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:

:$\backslash rho\_1(x\_1,\; x\_2)\; =\; \backslash frac\{x\_1\; -\; x\_2\}\{f(x\_1)\; -\; f(x\_2)\}$

:$\backslash rho\_2(x\_1,\; x\_2,\; x\_3)\; =\; \backslash frac\{x\_1\; -\; x\_3\}\{\backslash rho\_1(x\_1,\; x\_2)\; -\; \backslash rho\_1(x\_2,\; x\_3)\}\; +\; f(x\_2)$

:$\backslash rho\_n(x\_1,x\_2,\backslash ldots,x\_\{n+1\})=\backslash frac\{x\_1-x\_\{n+1\}\}\{\backslash rho\_\{n-1\}(x\_1,x\_2,\backslash ldots,x\_\{n\})-\backslash rho\_\{n-1\}(x\_2,x\_3,\backslash ldots,x\_\{n+1\})\}+\backslash rho\_\{n-2\}(x\_2,\backslash ldots,x\_\{n\})$