The infinite series representation of the sine integral (http://en.wikipedia.org/wiki/Trigonometric_integral, previous m.se question: Is there any infinite series representation of the sine integral?): $$\operatorname{Si}(x)= \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}=x-\frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}-\frac{x^7}{7! \cdot7}\pm\cdots$$
Is there a corresponding infinite product representation? I don't know how well the roots have been characterized, and sometimes digging through the literature for infinite product representations is tricky.