Bernoulli polynomial root symmetry

Since at least two people (myself and the asker of this question) have wondered about this at some point, this question will likely be useful to someone else eventually.

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I think your question is very interesting. What software are you using to compute the roots of the Bernoulli polynomials?

I used Maxima. We know that the Bernoulli polynomials are symmetric (for $n$ even) or anti-symmetric (for $n$ odd) around the axis $x=1/2$. This means that the roots are symmetric with respect to the $x=1/2$ axis. Now since the coefficients are real the complex roots are conjugate. Hence we expect symmetry both with respect to both, horizontal and vertical, axes.

Here is the maxima set of instructions:

\begin{verbatim} (%i1) fpprec:128$ (%i2) eqn : bernpoly(x,170)$ (%i3) slns : bfallroots(eqn)$ (%i4) slnlist : map(rhs, slns)$ (%i5) reals : realpart(slnlist)$ (%i6) imag : imagpart(slnlist)$ (%i7) plot2d ( [discrete, reals, imag], [style, points]); \end{verbatim}

Here is the plot:

enter image description here Thanks.

Tags:

Number Theory

Roots

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