A generalization of Halphén's formula for derivatives (Q509618)

Halphén's formula published in 1880 shows a compact expression for the derivative \[ \left ( f \left ( \frac 1x \right) g(x) \right )^{(n)}. \] This paper provides a generalization, when a formula for \[ \left ( \frac d{dz}\right )^{(n)} \frac{f(z^{-1})g(z)h^{n+1}(z)}{h(z)+z(1-zx^{-1})h'(z)}\bigg |_{z=x} \] is proved. The proof uses methods of complex analysis. For \(h(z)=1\), the author obtains the Halphén's formula.