chapter  2
Elementary Properties
Pages 1

The factorial satisfies some obvious, but very useful, rules of computation. The most important ones are

[x, d]k = [−x + (k − 1)d, d]k(−1)k x, d ∈ C, k ∈ Z, (2.1) [x, d]k = [x, d]h[x− hd, d]k−h x, d ∈ C, k, h ∈ Z, (2.2) [x, d]k = 1/[x− kd, d]−k x, d ∈ C, k ∈ Z, (2.3) [x, d]k = [x− d, d]k + kd[x− d, d]k−1 x, d ∈ C, k ∈ Z, (2.4) [xd, d]k = dk[x]k x, d ∈ C, k ∈ Z. (2.5)

Applying the difference operator (1.7), we can rewrite (2.4) for d = 1 as

∆[k]n = n[k]n−1.