# Fundamental Recurrence Formulas

In the theory of continued fractions, the fundamental recurrence formulas relate the partial numerators and the partial denominators with the numerators and denominators of the fraction's successive convergents. Let

be a general continued fraction, where the an (the partial numerators) and the bn (the partial denominators) are numbers. Denoting the successive numerators and denominators of the fraction by An and Bn, respectively, the fundamental recurrence formulas are given by

begin{align} A_0& = b_0& B_0& = 1\ A_1& = b_1 b_0 + a_1& B_1& = b_1\ A_{n+1}& = b_{n+1} A_n + a_{n+1} A_{n-1}& B_{n+1}& = b_{n+1} B_n + a_{n+1} B_{n-1}, end{align}

The continued fraction's successive convergents are then given by

These recurrence relations are due to John Wallis (1616-1703) and Leonhard Euler (1707-1783)

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Fundamental Recurrence Formulas - A Simple Example
... form that represents the golden ratio φ Applying the fundamental recurrence formulas we find that the successive numerators An are {1, 2, 3, 5, 8, 13...} and the successive denominators Bn are {1 ... the partial numerators in this example are equal to one, the determinant formula assures us that the absolute value of the difference between successive convergents ...

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