Some \(q\)-identities associated with Ramanujan's continued fraction (Q5940255)

The author derives some interesting identities reminiscent of formulae due to Ramanujan. NEWLINENEWLINENEWLINEThe main results are a representation using NEWLINE\[NEWLINE C(-q,q)=1+{(1+1/q)q\over 1+}{q^2\over 1+}{(1+1/q^2)q^3\over 1+}{q^4\over 1+}\cdots=1+{(q^2;q^4)_{\infty}^2\over (q^3;q^4)_{\infty}(q;q^4)_{\infty}} NEWLINE\]NEWLINE as NEWLINE\[NEWLINE \left[C(-q,q)-1\right]^2={(q^2;q^4)_{\infty}^2\over (q^4;q^4)_{\infty}^2} \left[\sum_{n=0}^{\infty} q^{4n^2+2n}{1+q^{4n+1}\over 1-q^{4n+1}} - \sum_{n=0}^{\infty} q^{4n^2+6n+2}{1+q^{4n+3}\over 1-q^{4n+3}} \right] NEWLINE\]NEWLINE and two identities for NEWLINE\[NEWLINE G(-q;q)=\sum_{n=0}^{\infty} q^{n(n-1)/2}(-q)_n/(q)_n = (-q,q)_{\infty}\left[{1\over (q^2;q^4)_{\infty}}+{(q^2;q^4)_{\infty}\over (q^3;q^4)_{\infty}(q;q^4)_{\infty}}\right] NEWLINE\]NEWLINE and NEWLINE\[NEWLINE H(-q;q)=\sum_{n=0}^{\infty} q^{n(n+1)/2}(-q)_n/(q)_n={(-q;q)_{\infty}\over (q^2;q^4)_{\infty}} NEWLINE\]NEWLINE of the form NEWLINE\[NEWLINE {(q^4;q^4)_{\infty}^2\over (-q;q)_{\infty}^2}\left[H(-q,q)\right]^2=\sum_{n=-\infty}^{\infty} {q^n\over 1-q^{4n+2}} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE {(q^4;q^4)_{\infty}^2\over (-q;q)_{\infty}^2}\left[G(-q,q)-H(-q,q)\right]^2=\sum_{n=-\infty}^{\infty} {q^n\over 1-q^{4n+1}} NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe connections between the three quantities introduced above are: NEWLINE\[NEWLINE C(-q,q)={G(-q,q)\over H(-q,q)},\;G(-q,q)-H(-q,q)={(-q;q)_{\infty}(q^2;q^4)_{\infty}\over (q;q^4)_{\infty}(q^3;q^4)_{\infty}} .NEWLINE\]