chapter  1
2 Generalized Factorials
Pages 3

An indeﬁnite sum or anti-diﬀerence of a function g(k) is defined as any solution f(k) to the equation

g(k) = ∆f(k).

(The anti-difference f(k) is uniquely determined up to a constant, or a periodic function with period 1, by the function g(k).)

We denote this indefinite sum by∑ g(k)δk := f(k). (1.8)

We use δx in analogy to the standard dx notation for integrals. For any indefinite sum (1.8) of a function g(k), a deﬁnite sum is defined

to be ∑b a g(k)δk := f(b)− f(a), (1.9)

and we remark that the connection to the usual step-by-step sum is

∑b a g(k)δk =

g(k). (1.10)

The harmonic numbers are defined by

Hn := n∑

1 k

= ∑n

0 [k]−1δk. (1.11)

The generalized harmonic numbers are defined for n,m ∈ N and c ∈ C by

H(m)c,n := n∑

1 (c + k)m

. (1.12)

Note that H(1)0,n = Hn.