Harmonic number

The harmonic number $H_{{n,1}}$ with $n=\lfloor {x}\rfloor$ (red line) with its asymptotic limit $\gamma +\ln(x)$ (blue line).

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

$H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.$

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.

Harmonic numbers were studied in antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.

The harmonic numbers roughly approximate the natural logarithm function^[1]:143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.

When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions in the Long Tail and the theory of network value.

Bertrand's postulate entails that, except for the case n = 1, the harmonic numbers are never integers.^[2]

The first 40 harmonic numbers

n	Harmonic number, Hn
	expressed as a fraction		decimal	relative size
1	1		1
2	3	/2	1.5
3	11	/6	~1.83333
4	25	/12	~2.08333
5	137	/60	~2.28333
6	49	/20	2.45
7	363	/140	~2.59286
8	761	/280	~2.71786
9	7 129	/2 520	~2.82897
10	7 381	/2 520	~2.92897
11	83 711	/27 720	~3.01988
12	86 021	/27 720	~3.10321
13	1 145 993	/360 360	~3.18013
14	1 171 733	/360 360	~3.25156
15	1 195 757	/360 360	~3.31823
16	2 436 559	/720 720	~3.38073
17	42 142 223	/12 252 240	~3.43955
18	14 274 301	/4 084 080	~3.49511
19	275 295 799	/77 597 520	~3.54774
20	55 835 135	/15 519 504	~3.59774
21	18 858 053	/5 173 168	~3.64536
22	19 093 197	/5 173 168	~3.69081
23	444 316 699	/118 982 864	~3.73429
24	1 347 822 955	/356 948 592	~3.77596
25	34 052 522 467	/8 923 714 800	~3.81596
26	34 395 742 267	/8 923 714 800	~3.85442
27	312 536 252 003	/80 313 433 200	~3.89146
28	315 404 588 903	/80 313 433 200	~3.92717
29	9 227 046 511 387	/2 329 089 562 800	~3.96165
30	9 304 682 830 147	/2 329 089 562 800	~3.99499
31	290 774 257 297 357	/72 201 776 446 800	~4.02725
32	586 061 125 622 639	/144 403 552 893 600	~4.05850
33	53 676 090 078 349	/13 127 595 717 600	~4.08880
34	54 062 195 834 749	/13 127 595 717 600	~4.11821
35	54 437 269 998 109	/13 127 595 717 600	~4.14678
36	54 801 925 434 709	/13 127 595 717 600	~4.17456
37	2 040 798 836 801 833	/485 721 041 551 200	~4.20159
38	2 053 580 969 474 233	/485 721 041 551 200	~4.22790
39	2 066 035 355 155 033	/485 721 041 551 200	~4.25354
40	2 078 178 381 193 813	/485 721 041 551 200	~4.27854

Identities involving harmonic numbers

By definition, the harmonic numbers satisfy the recurrence relation

$H_{n}=H_{n-1}+{\frac {1}{n}}.$

The harmonic numbers are connected to the Stirling numbers of the first kind:

$H_{n}={\frac {1}{n!}}\left[{n+1 \atop 2}\right].$

The functions

$f_{n}(x)={\frac {x^{n}}{n!}}(\log x-H_{n})$

satisfy the property

$f_{n}'(x)=f_{n-1}(x).$

In particular

$f_{1}(x)=x(\log x-1)$

is an integral of the logarithmic function.

The harmonic numbers satisfy the series identity

${\displaystyle \sum _{k=1}^{n}H_{k}=(n+1)[H_{n+1}-1].}$

Identities involving π

There are several infinite summations involving harmonic numbers and powers of π:^[3]

$\sum _{n=1}^{\infty }{\frac {H_{n}}{n\cdot 2^{n}}}={\frac {1}{12}}\pi ^{2};$

$\sum _{n=1}^{\infty }{\frac {H_{n}^{2}}{(n+1)^{2}}}={\frac {11}{360}}\pi ^{4};$

$\sum _{n=1}^{\infty }{\frac {H_{n}^{2}}{n^{2}}}={\frac {17}{360}}\pi ^{4};$

$\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{3}}}={\frac {1}{72}}\pi ^{4};$

Calculation

An integral representation given by Euler^[4] is

$H_{n}=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,dx.$

The equality above is obvious by the simple algebraic identity

${\frac {1-x^{n}}{1-x}}=1+x+\cdots +x^{n-1}.$

Using the simple integral transform x = 1−u, an elegant combinatorial expression for H_n is

${\begin{aligned}H_{n}&=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,dx\\&=-\int _{1}^{0}{\frac {1-(1-u)^{n}}{u}}\,du\\&=\int _{0}^{1}{\frac {1-(1-u)^{n}}{u}}\,du\\&=\int _{0}^{1}\left[\sum _{k=1}^{n}(-1)^{k-1}{\binom {n}{k}}u^{k-1}\right]\,du\\&=\sum _{k=1}^{n}(-1)^{k-1}{\binom {n}{k}}\int _{0}^{1}u^{k-1}\,du\\&=\sum _{k=1}^{n}(-1)^{k-1}{\frac {1}{k}}{\binom {n}{k}}.\end{aligned}}$

The same representation can be produced by using the third Retkes identity by setting $x_{1}=1,\ldots ,x_{n}=n$ and using the fact that $\Pi _{k}(1,\ldots ,n)=(-1)^{n-k}(k-1)!(n-k)!$

$H_{n}=H_{n,1}=\sum _{k=1}^{n}{\frac {1}{k}}=(-1)^{n-1}n!\sum _{k=1}^{n}{\frac {1}{k^{2}\Pi _{k}(1,\ldots ,n)}}=\sum _{k=1}^{n}(-1)^{k-1}{\frac {1}{k}}{\binom {n}{k}}.$

The Taylor series for the harmonic numbers is

${\displaystyle H_{x}=\sum _{k=2}^{\infty }(-1)^{k}\zeta (k)\;x^{k-1}\quad {\text{ for }}|x|<1}$

which comes from the Taylor series for the Digamma function.

Graph demonstrating a connection between harmonic numbers and the natural logarithm. The harmonic number H_n can be interpreted as a Riemann sum of the integral: $\int _{1}^{n+1}{\frac {1}{x}}\mathrm {d} x=\ln(n+1)$

The nth harmonic number is about as large as the natural logarithm of n. The reason is that the sum is approximated by the integral

$\int _{1}^{n}{1 \over x}\,dx$

whose value is ln(n).

The values of the sequence H_n - ln(n) decrease monotonically towards the limit

${\displaystyle \lim _{n\to +\infty }\left(H_{n}-\ln n\right)=\gamma ,}$

where γ ≈ 0.5772156649 is the Euler–Mascheroni constant. The corresponding asymptotic expansion as n → +∞ is

${\displaystyle H_{n}=\ln {n}+\gamma +{\frac {1}{2n}}-\sum _{k=1}^{\infty }{\frac {B_{2k}}{2kn^{2k}}}=\ln {n}+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\cdots ,}$

where $B_{k}$ are the Bernoulli numbers.

Special values for fractional arguments

There are the following special analytic values for fractional arguments between 0 and 1, given by the integral

$H_{\alpha }=\int _{0}^{1}{\frac {1-x^{\alpha }}{1-x}}\,dx\,.$

More values may be generated from the recurrence relation

$H_{\alpha }=H_{\alpha -1}+{\frac {1}{\alpha }}\,,$

or from the reflection relation

$H_{1-\alpha }-H_{\alpha }=\pi \cot {(\pi \alpha )}-{\frac {1}{\alpha }}+{\frac {1}{1-\alpha }}\,.$

For example:

$H_{\frac {1}{2}}=2-2\ln {2}$

$H_{\frac {1}{3}}=3-{\tfrac {\pi }{2{\sqrt {3}}}}-{\tfrac {3}{2}}\ln {3}$

$H_{\frac {2}{3}}={\tfrac {3}{2}}(1-\ln {3})+{\sqrt {3}}{\tfrac {\pi }{6}}$

$H_{\frac {1}{4}}=4-{\tfrac {\pi }{2}}-3\ln {2}$

$H_{\frac {3}{4}}={\tfrac {4}{3}}-3\ln {2}+{\tfrac {\pi }{2}}$

$H_{\frac {1}{6}}=6-{\tfrac {\pi }{2}}{\sqrt {3}}-2\ln {2}-{\tfrac {3}{2}}\ln {3}$

$H_{\frac {1}{8}}=8-{\tfrac {\pi }{2}}-4\ln {2}-{\tfrac {1}{\sqrt {2}}}\left\{\pi +\ln \left(2+{\sqrt {2}}\right)-\ln \left(2-{\sqrt {2}}\right)\right\}$

$H_{\frac {1}{12}}=12-3\left(\ln {2}+{\tfrac {\ln {3}}{2}}\right)-\pi \left(1+{\tfrac {\sqrt {3}}{2}}\right)+2{\sqrt {3}}\ln \left({\sqrt {2-{\sqrt {3}}}}\right)$

For positive integers p and q with p < q, we have:

${\displaystyle H_{\frac {p}{q}}={\frac {q}{p}}+2\sum _{k=1}^{\lfloor {\frac {q-1}{2}}\rfloor }\cos \left({\frac {2\pi pk}{q}}\right)\ln \left({\sin \left({\frac {\pi k}{q}}\right)}\right)-{\frac {\pi }{2}}\cot \left({\frac {\pi p}{q}}\right)-\ln \left(2q\right)}$

Asymptotic formulation

For every positive integer n, we have that

${\displaystyle \lim _{m\rightarrow +\infty }\left[H_{m}-H_{m+n}\right]=0\,.}$

Adding H_n to both sides gives

${\displaystyle {\begin{aligned}H_{n}&=\lim _{m\rightarrow +\infty }\left[H_{m}-(H_{m+n}-H_{n})\right]\\&=\lim _{m\rightarrow +\infty }\left[\left(\sum _{k=1}^{m}{\frac {1}{k}}\right)-\left(\sum _{k=1}^{m}{\frac {1}{n+k}}\right)\right]\\&=\sum _{k=1}^{+\infty }\left[{\frac {1}{k}}-{\frac {1}{n+k}}\right]=n\sum _{k=1}^{+\infty }{\frac {1}{k(n+k)}}\,.\end{aligned}}}$

Despite being derived for positive integers n, this last expression for H_n is well defined for any complex number n except the negative integers. The function H_n is the unique function of n for which (1) H₀ = 0, (2) H_n = H_n−1 + 1/n for all complex values n except the non-positive integers, and (3) lim_m→+∞ (H_m+n − H_m) = 0 for all complex values n.

Based on this last formula, it can be shown that:

$\int _{0}^{1}H_{x}\,dx=\gamma \,,$

where γ is the Euler–Mascheroni constant or, more generally, for every n we have:

$\int _{0}^{n}H_{x}\,dx=n\gamma +\ln {(n!)}\,.$

Generating functions

A generating function for the harmonic numbers is

$\sum _{n=1}^{\infty }z^{n}H_{n}={\frac {-\ln(1-z)}{1-z}},$

where ln(z) is the natural logarithm. An exponential generating function is

$\sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}H_{n}=-e^{z}\sum _{k=1}^{\infty }{\frac {1}{k}}{\frac {(-z)^{k}}{k!}}=e^{z}{\mbox{Ein}}(z)$

where Ein(z) is the entire exponential integral. Note that

${\mbox{Ein}}(z)={\mbox{E}}_{1}(z)+\gamma +\ln z=\Gamma (0,z)+\gamma +\ln z\,$

where Γ(0, z) is the incomplete gamma function.

Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function

$\psi (n)=H_{n-1}-\gamma .$

This relation is also frequently used to define the extension of the harmonic numbers to non-integer n. The harmonic numbers are also frequently used to define γ using the limit introduced earlier:

${\displaystyle \gamma =\lim _{n\rightarrow +\infty }{\left(H_{n}-\ln(n)\right)},}$

although

${\displaystyle \gamma =\lim _{n\to +\infty }{\left(H_{n}-\ln \left(n+{1 \over 2}\right)\right)}}$

converges more quickly.

In 2002, Jeffrey Lagarias proved^[5] that the Riemann hypothesis is equivalent to the statement that

$\sigma (n)\leq H_{n}+\ln(H_{n})e^{H_{n}},$

is true for every integer n ≥ 1 with strict inequality if n > 1; here σ(n) denotes the sum of the divisors of n.

The eigenvalues of the nonlocal problem

$\lambda \phi (x)=\int _{-1}^{1}{\frac {\phi (x)-\phi (y)}{|x-y|}}dy$

are given by $\lambda =2H_{n}$, where by convention, $H_{0}=0.$

Generalization

Generalized harmonic numbers

The generalized harmonic number of order n of m is given by

$H_{n,m}=\sum _{k=1}^{n}{\frac {1}{k^{m}}}.$

The limit as n tends to infinity exists if m > 1.

Other notations occasionally used include

$H_{n,m}=H_{n}^{(m)}=H_{m}(n).$

The special case of m = 0 gives $H_{n,0}=n$

The special case of m = 1 is simply called a harmonic number and is frequently written without the superscript, as

$H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}.$

Smallest natural number k such that kⁿ does not divide the denominator of generalized harmonic number H(k, n) nor the denominator of alternating generalized harmonic number H'(k, n) are

77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, ... (sequence A128670 in the OEIS)

In the limit of n → +∞, the generalized harmonic number converges to the Riemann zeta function

${\displaystyle \lim _{n\rightarrow +\infty }H_{n,m}=\zeta (m).}$

The related sum $\sum _{k=1}^{n}k^{m}$ occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of Stirling numbers.

Some integrals of generalized harmonic are

$\int _{0}^{a}H_{x,2}\,dx=a{\frac {\pi ^{2}}{6}}-H_{a}$

and

$\int _{0}^{a}H_{x,3}\,dx=aA-{\frac {1}{2}}H_{a,2},$ where A is the Apéry's constant, i.e. ζ(3).

and

$\sum _{k=1}^{n}H_{k,m}=(n+1)H_{n,m}-H_{n,m-1}$   for $m\geq 0$

Every generalized harmonic number of order m can be written as a function of harmonic of order m-1 using:

$H_{n,m}=\sum _{k=1}^{n-1}{\frac {H_{k,m-1}}{k(k+1)}}+{\frac {H_{n,m-1}}{n}}$   for example: $H_{4,3}={\frac {H_{1,2}}{1\cdot 2}}+{\frac {H_{2,2}}{2\cdot 3}}+{\frac {H_{3,2}}{3\cdot 4}}+{\frac {H_{4,2}}{4}}$

A generating function for the generalized harmonic numbers is

$\sum _{n=1}^{\infty }z^{n}H_{n,m}={\frac {\mathrm {Li} _{m}(z)}{1-z}},$

where $\mathrm {Li} _{m}(z)$ is the polylogarithm, and |z| < 1. The generating function given above for m = 1 is a special case of this formula.

Fractional argument for generalized harmonic numbers can be introduced as follows:

For every $p,q>0$ integer, and $m>1$ integer or not, we have from polygamma functions:

$H_{q/p,m}=\zeta (m)-p^{m}\sum _{k=1}^{\infty }{\frac {1}{(q+pk)^{m}}}$

where $\zeta (m)$ is the Riemann zeta function. The relevant recurrence relation is:

$H_{a,m}=H_{a-1,m}+{\frac {1}{a^{m}}}$

Some special values are:

$H_{{\frac {1}{4}},2}=16-8G-{\tfrac {5}{6}}\pi ^{2}$ where G is the Catalan's constant

$H_{{\frac {1}{2}},2}=4-{\tfrac {\pi ^{2}}{3}}$

$H_{{\frac {3}{4}},2}=8G+{\tfrac {16}{9}}-{\tfrac {5}{6}}\pi ^{2}$

$H_{{\frac {1}{4}},3}=64-27\zeta (3)-\pi ^{3}$

$H_{{\frac {1}{2}},3}=8-6\zeta (3)$

$H_{{\frac {3}{4}},3}={({\tfrac {4}{3}})}^{3}-27\zeta (3)+\pi ^{3}$

Multiplication formulas

The multiplication theorem applies to harmonic numbers. Using polygamma functions, we obtain

$H_{2x}={\frac {1}{2}}\left(H_{x}+H_{x-{\frac {1}{2}}}\right)+\ln {2}$

$H_{3x}={\frac {1}{3}}\left(H_{x}+H_{x-{\frac {1}{3}}}+H_{x-{\frac {2}{3}}}\right)+\ln {3},$

or, more generally,

$H_{nx}={\frac {1}{n}}\left(H_{x}+H_{x-{\frac {1}{n}}}+H_{x-{\frac {2}{n}}}+\cdots +H_{x-{\frac {n-1}{n}}}\right)+\ln {n}.$

For generalized harmonic numbers, we have

$H_{2x,2}={\frac {1}{2}}\left(\zeta (2)+{\frac {1}{2}}\left(H_{x,2}+H_{x-{\frac {1}{2}},2}\right)\right)$

$H_{3x,2}={\frac {1}{9}}\left(6\zeta (2)+H_{x,2}+H_{x-{\frac {1}{3}},2}+H_{x-{\frac {2}{3}},2}\right),$

where $\zeta (n)$ is the Riemann zeta function.

Generalization to the complex plane

When | x−1 | < 1, we can write the integrand (1−x^s)/(1−x) as an infinite series. We start by writing

${\displaystyle x^{s}=(1+(x-1))^{s}=\sum _{k=0}^{+\infty }{s \choose k}(x-1)^{k}\,,}$

which is the binomial expansion for the suitably extended binomial coefficients. Then

${\displaystyle {\frac {1-x^{s}}{1-x}}={\frac {1-\sum _{k=0}^{+\infty }{s \choose k}(x-1)^{k}}{1-x}}=\sum _{k=1}^{+\infty }{s \choose k}(x-1)^{k-1}\,.}$

The integral from some value a ∈ (0, 1) is then

${\displaystyle \int _{a}^{1}{\frac {1-x^{s}}{1-x}}\,dx=-\sum _{k=1}^{+\infty }{s \choose k}{\frac {(a-1)^{k}}{k}}\,.}$

By choosing a = 0,

${\displaystyle H_{s}=\int _{0}^{1}{\frac {1-x^{s}}{1-x}}\,dx=-\sum _{k=1}^{+\infty }{s \choose k}{\frac {(-1)^{k}}{k}}}$

we get both an integral and a series representation for a function that interpolates the harmonic numbers and, via analytic continuation, extends the definition to the complex plane other than the negative integers s. The interpolating function is in fact the digamma function

$H_{s}=\psi (s+1)+\gamma =\int _{0}^{1}{\frac {1-x^{s}}{1-x}}\,dx,$

where ψ(x) is the digamma, and γ is the Euler-Mascheroni constant. The integration process may be repeated to obtain

$H_{s,2}=-\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{s \choose k}H_{k}.$

Relation to the Riemann zeta function

Some derivatives of fractional harmonic numbers are given by:

${\frac {d^{n}H_{x}}{dx^{n}}}=(-1)^{n+1}n!\left[\zeta (n+1)-H_{x,n+1}\right]$

${\frac {d^{n}H_{x,2}}{dx^{n}}}=(-1)^{n+1}(n+1)!\left[\zeta (n+2)-H_{x,n+2}\right]$

${\frac {d^{n}H_{x,3}}{dx^{n}}}=(-1)^{n+1}{\frac {1}{2}}(n+2)!\left[\zeta (n+3)-H_{x,n+3}\right].$

And using Maclaurin series, we have for x < 1:

$H_{x}=\sum _{n=1}^{\infty }(-1)^{n+1}x^{n}\zeta (n+1)$

$H_{x,2}=\sum _{n=1}^{\infty }(-1)^{n+1}(n+1)x^{n}\zeta (n+2)$

$H_{x,3}={\frac {1}{2}}\sum _{n=1}^{\infty }(-1)^{n+1}(n+1)(n+2)x^{n}\zeta (n+3).$

For fractional arguments between 0 and 1, and for a > 1:

$H_{\frac {1}{a}}={\frac {1}{a}}\left(\zeta (2)-{\frac {1}{a}}\zeta (3)+{\frac {1}{a^{2}}}\zeta (4)-{\frac {1}{a^{3}}}\zeta (5)+\cdots \right)$

$H_{{\frac {1}{a}},2}={\frac {1}{a}}\left(2\zeta (3)-{\frac {3}{a}}\zeta (4)+{\frac {4}{a^{2}}}\zeta (5)-{\frac {5}{a^{3}}}\zeta (6)+\cdots \right)$

$H_{{\frac {1}{a}},3}={\frac {1}{2a}}\left(2\cdot 3\zeta (4)-{\frac {3\cdot 4}{a}}\zeta (5)+{\frac {4\cdot 5}{a^{2}}}\zeta (6)-{\frac {5\cdot 6}{a^{3}}}\zeta (7)+\cdots \right).$

Hyperharmonic numbers

The next generalization was discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers.^[1]:258 Let

$H_{n}^{(0)}={\frac {1}{n}}.$

Then the nth hyperharmonic number of order r (r>0) is defined recursively as

$H_{n}^{(r)}=\sum _{k=1}^{n}H_{k}^{(r-1)}.$

In special, $H_{n}=H_{n}^{(1)}$.

See also

• Watterson estimator
• Tajima's D
• Coupon collector's problem
• Jeep problem
• Riemann zeta function

Notes

1. John H., Conway; Richard K., Guy (1995). The book of numbers. Copernicus.
2. Ronald L., Graham; Donald E., Knuth; Oren, Patashnik (1994). Concrete Mathematics. Addison-Wesley.
3. Sondow, Jonathan and Weisstein, Eric W. "Harmonic Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicNumber.html
4. Sandifer, C. Edward (2007), How Euler Did It, MAA Spectrum, Mathematical Association of America, p. 206, ISBN 9780883855638.
5. Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". Amer. Math. Monthly. 109: 534–543. arXiv:math.NT/0008177. doi:10.2307/2695443.

References

• Arthur T. Benjamin; Gregory O. Preston; Jennifer J. Quinn (2002). "A Stirling Encounter with Harmonic Numbers" (PDF). Mathematics Magazine. 75 (2): 95–103. doi:10.2307/3219141.
• Donald Knuth (1997). "Section 1.2.7: Harmonic Numbers". The Art of Computer Programming. Volume 1: Fundamental Algorithms (Third ed.). Addison-Wesley. pp. 75–79. ISBN 0-201-89683-4.
• Ed Sandifer, How Euler Did It — Estimating the Basel problem (2003)
• Paule, Peter; Schneider, Carsten (2003). "Computer Proofs of a New Family of Harmonic Number Identities" (PDF). Adv. Appl. Math. 31 (2): 359–378. doi:10.1016/s0196-8858(03)00016-2.
• Wenchang Chu (2004). "A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers" (PDF). The Electronic Journal of Combinatorics. 11: N15.
• Ayhan Dil; István Mező (2008). "A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers". Applied Mathematics and Computation. 206 (2): 942–951. doi:10.1016/j.amc.2008.10.013.
• Zoltán Retkes (2008). "An extension of the Hermite–Hadamard Inequality". Acta Sci. Math. (Szeged). 74: 95–106.

External links

• Weisstein, Eric W. "Harmonic Number". MathWorld.

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