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

单词

Harmonic Number

释义

Harmonic Number

A number of the form

(1)

This can be expressed analytically as

(2)

where

is the Euler-Mascheroni Constant and

is the Digamma Function. Thenumber formed by taking alternate signs in the sum also has an analytic solution

			(3)
			(4)

The first few harmonic numbers

are 1,

, ... (Sloane's A001008and A002805).The Harmonic Number

is never an Integer (except for

), which can be proved by using the strongtriangle inequality to show that the 2-adic value of

is greater than 1 for

. Theharmonic numbers have Odd Numerators and Even Denominators. The

thharmonic number is given asymptotically by

(5)

where

is the Euler-Mascheroni Constant (Conway and Guy 1996). Gosper gave the interestingidentity

(6)

where

is the incomplete Gamma Function and

is the Euler-Mascheroni Constant.Borwein and Borwein (1995) show that

			(7)
			(8)
			(9)

where

is the Riemann Zeta Function. The first of these had been previously derived byde Doelder (1991), and the last by Euler (1775). These identities are corollaries of the identity

(10)

(Borwein and Borwein 1995). Additional identities due to Euler are

(11)

(12)

for

, 3, ... (Borwein and Borwein 1995), where

is Apéry's Constant. These sums are related to so-called Euler Sums.

Conway and Guy (1996) define the second harmonic number by

(13)

the third harmonic number by

(14)

and the

th harmonic number by

(15)

A slightly different definition of a two-index harmonic number

is given by Roman (1992) in connection with theHarmonic Logarithm. Roman (1992) defines this by

			(16)
			(17)

plus the recurrence relation

(18)

For general

and

, this is equivalent to

(19)

and for

, it simplifies to

(20)

For

, the harmonic number can be written

(21)

where

is the Roman Factorial and

is a Stirling Number of the First Kind.

A separate type of number sometimes also called a ``harmonic number'' is a Harmonic Divisor Number (or Ore Number).

See also Apéry's Constant, Euler Sum, Harmonic Logarithm, Harmonic Series,Ore Number

References

Borwein, D. and Borwein, J. M. ``On an Intriguing Integral and Some Series Related to .'' Proc. Amer. Math. Soc. 123, 1191-1198, 1995.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 143 and 258-259, 1996.

de Doelder, P. J. ``On Some Series Containing and for Certain Values of and .'' J. Comp. Appl. Math. 37, 125-141, 1991.

Roman, S. ``The Logarithmic Binomial Formula.'' Amer. Math. Monthly 99, 641-648, 1992.

Sloane, N. J. A. SequencesA001008/M2885and A002805/M1589in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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