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I have been searching for identities involving generalized harmonic numbers \begin{equation*}H_n^{(p)}=\sum_{k=1}^{n}\frac{1}{k^p}\end{equation*} I found several identities in terms of $H_n^{(1)}$, but I am looking for some interesting identities for $H_n^{(2)}$. Does anyone know of any identities know of any nontrivial identities for $H_n^{(2)}$? I found some listed on Wikipedia, but this list is not comprehensive. Thanks for your help.

  1. integral identities
  2. summation identities
  3. recursive identities
  4. in terms of another function
Hatchet
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2 Answers2

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There is a nice list and a set of references at mathworld. Additionally, I discovered this one while writing a thesis on the Riemann Zeta function.

$$\sum_{n=1}^\infty \frac{H_n^{(s)}} {n^s}=\frac{\zeta(s)^2+\zeta(2s)}{2}.$$

John Molokach
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  • You wouldn't happen to have reference to a proof of that identity would you? – user3002473 Jul 07 '15 at 01:36
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    My Master's Thesis, pages 68-72. http://thescholarship.ecu.edu/bitstream/handle/10342/4703/Molokach_ecu_0600O_11345.pdf?sequence=1 – John Molokach Jul 07 '15 at 01:48
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    @user3002473: See the answer to this question. – Lucian Jul 07 '15 at 02:09
  • @John: Thanks for the material, I am looking forward to reading it. On a similar topic, do you know of series involving the digamma function? – Hatchet Jul 07 '15 at 09:30
  • Check the mathworld site for the digamma function. The last reference there is also a good resource. http://www.amazon.com/exec/obidos/ASIN/0387488065/ref=nosim/weisstein-20 – John Molokach Jul 07 '15 at 10:09
  • Today's post by John D. Cook is of note. http://www.johndcook.com/blog/2015/07/19/numerators-of-harmonic-numbers/?utm_source=feedburner&utm_medium=email&utm_campaign=Feed%3A+TheEndeavour+%28The+Endeavour%29 – John Molokach Jul 20 '15 at 11:59
  • @John Molokach I like your master's thesis but I'm afraid your nice formula is wrong. Consider s = 2. We have on the one hand from http://mathworld.wolfram.com/HarmonicNumber.html formula (18): $\sum _{n=1}^{\infty } (\frac{H_n}{n})^2=\frac{17 \pi ^4}{360}$, on the other hand your formula gives $\frac{1}{2} \left(\zeta (2)^2+\zeta (4)\right)=\frac{7 \pi ^4}{360}$ which is a different result. And, IMHO, I don't think that the sum for general real $s$ has a simple closed form. – Dr. Wolfgang Hintze Sep 25 '17 at 23:04
  • @John Molokach Don't worry, the formula is ok (at least for s=2), if you correct a typo: you wrote here $H_n^s$ whereas you meant $H_n^{(s)}$. Sometimes the small things matter. – Dr. Wolfgang Hintze Sep 25 '17 at 23:44
  • @Dr. Wolfgang Hintze, I didn't think to use the parentheses because of the notation in various literature (including MathWorld) doesn't use it. (see http://mathworld.wolfram.com/HarmonicNumber.html for example, where the superscript is used NOT as an exponent.) – John Molokach Oct 05 '17 at 00:43
  • @John Molokach In Mathworld notations are used for functions related to the harmonic number: $H_{n,r}$ for the generalized harmonic number, and $H_n^{(r)}$ for the harmonic number of the order $r$, introduced by Conway. We have been talking here about $H_{n,r}$. I tend to use this notation in the future. The only notation which should definitely not be used - and I hope you agree - is the simple power $H_n^r$ for anything else than the r-th power of $H_n$. – Dr. Wolfgang Hintze Oct 05 '17 at 10:31
  • @JohnMolokach Can you gentlemen help me answer the question I posed here?: https://math.stackexchange.com/questions/3216588/expressions-approximating-generalized-harmonic-number-truncated-polynomials-wit – user3108815 May 07 '19 at 00:49
  • @Dr.WolfgangHintze What about you Dr? – user3108815 May 07 '19 at 00:49
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I discovered a Generalized Harmonic Summation Identity some time ago.

$$ \sum_{r=1}^{n} \dfrac{H_{r} ^{(m)}}{r^m} = \dfrac{1}{2} \left( [H_{n}^{(m)}]^2 + H_{n}^{(2m)} \right) \quad ; \quad m \geq 1 $$

I have posted a proof here.

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MathGod
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