Thiele's Interpolation Formula

Let $\rho$ be a Reciprocal Difference. Then Thiele's interpolation formula is the Continued Fraction

$$f(x)=f(x_1)+{x-x_1\over\rho(x_1, x_2)+} {x-x_2\over\rho_2(x_... ...{x-x_3\over \rho_3(x_1, x_2, x_3, x_4)-\rho(x_1, x_2)+\ldots}.$$

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 881, 1972.

Milne-Thomson, L. M. The Calculus of Finite Differences. London: Macmillan, 1951.