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Generalized Continued Fraction

Continued fraction definition

$There are no fewer than two distinct continued fraction concepts described as generalized continued fraction. Perhaps most commonly, a numerical continued fractions ξ is described as "generalized" provided ξ is of the form ξ = b_0 + a_1/(b_1 + a_2/(b_2 + a_3/(b_3 + ...))) where the partial numerators a_1, a_2, ... are allowed to be arbitrary. This is in contrast to the case where a_k = 1 for k = 1, 2, ..., whereby the resulting continued fraction is considered regular. At least one other source defines a generalized continued fraction to be any continued fraction with elements consisting of arbitrary mathematical objects such as vectors in C^n, C-valued square matrices, Hilbert space operators, multivariate expressions, other continued fractions, etc. As it is written, a numerical continued fraction can be used to construct one of these generalized fractions in the following way: Given a continued fraction of the form ξ = b_0 + continued fraction k _(n=1)^∞ a_n/b_n with associated second-order recursion A_n = b_n A_(n - 1) + a_n A_(n - 2), B_n = b_n B_(n - 1) + a_n B_(n - 2), n = 1, 2, 3, ..., subject to the initial conditions B_(-1) = 0, A_0 = b_0, A_(-1) = B_0 = 1, define an nth order recursion among the elements of ξ. The result of this will be a continued fraction ξ^^ which is said to be generalized due to the fact that each of the approximants (A_n)^^/(B_n)^^ of ξ^^ are n-dimensional vectors rather than numerical constants.$

Details

partial denominator | partial numerator | sequence

Timeline

References

Annie A.M. Cuyt, Vigdis Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones. Handbook of Continued Fractions for Special Functions. p. 28, 2008.

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