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Date: 22-6-2019
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Date: 12-10-2018
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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.
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