Based on what I read in a very old book, we show
We can divide the sum into even and odd contributions
One sees that
In order to evaluate the right hand side of this equation we consider the function
, such that
We now take a small detour and consider the finite sum and its derivatives
Multiplying the last equation by and using
Now one can solve for
One can think of integration by parts and notice that the second term vanishes when . Now
In other words,
from which it is immediate
Now note and therefore
Finally, the identity establishes the result .
Happy new year to evri1!