The construction of rational iterating functions

Suppose integers $n⩾1$ and $\sigma ⩾2$ are given, together with n distinct points $z_2,\dots ,z_n$ , in the complex plane. Define ${\mathrm{\Phi }}_M={\mathrm{\Phi }}_M(\sigma ;z_1,\dots ,z_n)$ to be the class of rational functions ${\varphi }_{p,q}(z)=g_p(z)/h_q(z)$ (where g and h are polynomials of degree $p⩾1$ and $q⩾1$ , respectively) such that $(1)p+q+1=M,(2)\varphi$ when iterated converges with order $\sigma$ at each $z_i,i=1,\dots ,n$ . Then if $M>\sigma n,{\mathrm{\Phi }}_M$ is null; if $M=\sigma n$ ${\mathrm{\Phi }}_M$ contains exactly $\sigma n$ elements. For every $M⩾\sigma n$ , we show how to construct all the elements of ${\mathrm{\Phi }}_M$ by expressing, for each choice of p and q which satisfies $p+q+1=M$ , the coefficients of $g_p$ and $h_q$ in terms of $M-\sigma n$ arbitrarily chosen values. In fact, these coefficients are expressed in terms of generalized Newton sums $S_n^{j,k}=S_n^{j,k}(z_1,\dots ,z_n)$ , $1⩽j⩽n,k⩾n$ , which we show may be calculated by recursion from the normal Newton sums $S_n^{j,n}$ . Hence, given a polynomial $f_n(z)$ with n distinct (unknown) zeros $z_1,\dots ,z_n$ , we may construct all ${\varphi }_{p,q}(z)$ which converge to the $z_i$ with order $\sigma$ in the case $\sigma =2$ , the choice $p=n$ , $q=n-1$ , yields the Newton-Raphson iteration ${\varphi }_{n,n-1}\in {\mathrm{\Phi }}_{2n}$ ; the Schröder and König iterations are shown to be elements of ${\mathrm{\Phi }}_{2(2\sigma -3)(n-1)+2}$ and ${\mathrm{\Phi }}_{2(\sigma -1)(n-1)+2}$ , respectively. We show by example that there exist cases in which ${\varphi }_{n,n-1}$ has an undesirable property (attractive cycles) not shared by other iterating functions in the same class ${\mathrm{\Phi }}_{2n}$ .