical Reviews number (MathSciNet)
MR3086962

Zentralblatt MATH identifier
1272.26007

Subjects
Primary: 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
Secondary: 11B68: Bernoulli and Euler numbers and polynomials

Keywords
Ramanujan polynomials Riemann zeta function values reciprocal polynomials roots on the unit circle Bernoulli numbers Euler numbers.

Citation

Lalín, Matilde N.; Rogers, Mathew D. Variations of the Ramanujan polynomials and remarks on $\zeta(2j+1)/\pi^{2j+1}$. Funct. Approx. Comment. Math. 48 (2013), no. 1, 91--111. doi:10.7169/facm/2013.48.1.8. https://projecteuclid.org/euclid.facm/1364222831


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