1 Erratum to: Comput. Methods Funct. Theory (2017) 17:19–45 DOI 10.1007/s40315-016-0174-y

In the proof of Lemma 2.6 Case 2 has to be replaced as follows.

Case 2 Let \(A=\omega ^l\) for some l such that \(0\le l\le m-1\). Then also \(F(t_0)=0=F^{'}(t_0)\) implies \(e^{nt_0}=A\) and \(e^{mt_0}=1\). Now, if possible, suppose that there exist more than one \(t_0\) such that \(e^{mt_0}=1\) and \(e^{nt_0}=A\), i.e., there exist \(t_{0p}\), \(t_{0q}\) with \(e^{t_{0p}}\ne e^{t_{0q}}\) such that \(e^{mt_{0p}}=1=e^{mt_{0q}}\) and \(e^{nt_{0p}}=A=e^{nt_{0q}}\), i.e., \(e^{m(t_{0p}-t_{0q})}=1\) and \(e^{n(t_{0p}-t_{0q})}=1\), i.e., \(m(t_{0p}-t_{0q})=2k_1\pi i\) for some \(k_1\in \mathbb {Z}\) and \(n(t_{0p}-t_{0q})=2k_2\pi i\) for some \(k_2\in \mathbb {Z}\). Since \(\gcd (m,n)=1\), so there exists \(x, y\in \mathbb {Z}\) such that \(mx+ny=1\), i.e., \(m(t_{0p}-t_{0q})x+n(t_{0p}-t_{0q})y=(t_{0p}-t_{0q})\), i.e., \(2k_{1}\pi ix+2k_{2}\pi iy=(t_{0p}-t_{0q})\), i.e., \(2\pi i(xk_{1}+yk_{2})=(t_{0p}-t_{0q})\), i.e., \(2s\pi i=(t_{0p}-t_{0q})\), where \(s=xk_{1}+yk_{2}\in \mathbb {Z}\).

Therefore \(e^{t_{0p}}=e^{t_{oq}}\), which is a contradiction to \(e^{t_{0p}}\not =e^{t_{0q}}\). Therefore \(\phi (e^t)\), hence \(\phi (z)\), has exactly one multiple zero \(\omega ^j\), where \(0\le j\le m-1\) and \(\omega ^{mj}=1\), \(\omega ^{nj}=\omega ^l\) and that is of multiplicity 4. Now in particular if \(A=1\), then we have \(\omega ^j\) is the multiple zero of \(\phi (z)\) for some \(j\in \{0,1,\ldots ,m-1\}\) such that \(\omega ^{mj}=1\) and \(\omega ^{nj}=1\) i.e., \(\omega ^j=1\) as \(\gcd (m,n)=1\).