Reciprocal difference

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In mathematics, the reciprocal difference of a finite sequence of numbers (x0,x1,...,xn) on a function f(x) is defined inductively by the following formulas:

\rho_1(x_0, x_1) = \frac{x_0 - x_1}{f(x_0) - f(x_1)}
\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)
\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})

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