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})

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