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
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).
See also Logarithm, Roman Factorial
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.
© 1996-9 Eric W. Weisstein
1999-05-25