Reciprocal difference

In mathematics, the reciprocal difference of a finite sequence of numbers ${\displaystyle (x_{0},x_{1},...,x_{n})}$ on a function ${\displaystyle f(x)}$ is defined inductively by the following formulas:

${\displaystyle \rho _{1}(x_{0},x_{1})={\frac {x_{0}-x_{1}}{f(x_{0})-f(x_{1})}}}$

${\displaystyle \rho _{2}(x_{0},x_{1},x_{2})={\frac {x_{0}-x_{2}}{\rho _{1}(x_{0},x_{1})-\rho _{1}(x_{1},x_{2})}}+f(x_{1})}$

${\displaystyle \rho _{n}(x_{0},x_{1},\ldots ,x_{n})={\frac {x_{0}-x_{n}}{\rho _{n-1}(x_{0},x_{1},\ldots ,x_{n-1})-\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})}}+\rho _{n-2}(x_{1},\ldots ,x_{n-1})}$