Evaluations of a continued fraction of Ramanujan (Q2355995)

Let \((a;q)_k:=\prod_{n=0}^{k-1}(1-aq^n) \). The Ramanujan eta function is defined as \(f(-q):=(q;q)_\infty\) and the Weber function is defined as \(\Phi(-q):=(-q;q)_\infty\). The elliptic integral of the first kind is denoted \[ K(x)=\int\limits_0^{\pi/2} \frac{1}{\sqrt{1-x^2 \sin^2 t}}\, dt. \] It is well known that if \(q,a,b\) are complex numbers with \(|q|<1\) or if \(q,a,b\) are complex numbers with \(a=bq^m\) for same integer \(m\) then \[ U=U(a,b;q)=\frac{(-a;q)_\infty\, (b;q)_\infty-(a;q)_\infty\, (-b;q)_\infty}{(-a;q)_\infty\, (b;q)_\infty+(a;q)_\infty\, (-b;q)_\infty}= \] \[ =\frac{a-b}{1-q+}\; \frac{(a-bq)(aq-b)}{1-q^3+}\;\frac{q(a-bq^2)(aq^2-b)}{1-q^5+}\;\frac{q^2(a-bq^3)(aq^3-b)}{1-q^7+}\; \dots \] The main concern in the present article is to simplify and evaluate \(U\) for some specific cases of \(a,b\) and \(q\). Eighteen propositions witch are related functions \(f(-q),\Phi(-q), K(x) \) and others are proven in the article. The bibliography contains 8 items.