For all integers and nonnegative integers , the harmonic logarithms of order and degree are defined as the unique functions satisfying
1. ,
2. has no constant term except ,
3. ,
where the "Roman symbol" is defined by
(1)
|
(Roman 1992). This gives the special cases
(2)
| |||
(3)
|
where is a harmonic number. The harmonic logarithm has the integral
(4)
|
The harmonic logarithm can be written
(5)
|
where is the differential operator, (so is the th integral). Rearranging gives
(6)
|
This formulation gives an analog of the binomial theorem called the logarithmic binomial theorem. Another expression for the harmonic logarithm is
(7)
|
where is a Pochhammer symbol and is a two-index harmonic number (Roman 1992).