Pi is the scaling factor of the diameter of a circle to its circumference; there's an infinite set of such scaling factors: one for each ellipse (the circle is a special case). I wonder which/what sort of infinite series arise from/for the generalized elliptic scaling factors?
I rather suspect that the generalised elliptic scaling factor is a continuous function, so the answer may be a bit boring. For any infinite series with a finite sum I would be able to give you an ellipse (indeed, probably an infinite number of ellipses) whose scaling factor is a rational multiple of the sum.
