On some arithmetical properties of Rogers-Ramanujan continued fraction
Define a general Rogers-Ramanujan continued fraction by \[ R(z;x)=1+\frac{zx}{1}{\atop +}\frac{z^2 x}{1}{\atop +} \frac{z^3 x}{1}{\atop +}\dots\quad(|z|<1). \] The authors prove that \(R(1/q; a/b)\) is not a quadratic number if \(a/b\) is a non-zero rational number, \(q\) is an integer with \(|q|\geq 2\) and \(a^4<|q|\). This main theorem is a direct consequence of the general result on continued fractions with rational coefficients. They also establish a more general result on the linear independence related to Rogers-Ramanujan identities, which implies that \(R(1/q;1)\) is not quadratic for an integer \(q\) with \(|q|\geq 2\). The same technique is used as in the paper of the first author [C. R. Acad. Sci., Paris, Sér. I 320, 1041-1044 (1995; Zbl 0836.11024)].
scientific article; zbMATH DE number 1546693