Now we derive a new recursive expression for LIF by using the power series expansion of the ideal transfer function and by identifying the result to a maximally flat condition LIF.
Let consider function 
defined over
the 
plane minus the real semi-axis 
.
Since this function is analytic on its definition domain, it permits a
power series expansion (eq. 4).
It is clear that the family of FIR filters, whose transfer function
is defined in equation 5, verifies the maximally
flat condition (eq. 3) since they actually are partial
series expansions for 
around z=1
(eq. 4). These transfer functions are new expressions
for LIFs.
This expression may be rewritten recursively (eq. 6). Another nested expression will be useful for the implementation of LIFs (eq. 7).
