Z-Transform

The -transform of is defined by

 (1)

where
 (2)

is the Delta Function, is the sampling period, and is the Laplace Transform. An alternative definition is
 (3)

where
 (4)

The inverse -transform is
 (5)

It satisfies
 (6) (7) (8) (9) (10) (11) (12) (13) (14)

Transforms of special functions (Beyer 1987, pp. 426-427) include
 (15) (16) (17) (18) (19) (20) (21) (22) (23) (24)

where is the Heaviside Step Function. In general,
 (25) (26)

where the are Eulerian Numbers. Amazingly, the Z-transforms of are therefore generators for Euler's Triangle.