A question similar to: How to derive the Golden mean by using properties of Gamma function?
The limiting case of Gordon's continued fraction when $q$ approaches $i$ yields:
$$\sqrt2 + 1 = \frac{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdot\ldots}{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdot\ldots} $$
How to derive this by using properties of the Gamma function?
Any help is appreciated!