What is the closed form value of the infinite product $$ \prod_{n=2}^\infty\left\{\frac{1}{e^{2qx}}\left(1+\frac{x}n\right)^{2nq+qx}\left(1-\frac{x}{2n-1}\right)^p\left(1+\frac{x}{2n+1}\right)^p\right\} ? $$ In particular, what are the values for the cases:
- $q=1$, $p=2$, $x=1$
- $q=1$, $p=-2$, $x=1$
- $q=1$, $p=-2$, $x=2$ ?
Notes:
As asked by a fellow member the problem is a generalization of the problem found in Mathproblems journal, Volume 4, Issue 2 (2014), p. 264, Problem 99, proposed by Li Yin.