Can someone please assist me with the missing steps in the proof of 'proposition 3' in M. Kaneko and D. Zagier paper (https://people.mpim-bonn.mpg.de/zagier/files/progmath/129/165/fulltext.pdf, pg. 4). I am struggling to understand this brief proof (only 3 lines), line 2 to 3 in particular, viz: $$ H(w, wq, w^{\frac{1}{2}}q\zeta) = -w^{-\frac{1}{24}}\frac{1-w^{-\frac{1}{8}}q^{-\frac{1}{12}}\zeta^{-1}}{1-w^{\frac{1}{8}}q^{\frac{1}{12}}\zeta}H(w, q,\zeta). \tag{2}$$ $$ H(w, wq, w^{\frac{1}{2}}q\zeta) = -w^{-\frac{1}{6}}q^{-\frac{1}{2}}\zeta^{-1}H(w,q,\zeta). \tag{3}$$ Here $$ H(w,q,\zeta) := q^{\frac1{24}} \prod_{m=0}^{\infty} {(1-q^{\frac{m}2}w^{\frac{m^2}8}}\zeta)(1-q^{\frac{m}2}w^{-\frac{m^2}8}\zeta^{-1}). $$
I am trying to get an expression for $H(w, w^{\frac{1}{2}}q^{2}, w^{\frac{1}{8}}q\zeta)$. To do this I need to understand the said proof. That is, the above two lines.
Thank you in advance.
Shaka!
This is really about simplification of the term $$\frac{1-w^{-\frac{1}{8}} q^{-\frac{1}{12}} \zeta^{-1}} {1-w^{\frac{1}{8}}q^{\frac{1}{12}}\zeta}.$$ Everything else just happens to be at the periphery, and I ignore everything else completely in this answer.
Let $X$ denote $$ w^{-1/8} q^{-1/12} \zeta^{-1},$$ so that the term can be written $$ \frac{1 - X}{1 - X^{-1}}.$$
Then the proof from $(2)$ to $(3)$ in your OP follows from noticing $$ \frac{1 - X}{1 - X^{-1}} = \frac{1}{X^{-1}}\frac{1 - X}{X - 1} = - X.$$
I note that there is a small typo in $(3)$: it should read $$H(w, wq, w^{\frac{1}{2}}q\zeta) = -w^{-\frac{1}{6}}q^{-\frac{1}{12}}\zeta^{-1}H(w,q,\zeta),$$ where the difference is the exponent $- \frac{1}{12}$ of $q$.