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 =
The harmonic numbers are defined by
Hn := 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
Note that H(1)0,n = Hn.