Finding a function by infinite product

Question

We know that the product $$x\prod_{n=1}^{\infty}\bigg(1-\frac{x^2}{n^2\pi^2}\bigg)=\sin x$$ I would like to know if we can find a function for $$x\prod_{n=1}^{\infty}\bigg(1-\frac{x^{2k}}{n^{2k}\pi^{2k}}\bigg)$$ For some positive integer $k$.

Answer 1

For any value of $k$, closed form expressions do exist. The only problem is that they invoke trigonometric functions with complex arguments and they quickly become quite messy.

Let $$f_k=x\prod_{n=1}^{\infty}\bigg(1-\frac{x^{2k}}{n^{2k}\pi^{2k}}\bigg)$$

The very first ones are simple

$$f_2=\frac{\sin (x) \sinh (x)}{x}$$ $$f_3=\frac{\sin (x) \left(\cosh \left(\sqrt{3} x\right)-\cos (x)\right)}{2 x^2}$$ $$f_4=\frac{\sin (x) \sinh (x) \left(\cosh \left(\sqrt{2} x\right)-\cos \left(\sqrt{2} x\right)\right)}{2 x^3}$$

For $k > 4$, they become more than messy.