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Harmonic Logarithm

For all integers n and nonnegative integers t, the harmonic logarithms lambda_n^((t))(x) of order t and degree n are defined as the unique functions satisfying

1. lambda_0^((t))(x)=(lnx)^t,

2. lambda_n^((t))(x) has no constant term except lambda_0^((0))(x)=1,

3. d/(dx)lambda_n^((t))(x)=|_n]lambda_(n-1)^((t))(x),

where the "Roman symbol" |_n] is defined by

|_n]={n for n!=0; 1 for n=0
(1)

(Roman 1992). This gives the special cases

lambda_n^((0))(x)={x^n for n>=0; 0 for n<0
(2)
lambda_n^((1))(x)={x^n(lnx-H_n) for n>=0; x^n for n<0,
(3)

where H_n is a harmonic number. The harmonic logarithm has the integral

intlambda_n^((1))(x)dx=1/(|_n+1])lambda_(n+1)^((1))(x).
(4)

The harmonic logarithm can be written

lambda_n^((t))(x)=|_n]!D^~^(-n)(lnx)^t,
(5)

where D^~ is the differential operator, (so D^~^(-n) is the nth integral). Rearranging gives

D^~^klambda_n^((t))(x)=|_(|_n]!)/(|_n-k])]!lambda_(n-k)^((t))(x).
(6)

This formulation gives an analog of the binomial theorem called the logarithmic binomial theorem. Another expression for the harmonic logarithm is

lambda_n^((t))(x)=x^nsum_(j=0)^t(-1)^j(t)_jc_n^((j))(lnx)^(t-j),
(7)

where (t)_j=t(t-1)...(t-j+1) is a Pochhammer symbol and c_n^((j)) is a two-index harmonic number (Roman 1992).

See also

Logarithm, Roman Factorial

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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.

Referenced on Wolfram|Alpha

Harmonic Logarithm

Cite this as:

Weisstein, Eric W. "Harmonic Logarithm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicLogarithm.html

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