4 Removed duplicate paragraph that was carried over from earlier edit

[Edited again to give a second identity relating $E$ to eta products]

Continued fraction or not, an expression $q^{\frac{(r-s)^2}{8(r+s)}} f(\pm q^r, \pm q^s)$ is a modular form of weight $1/2$ for all integers $r,s$ with $r+s>0$, because it is a sum $\sum_{n=-\infty}^\infty \pm q^{(cn+d)^2}$ with rational $c,d$ and periodic signs. Therefore the quotient of two such expressions is a modular function, and takes algebraic valus at quadratic imaginary values.

The quotient $$E(q) = q - q^2 + q^6 - q^7 + q^8 - q^9 + q^{11} - 2q^{12} + 2q^{13} - 2q^{14} + 2q^{15} \cdots$$ looks like a modular unit $-$ its logarithmic derivative has small coefficients $-$ but not quite an eta product; instead it seems to be a quotient of Klein forms: $$E(q) = q \prod_{n=1}^\infty (1-q^n)^{\chi(n)},$$ where $\chi$ is the Dirichlet character of conductor $12$, given by

$$\chi(n) = \cases{ +1,& if n \equiv \pm 1 \bmod 12; \cr -1,& if n \equiv \pm 5 \bmod 12; \cr 0,& otherwise. }$$ One identity relating $E$ to an $\eta$ product, similar to but somewhat more complicated than the ones you give for $A,B,C,D,$ is $$\frac1{E(q)} - E(q) = \frac{\eta(2\tau)^2 \eta(6\tau)^4}{\eta(\tau)\eta(3\tau)\eta(12\tau)^4}.$$ Two identities relating $E$ to $\eta$ products, similar to but somewhat more complicated than the ones you give for $A,B,C,D,$ are $$\frac1{E(q)} - E(q) = \frac{\eta(2\tau)^2 \eta(6\tau)^4}{\eta(\tau)\eta(3\tau)\eta(12\tau)^4},$$ and (a bit simpler) $$\frac1{E(q)} + E(q) = \frac{\eta(4\tau)}{\eta(\tau)} \Bigl(\frac{\eta(3\tau)}{\eta(12\tau)}\Bigr)^3.$$

3 Added a second identity relating E to an eta product

[Edited again to give a second identity relating $E$ to eta products]

Continued fraction or not, an expression $q^{\frac{(r-s)^2}{8(r+s)}} f(\pm q^r, \pm q^s)$ is a modular form of weight $1/2$ for all integers $r,s$ with $r+s>0$, because it is of the form a sum $\sum_{n=-\infty}^\infty \pm q^{(cn+d)^2}$ with rational $c,d$ and periodic signs. Therefore the quotient of two such expressions is a modular function, and takes algebraic valus at quadratic imaginary values.

The quotient $$E(q) = q - q^2 + q^6 - q^7 + q^8 - q^9 + q^{11} - 2q^{12} + 2q^{13} - 2q^{14} + 2q^{15} \cdots$$ looks like a modular unit $-$ its logarithmic derivative has small coefficients $-$ but not quite an eta product; instead it seems to be a quotient of Klein forms: $$E(q) = q \prod_{n=1}^\infty (1-q^n)^{\chi(n)},$$ where $\chi$ is the Dirichlet character of conductor $12$, given by

$$\chi(n) = \cases{ +1,& if n \equiv \pm 1 \bmod 12; \cr -1,& if n \equiv \pm 5 \bmod 12; \cr 0,& otherwise. }$$ One identity relating $E$ to an $\eta$ product, similar to but somewhat more complicated than the ones you give for $A,B,C,D,$ is $$\frac1{E(q)} - E(q) = \frac{\eta(2\tau)^2 \eta(6\tau)^4}{\eta(\tau)\eta(3\tau)\eta(12\tau)^4}.$$ Two identities relating $E$ to $\eta$ products, similar to but somewhat more complicated than the ones you give for $A,B,C,D,$ are $$\frac1{E(q)} - E(q) = \frac{\eta(2\tau)^2 \eta(6\tau)^4}{\eta(\tau)\eta(3\tau)\eta(12\tau)^4},$$ and (a bit simpler) $$\frac1{E(q)} + E(q) = \frac{\eta(4\tau)}{\eta(\tau)} \Bigl(\frac{\eta(3\tau)}{\eta(12\tau)}\Bigr)^3.$$

2 Insert missing factor of q in product formula for E

Continued fraction or not, an expression $q^{\frac{(r-s)^2}{8(r+s)}} f(\pm q^r, \pm q^s)$ is a modular form of weight $1/2$ for all integers $r,s$ with $r+s>0$, because it is of the form $\sum_{n=-\infty}^\infty \pm q^{(cn+d)^2}$ with rational $c,d$ and periodic signs. Therefore the quotient of two such expressions is a modular function, and takes algebraic valus at quadratic imaginary values.

The quotient $$E(q) = q - q^2 + q^6 - q^7 + q^8 - q^9 + q^{11} - 2q^{12} + 2q^{13} - 2q^{14} + 2q^{15} \cdots$$ looks like a modular unit $-$ its logarithmic derivative has small coefficients $-$ but not quite an eta product; instead it seems to be a quotient of Klein forms: $$E(q) = q \prod_{n=1}^\infty (1-q^n)^{\chi(n)},$$ where $\chi$ is the Dirichlet character of conductor $12$, given by

$$\chi(n) = \cases{ +1,& if n \equiv \pm 1 \bmod 12; \cr -1,& if n \equiv \pm 5 \bmod 12; \cr 0,& otherwise. }$$ One identity relating $E$ to an $\eta$ product, similar to but somewhat more complicated than the ones you give for $A,B,C,D,$ is $$\frac1{E(q)} - E(q) = \frac{\eta(2\tau)^2 \eta(6\tau)^4}{\eta(\tau)\eta(3\tau)\eta(12\tau)^4}.$$

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