On polynomials with a prescribed zero (Q5906518)

Let \(p(z)= \sum^ n_{j=0} a_ j z^ j\) be a polynomial of degree at most ``\(n\)'' vanishing at \(z= \zeta\) \((\zeta^{n+1}\neq 1)\). The authors use Lagrange's interpolation formula to prove that for every complex number \(\lambda\) and \(k= 0,1,2,\dots,n\), \[ \begin{multlined} a_ k= \bigl(1/k!(n- k+ 1)\bigr)\sum^{n-k}_{j=0} p^{(k)}\bigl(e^{2\pi ji/(n- k+1)}\bigr)-\\ \bigl(\lambda/(n+ 1)\bigr)\sum^ n_{j=0} p\bigl(e^{2\pi ji/(n+ 1)}\bigr)\bigl/\bigl(\zeta e^{-2\pi ji/(n+1)}- 1\bigr).\end{multlined} \] {}.