Harmonic Number

A number of the form

$$H_n=\sum_{k=1}^n {1\over k}.$$ (1)

This can be expressed analytically as

$$H_n = \gamma+\psi_0(n+1),$$ (2)

where $\gamma$ is the Euler-Mascheroni Constant and $\Psi(x)=\psi_0(x)$ is the Digamma Function. The number formed by taking alternate signs in the sum also has an analytic solution

$$H'_n = \sum_{k=1}^n {(-1)^{k+1}\over k}$$ (3)

$$= \ln 2+{\textstyle{1\over 2}}(-1)^n[\psi_0({\textstyle{1\over 2}}n+{\textstyle{1\over 2}})-\psi_0({\textstyle{1\over 2}}n+1)].$$ (4)

The first few harmonic numbers $H_n$ are 1, $3/2$, $11/6$, $25/12$, $137/60$, ... (Sloane's A001008 and A002805). The Harmonic Number $H_n$ is never an Integer (except for $H_1$), which can be proved by using the strong triangle inequality to show that the 2-adic value of $H_n$ is greater than 1 for $n>1$. The harmonic numbers have Odd Numerators and Even Denominators. The $n$th harmonic number is given asymptotically by

$$H_n\sim \ln n+\gamma+{1\over 2n},$$ (5)

where $\gamma$ is the Euler-Mascheroni Constant (Conway and Guy 1996). Gosper gave the interesting identity

$$\sum_{i=0}^\infty {z^i H_i\over i!} = -e^z\sum_{k=1}^\infty {(-z)^k\over kk!} = e^z[\ln z+\Gamma(0,z)+\gamma],$$ (6)

where $\Gamma(0,z)$ is the incomplete Gamma Function and $\gamma$ is the Euler-Mascheroni Constant. Borwein and Borwein (1995) show that

$$\sum_{n=1}^\infty {{H_n}^2\over(n+1)^2} = {\textstyle{11\over 4}}\zeta(4)={\textstyle{11\over 360}}\pi^4$$ (7)

$$\sum_{n=1}^\infty {{H_n}^2\over n^2} = {\textstyle{17\over 4}}\zeta(4)={\textstyle{17\over 360}}\pi^4$$ (8)

$$\sum_{n=1}^\infty {H_n\over n^3} = {\textstyle{5\over 4}}\zeta(4)={\textstyle{1\over 72}}\pi^4,$$ (9)

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

$${1\over\pi}\int_0^\pi x^2\{\ln[2\cos({\textstyle{1\over 2}}x... ...{\textstyle{11\over 2}}\zeta(4)={\textstyle{11\over 180}}\pi^4$$ (10)

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

$$\sum_{n=1}^\infty {H_n\over n^2}=2\zeta(3)$$ (11)

$$2\sum_{n=1}^\infty {H_n\over n^m}=(m+2)\zeta(m+1)-\sum_{n=1}^{m-2} \zeta(m-n)\zeta(n+1)$$ (12)

for $m=2$, 3, ... (Borwein and Borwein 1995), where $\zeta(3)$ is Apéry's Constant. These sums are related to so-called Euler Sums.

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

$$H_n^{(2)}\equiv \sum_{i=1}^n H_i = (n+1)(H_{n+1}-1)=(n+1)(H_{n+1}-H_1),$$ (13)

the third harmonic number by

$$H_n^{(3)}\equiv \sum_{i=1}^n H_i^{(2)} = {n+2\choose 2}(H_{n+2}-H_2),$$ (14)

and the $n$th harmonic number by

$$H_n^{(k)} = {n+k-1\choose k-1}(H_{n+k-1}-H_{k-1}).$$ (15)

A slightly different definition of a two-index harmonic number $c^{(j)}_n$ is given by Roman (1992) in connection with the Harmonic Logarithm. Roman (1992) defines this by

$$c_n^{(0)} = \left\{\begin{array}{ll} 1 & \mbox{for $n\geq 0$}\\ 0 & \mbox{for $n<0$}\end{array}\right.$$ (16)

$$c_0^{(j)} = \left\{\begin{array}{ll} 1 & \mbox{for $j=0$}\\ 0 & \mbox{for $j\not=0$}\end{array}\right.$$ (17)

plus the recurrence relation

$$cn_n^{(j)}=c_n^{(j-1)}+nc_{n-1}^{(j)}.$$ (18)

For general $n>0$ and $j>0$, this is equivalent to

$$c_n^{(j)}=\sum_{i=1}^n {1\over i}c_i^{(j-1)},$$ (19)

and for $n>0$, it simplifies to

$$c_n^{(j)}=\sum_{i=1}^n{n\choose i}(-1)^{i-1}i^{-j}.$$ (20)

For $n<0$, the harmonic number can be written

$$c_n^{(j)}=(-1)^j\left\lfloor{n}\right\rceil !s(-n,j),$$ (21)

where $\left\lfloor{n}\right\rceil !$ is the Roman Factorial and $s$ 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 $\zeta(4)$.'' 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 $\Psi(x)-\Psi(y)$ and $(\Psi(x)-\Psi(y))^2$ for Certain Values of $x$ and $y$.'' 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. Sequences A001008/M2885 and A002805/M1589 in ``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.