Harmonic Logarithm
المؤلف:
Loeb, D. and Rota, G.-C.
المصدر:
"Formal Power Series of Logarithmic Type." Advances Math. 75,
الجزء والصفحة:
...
23-6-2019
1452
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. ![d/(dx)lambda_n^((t))(x)=|_n]lambda_(n-1)^((t))(x)](http://mathworld.wolfram.com/images/equations/HarmonicLogarithm/Inline9.gif)
,
where the "Roman symbol" ![|_n]](http://mathworld.wolfram.com/images/equations/HarmonicLogarithm/Inline10.gif)
is defined by
![|_n]=<span style=]() {n for n!=0; 1 for n=0 " src="http://mathworld.wolfram.com/images/equations/HarmonicLogarithm/NumberedEquation1.gif" style="height:41px; width:117px" /> |
(1)
|
(Roman 1992). This gives the special cases
where 
is a harmonic number. The harmonic logarithm has the integral
![intlambda_n^((1))(x)dx=1/(|_n+1])lambda_(n+1)^((1))(x).](http://mathworld.wolfram.com/images/equations/HarmonicLogarithm/NumberedEquation2.gif) |
(4)
|
The harmonic logarithm can be written
![lambda_n^((t))(x)=|_n]!D^~^(-n)(lnx)^t,](http://mathworld.wolfram.com/images/equations/HarmonicLogarithm/NumberedEquation3.gif) |
(5)
|
where 
is the differential operator, (so 
is the 
th integral). Rearranging gives
![D^~^klambda_n^((t))(x)=|_(|_n]!)/(|_n-k])]!lambda_(n-k)^((t))(x).](http://mathworld.wolfram.com/images/equations/HarmonicLogarithm/NumberedEquation4.gif) |
(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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