Explicit evaluations of a level 13 analogue of the Rogers-Ramanujan continued fraction
The paper deals with the results concerning the level \(13\) analogue of the Rogers-Ramanujan continued fractions. Set \(R(q)=q\prod_{j=1}^\infty (1-q^j)^{(\frac j{13})}\), \(F(q)=\frac 1{R(q)}-3-R(q)\) and \(T(q)=\frac {F(q)}{F^2(q)+F(q)+13}\) where \(\mid q\mid <1\) and \((\frac ab)\) is the Legendre symbol. Then the authors prove \[ \left(\frac{\sqrt{13}+3}2+R(e^{-2\pi\alpha})\right)\left(\frac{\sqrt{13}+3}2+R(e^{-2\pi\beta})\right)=\frac{13+3\sqrt{13}}2 \] \[ \left(\frac{\sqrt{13}-3}2-R(e^{-\pi\alpha})\right)\left(\frac{\sqrt{13}-3}2-R(e^{-\pi\beta})\right)=\frac{13-3\sqrt{13}}2 \] where \(\alpha\beta =\frac 1{13}\) and \(\alpha ,\beta >0\), \[ F(e^{-2\pi\sqrt{\frac 2{13}}})=u,\;F(-e^{-\pi\sqrt{\frac 2{13}}})=v_1,\;F(e^{-\pi\sqrt{\frac 2{13}}})=v_2,\;F(e^{-4\pi\sqrt{\frac 2{13}}})=v_3 \] where \[ t=\frac{17}{24}-\frac 1{24}(624\sqrt{78}+5239)^{\frac 13}+\frac 1{24}(624\sqrt{78}-5239)^{\frac 13}, \;u=\frac{1-6t+\sqrt{1-12t-16t^2}}{2t} \] and \(v_1\), \(v_2\) and \(v_3\) are the roots of the cubic polynomial \(x^3-(u^2+4u)x^2-(4u^2+13u)x +u^3\) with \(v_1<0<v_2<v_3\), \[ T(e^{-2\pi\sqrt{\frac 3{13}}})=\frac{7\sqrt{13}-25}6,\;T(-e^{-\pi\sqrt{\frac 3{13}}})=\frac{1-\sqrt{13}}6,\;T(-e^{-\pi\sqrt{\frac 7{13}}})=-\frac 17, \] \[ T(e^{-2\pi\sqrt{\frac 7{13}}})=\frac{17}{63}-\frac 2{189}(4563+273\sqrt{273})^{\frac 13}+\frac 2{189}(4563-273\sqrt{273})^{\frac 13}. \] They give the explicit evaluation of \(R(e^{-\pi\sqrt{\frac m{13}}})\) for \(m=15, 31, 55, 231, 255\) and also evaluate \(R(q)\) using Ramanujan-Weber class invariants. Finally they prove that \(R(e^{-\pi\sqrt n})\) is a unit for every positive rational number \(n\).
scientific article; zbMATH DE number 6308464
scientific article