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:

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

${\displaystyle \rho _{2}(x_{1},x_{2},x_{3})={\frac {x_{1}-x_{3}}{\rho _{1}(x_{1},x_{2})-\rho _{1}(x_{2},x_{3})}}+f(x_{2})}$

${\displaystyle \rho _{n}(x_{1},x_{2},\ldots ,x_{n+1})={\frac {x_{1}-x_{n+1}}{\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})-\rho _{n-1}(x_{2},x_{3},\ldots ,x_{n+1})}}+\rho _{n-2}(x_{2},\ldots ,x_{n})}$