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. 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).

REFERENCES:

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.