Evaluation of the Multiple Sine Function...

DreamWeaver

Well-known member
Part 1:

Define the Multiple Sine Function by

$$\displaystyle \mathcal{S}_m(x)=\text{exp}\left(\frac{x^{m-1}}{m-1}\right) \prod_{k=1}^{\infty}\left(\mathcal{P}_m\left(\frac{x}{k}\right)\mathcal{P}_m\left(-\frac{x}{k}\right)^p\right)^q$$

Where $$\displaystyle p=(-1)^{m-1}\,$$ and $$\displaystyle q=k^{m-1}\,$$ (these exponents weren't showing up so well in the above, hence the abbreviations). The function $$\displaystyle \mathcal{P}_n(z)\,$$ is defined by:

$$\displaystyle \mathcal{P}_n(z)=(1-z)\,\text{exp}\left(z+\frac{z^2}{2}+\frac{z^3}{3}+\, \cdots \, +\frac{z^n}{n}\right)$$

The challenge is this: for $$\displaystyle 0\le\theta <\pi\,$$ and $$\displaystyle m\ge 2\,$$, prove that:

$$\displaystyle \int_0^{\theta}x^{m-2}\log(\sin x)\,dx=\frac{\theta^{m-1}}{(m-1)}\log(\sin \theta)-\frac{\pi^{m-1}}{(m-1)}\log\mathcal{S}_m\left(\frac{\theta}{\pi}\right)$$