The following formula is attributed to Euler: $$\frac{\sin(2 k)}{2 k} = \prod_{n=0}^\infty \cos(k \frac{1}{2^n})$$
This can be shown through $m$ applications of $\sin(x) = 2 \sin(x/2) \cos(x/2)$ to find $\sin(x) = 2^m \sin(x/2^m) \prod_{n=1}^m \cos(\frac{x}{2^n})$. Taking $m \rightarrow \infty$ and replacing $x$ with $2k$ completes the proof.
There are simple generalizations for other powers of $2$ inside the parentheses, such as
$$\frac{\sin(2 k)}{2 k}\frac{\sin(\sqrt{2} k)}{\sqrt{2} k} = \prod_{n=0}^\infty \cos( k\frac{1}{\sqrt{2}^n})$$
which follow from separating the terms with even and odd powers $n$.
There are yet more generalizations that start from
$$\frac{\sin(x)}{x} = \prod_{n=1}^{\infty} \frac{\sin(\frac{x}{q^{n-1}})}{n \sin(\frac{x}{q^{n}})}$$ and are most easily seen by taking the upper bound on the product to be $m$, telescoping the terms in the product, and then taking $m \rightarrow \infty$.
Now for the question. Consider the following function:
$$f_{\lambda, r}(k) = \prod_{n=0}^\infty\left(\cos(k\lambda^n)+ i r \sin(k\lambda^n) \right)$$
Note that it is clear that when $r=1$, the formula above reduces to $e^{i \frac{k}{1-\lambda}}$, and when $r=0$ and $\lambda=1/2, 1/\sqrt{2}$, the formula reduces to the two cases at the top of the question.
Are there specific cases of $\lambda$ and $r$ when one can simplify $f_{\lambda, r}(k)$ to some simple function of $k$? In particular, I would like the formula to hold for all $k$, but I am happy with fixed, nontrivial choices of $\lambda$ and $r$ ($\lambda \neq -1,0,1$, $r\neq -1,0,1$). As an example, an evaluation for the case $r=1/2$ and $\lambda = 1/2$ would suffice.
For $r=\lambda=1/2$, $f_{r,\lambda}$ is the Fourier transform of the measure $$\sum_{x\in F} c_x\delta_x $$ where $F =\{\sum_{n=0}^{\infty} 2^{-n}\epsilon_n : \epsilon_n =\pm 1 \}$ and $$x = \sum_{n=0}^{\infty}\frac{\epsilon_n}{2^n} \Rightarrow c_x = \lim_{n\to\infty} \frac{3^{\#\{k\leq n: \epsilon_k=1 \} } }{2^n}$$ Here $\delta_x$ is the delta measure with the unit mass at $\{x\}$. Generally, $f_{r,\lambda}$ is the Fourier transform of the measure $$\sum_{x\in F} c_x\delta_x $$ where $F =\{\sum_{n=0}^{\infty} \epsilon_n\lambda^n : \epsilon_n =\pm 1 \}$ and $$x = \sum_{n=0}^{\infty}\epsilon_n\lambda^n \Rightarrow c_x = \lim_{n\to\infty} (1+r)^{z_n}(1-r)^{n-z_n}$$ where $z_n = \#\{k\leq n: \epsilon_k=1 \}$.