Symmetry and folding of continued fractions (Q558134)

Denote the \(h\)-th convergent of the continued fraction expansion of \([a_0,a_1,a_2,\dots]\) by \(x_h/y_h=[a_0,a_1,\dots,a_h]=[a_0, w_h]\). Then the Folding Lemma by \textit{M. Mendès France} [Acta Arith. 23, 207--215 (1973; Zbl 0228.10007)] is stated as \(x_h/y_h+(-1)^h/cy_h^2=[a_0,w_h,c-y_{h-1}/y_h]=[a_0,w_h,c,-\overleftarrow w_h]\) where \(-\overleftarrow w_h=-a_h,-a_{h-1},\dots,-a_1\). The Folding Lemma makes it easy to adjust a continued fraction expansion for an appended term. The author describes the relationship between the expansion of a quadratic irrational integer and the Folding Lemma. The author shows that if \(\delta\) is the larger root of \(\delta^2-t\delta+n=0\), where \(t^2-4n>0\) is not a square, then \(\delta=[a_0,\overline{w,u,\overleftarrow w,2a_0-t}]\), where \(u\) is some integer. The result here has been already known. However, the proof can be achieved elegantly by the Folding Lemma.