Reciprocal difference
# Reciprocal difference

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

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