Extra scale term in normalizing constant?

[jmp448]

Hello,

I have a question about the implementation of the normalizing constant of the von Mises Fisher distribution. Digging into the code, I saw that the normalizing constant seems to be computed here:

which seems to have an extra scale term (in the third term of the output) compared to what I found in the paper:

Could you help me understand where this scaling term comes from?

Many thanks!

Josh