Theorems stated by Ramanujan. VII: Theorems on continued fractions. (Q1434884)

Verf. behandelt folgende mit gewissen unendlichen Produkten und den Eigenschaften gewisser Modulfunktionen zusammenhängende Sätze: (1) Mit \[ u = \dfrac x{1+\lower1.7ex\hbox{\(\dfrac{x^5}{1}\)}} \lower4.5ex\hbox{\({}+{}\)} \lower6.2ex\hbox{\(\dfrac{x^{10}}{1}\)} \lower7.8ex\hbox{\({}+{}\)} \lower9.5ex\hbox{\(\dfrac{x^{15}}{1}\)} \lower11.1ex\hbox{\({}+{}\)} \lower12.8ex\hbox{\(\dfrac{x^{20}}{1}\)} \lower14.3ex\hbox{\({}+\cdots\)} \] und \[ v = \dfrac{x^{\frac15}}{1} \lower1.5ex\hbox{\({}+{}\)} \lower3.0ex\hbox{\(\dfrac{x}{1}\)} \lower4.5ex\hbox{\({}+{}\)} \lower6.2ex\hbox{\(\dfrac{x^{2}}{1}\)} \lower7.8ex\hbox{\({}+{}\)} \lower9.5ex\hbox{\(\dfrac{x^{3}}{1}\)} \lower11.1ex\hbox{\({}+{}\)} \lower12.8ex\hbox{\(\dfrac{x^{4}}{1}\)} \lower14.3ex\hbox{\({}+\cdots\)} \] ist \[ v^5=u\cdot\frac{1-2u+4u^2-3u^3+u^4}{1+3u+4u^2+2u^2+u^4}. \] (2) \( \dfrac{1}{1} \lower1.5ex\hbox{}{}+{}\)