Harmonic Logarithm

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
 (4)

The harmonic logarithm has the Integral
 (5)

The harmonic logarithm can be written
 (6)

where is the Differential Operator, (so is the th Integral). Rearranging gives
 (7)

This formulation gives an analog of the Binomial Theorem called the Logarithmic Binomial Formula. Another expression for the harmonic logarithm is
 (8)

where is a Pochhammer Symbol and is a two-index Harmonic Number (Roman 1992).

Loeb, D. and Rota, G.-C. Formal Power Series of Logarithmic Type.'' Advances Math. 75, 1-118, 1989.
Roman, S. The Logarithmic Binomial Formula.'' Amer. Math. Monthly 99, 641-648, 1992.