TITLE: Infinite product of sinc functions
QUESTION [8 upvotes]: Calculate the infinite product
$f_q(x):=\prod_{n=0}^\infty\frac{\sin(q^n x)}{q^n x}$,
where $x$ is real and $0<q<1$.
In other words, $f_q$ must satisfy the functional equation $f_q(x)=f_q(qx)\operatorname{sinc}(x)$ with an initial condition $f_q(0)=1$,
where $\operatorname{sinc}(x):=\frac{\sin(x)}{x}$.
Using the Euler's factorization of sinc one can rewrite $f_q(x)$ as a certain infinite product of $q$-Pochhammer symbols. Therefore I hope that $f_q$ can be expressed in terms of some hypergeometric functions. The Fourier transform of $f_q$ is a smooth function with compact support.

REPLY [2 votes]: I guess, expression $(10)$ from here is the most fresh suggestion on the original question up to this moment for the case, when $q=2.$ Substitute $2\rightarrow q$ and obtain the following expression
$$R(x) = \prod\limits_{n=0}^{m-1}\mathrm{sinc}\dfrac{\pi x}{q^n}\cdot\dfrac{\left(x^2, \dfrac1q\right)_\infty}{\left(x^2, \dfrac1q\right)_m}\cdot\exp\left(\sum\limits_{k=1}^\infty\ \dfrac{c_k}{1-q^{-mk}} \left(\dfrac{x}{q^m}\right)^{2k}\right), \quad |x| < q^{m+1}.$$