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Bauer-Muir Transformation

Continued fraction definition

$Given a sequence w = {w_n} of complex numbers, the Bauer-Muir transformation of a generalized continued fraction ξ of the form ξ = b_0 + a_1/(b_1 + a_2/(b_2 + a_3/(b_3 + ...))) with respect to w is the continued fraction ζ of the form ζ = d_0 + continued fraction k _(m=1)^∞ c_m/d_m whose canonical numerators C_n, respectively canonical denominators D_n, are defined by the recursion relations C_(-1) = 1, C_n = A_n + w_n A_(n - 1), D_(-1) = 0, and D_n = B_n + w_n B_(n - 1) for n = 1, 2, 3, .... Here, A_n/B_n denotes the canonical nth convergents of ξ. One well-know result concerning the Bauer-Muir transformation is a characterization of its existence. In particular, given a generalized continued fraction ξ of the form stated above and a corresponding complex sequence w = {w_n}, the Bauer-Muir transformation of ξ with respect to w exists if and only if λ_n !=0 where here, λ_n = a_n - w_n(b_n + w_n) for n = 1, 2, 3, .... Moreover, Lorentzen and Waadeland showed that if it exists, the Bauer-Muir transformation of ξ with respect to w has the form ζ = b_0 + w_0 + λ_1/(b_1 + w_1 + c_2/(d_2 + c_3/(d_3 + ...))) where c_n = a_(n - 1) q_(n - 1) and d_n = b_n + w_n - w_(n - 2) q_(n - 1) for q_n = λ_(n + 1)/λ_n, n = 1, 2, 3, .... More specific properties of the Bauer-Muir transformation have also been studied in relation to various other topics including but not limited to the Rogers-Ramanujan continued fraction.$

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