Define the backwards difference operator by
and the factorial polynomials (aka rising factorials or Pochhammer symbol) by
These give a discrete-time counterpart to , viz.,
In these terms, a discrete-time Taylor series about can be defined:
th-order Lagrange interpolation via truncated discrete-time Taylor series expansion about time :
Each term in the expansion can be computed recursively from the previous term:
This gives the same efficient computational form found previously:
where is the desired delay for fractional-delay filtering, and is the output signal for th-order Lagrange interpolation (modular!). See also Newton's divided difference interpolation formula.