“Harmonic Logarithm”的概念、定义、习题解题方法及翻译-数学辞典

单词

Harmonic Logarithm

释义

Harmonic Logarithm

For all Integers and Nonnegative Integers , the harmoniclogarithms of order and degree are defined as the unique functions satisfying

where the ``Roman Symbol''

is defined by

(1)

(Roman 1992). This gives the special cases

			(2)
			(3)

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). Rearranginggives

(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.

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