I'm trying to solve an exercise in the book The Arithmetic of Elliptic Curves by Joseph H. Silverman on the page 106. The exercise asks to prove that \begin{equation} nP=\left( x-\frac{\psi_{n-1}\psi_{n+1}}{\psi_{n}^2},\frac{\psi_{n+2}\psi_{n-1}^2-\psi_{n-2}\psi_{n+1}^2}{4y \psi_n^3}\right) \end{equation} where P is a point of an elliptic curve E given by $y^2=x^3+ax+b$ and \begin{align*} \psi_0 & =0\\ \psi_1 & =1\\ \psi_2 & =2y\\ \psi_3 & =3x^4+6ax^2+12bx-a^2\\ \psi_4 & =4y(x^6+5ax^4+20bx^3-5a^2x^2-4abx-8b^2-a^3)\\ \psi_{2m+1} & =\psi_{m+2}\psi_{m}^3-\psi_{m-1}\psi_{m+1}^3 \ \ m\geqslant2\\ \psi_{2m} & =(2y)^{-1}(\psi_m)(\psi_{m+2}\psi_{m-1}^2-\psi_{m-2}\psi_{m+1}^2) \ m\geqslant3\\ \end{align*} Does anyone have an idea how to prove this? I tried to do some kind of recursive reasoning however I did not really succeed.
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