A summed set of (negative) single impulses is given by
$-\sum_{R=0}^m \frac{\sin \pi (x-(R+1))}{\pi (x-(R+1))}$
Mathematica simplifies this to a function involving the Lerch Transcendent $\phi (z,s,a)$:
$\frac{\sin \pi x \bigl(\phi (-1,1,2-x)+(-1)^{m+1}\phi (-1,1,2+m-x)\bigr)}{\pi}$
But a plot of the two formulae with (randomly chosen) $m=25$ produces very different curves - blue is the single impulse, orange is the $\phi$ function:
Can anyone help explain this? Is it to do with transcendence? Is the 'simplification' algebraically manipulable?