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Euler Sum

In response to a letter from Goldbach, Euler considered Double Sums of the form

$\displaystyle s(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^m (k+1)^{-n}$ (1)
  $\textstyle =$ $\displaystyle \sum_{k=1}^\infty [\gamma+\psi_0(k+1)]^m (k+1)^{-n}$ (2)

with $m\geq 1$ and $n\geq 2$ and where $\gamma$ is the Euler-Mascheroni Constant and $\Psi(x)=\psi_0(x)$ is the Digamma Function. Euler found explicit formulas in terms of the Riemann Zeta Function for $s(1,n)$ with $n\geq 2$, and E. Au-Yeung numerically discovered
\begin{displaymath}
\sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^2 k^{-2} = {\textstyle{17\over 4}}\zeta(4),
\end{displaymath} (3)

where $\zeta(z)$ is the Riemann Zeta Function, which was subsequently rigorously proven true (Borwein and Borwein 1995). Sums involving $k^{-n}$ can be re-expressed in terms of sums the form $(k+1)^{-n}$ via
$\sum_{k=1}^\infty\left({1+{1\over 2^m}+\ldots+{1\over k^m}}\right)k^{-n}$
$\quad = \sum_{k=0}^\infty \left[{1+{2\over 2^m}+\ldots+{1\over(k+1)^m}}\right](k+1)^{-n}$
$\quad = \sum_{k=1}^\infty \left({1+{1\over 2^m}+\ldots+{1\over k^m}}\right)(k+1)^{-n}+\sum_{k=1}^\infty k^{-(m+n)}$
$\quad \equiv \sigma_h(m,n)+\zeta(m+n)$ (4)
and


\begin{displaymath}
\sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^2 k^{-n}=s_h(2,n)+2s_h(1,n+1)+\zeta(n+2),
\end{displaymath} (5)

where $\sigma_h$ is defined below.


Bailey et al. (1994) subsequently considered sums of the forms


$\displaystyle s_h(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^m (k+1)^{-n}$ (6)
$\displaystyle s_a(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left[{1-{1\over 2}+\ldots+{(-1)^{k+1}\over k}}\right]^m (k+1)^{-n}$ (7)
$\displaystyle a_h(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^m (-1)^{k+1}(k+1)^{-n}$ (8)
$\displaystyle a_a(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1-{1\over 2}+\ldots+{(-1)^{k+1}\over k}}\right)^m (-1)^{k+1}(k+1)^{-n}$  
      (9)
$\displaystyle \sigma_h(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1+{1\over 2^m}+\ldots+{1\over k^m}}\right)(k+1)^{-n}$ (10)
$\displaystyle \sigma_a(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1-{1\over 2^m}+\ldots+{(-1)^{k+1}\over k^m}}\right)(k+1)^{-n}$ (11)
$\displaystyle \alpha_h(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1+{1\over 2^m}+\ldots+{1\over k^m}}\right)(-1)^{k+1}(k+1)^{-n}$ (12)
      (13)
$\displaystyle \alpha_a(m,n)$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \left({1-{1\over 2^m}+\ldots+{(-1)^{k+1}\over k^m}}\right)(-1)^{k+1}(k+1)^{-n},$  
      (14)

where $s_h$ and $s_a$ have the special forms


$\displaystyle s_h$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty [\gamma+\psi_0(n+1)]^m {k+1}^{-n}$ (15)
$\displaystyle a_a$ $\textstyle =$ $\displaystyle \sum_{k=1}^\infty \{\ln 2+{\textstyle{1\over 2}}(-1)^n[\psi_0({\t...
...2}}n+{\textstyle{1\over 2}})-\psi_0({\textstyle{1\over 2}}n+1)]\}^m (k+1)^{-m}.$ (16)

Analytic single or double sums over $\zeta(z)$ can be constructed for

$s_h(1,n)={\textstyle{1\over 2}}n\zeta(n+1)-{\textstyle{1\over 2}}\sum_{k=1}^{n-2} \zeta(n-k)\zeta(k+1)$ (17)
$s_h(2,n)={\textstyle{1\over 3}}n(n+1)\zeta(n+2)+\zeta(2)\zeta(n)-{\textstyle{1\over 2}}n\sum_{k=0}^{n-2}\zeta(n-k)\zeta(k+2)$
$\phantom{=}+{\textstyle{1\over 3}}\sum_{k=2}^{n-2}\zeta(n-k)\sum_{j=1}^{k-1} \zeta(j+1)\zeta(k+1-j)+\sigma_h(2,n)$ (18)
$s_h(2,2n-1)={\textstyle{1\over 6}}(2n^2-7n-3)\zeta(2n+1)+\zeta(2)\zeta(2n-1)$
$\phantom{=}-{\textstyle{1\over 2}}\sum_{k=1}^{n-2} (2k-1)\zeta(2n-1-2k)\zeta(2k+2)$
$\phantom{=}+{\textstyle{1\over 3}} \sum_{k=1}^{n-2} \zeta(2k+1) \sum_{j=1}^{n-2-k} \zeta(2j+1)\zeta(2n-1-2k-2j)$ (19)
$\sigma_h(1,n)=s_h(1,n)$ (20)
$\sigma_h(2,2n-1)=-{\textstyle{1\over 2}}(2n^2+n+1)\zeta(2n+1)+\zeta(2)\zeta(2n-1)$
$\phantom{=}+\sum_{k=1}^{n-1} 2k\zeta(k+1)\zeta(2n-2k)$ (21)
$\sigma_h(m{\rm\ even},n{\rm\ odd})={\textstyle{1\over 2}}\left[{{m+n\choose m}-...
...a(m)\zeta(n)-\sum_{j=1}^{m+n}\left[{{2j-2\choose m-1}+{2j-2\choose n-1}}\right]$
$\phantom{==}\times\zeta(2j-1)\zeta(m+n-2j+1)$ (22)
$\sigma_h(m{\rm\ odd},n{\rm\ even})=-{\textstyle{1\over 2}}\left[{{m+n\choose m}...
...]\zeta(m+n)+\sum_{k=1}^{m+n} \left[{{2j-2\choose m-1}+{2j-2\choose n-1}}\right]$
$\phantom{==}\times\zeta(2j-1)\zeta(m+n-2j+1),$ (23)

where ${n\choose m}$ is a Binomial Coefficient. Explicit formulas inferred using the PSLQ Algorithm include


$\displaystyle s_h(2,2)$ $\textstyle =$ $\displaystyle {\textstyle{3\over 2}}\zeta(4)+{\textstyle{1\over 2}}[\zeta(2)]^2$ (24)
  $\textstyle =$ $\displaystyle {\textstyle{11\over 360}}\pi^4$ (25)
$\displaystyle s_h(2,4)$ $\textstyle =$ $\displaystyle {\textstyle{2\over 3}}\zeta(6)-{\textstyle{1\over 3}}\zeta(2)\zeta(4)+{\textstyle{1\over 3}}[\zeta(2)]^3-[\zeta(3)]^2$ (26)
  $\textstyle =$ $\displaystyle {\textstyle{37\over 22680}}\pi^6-[\zeta(3)]^2$ (27)
$\displaystyle s_h(3,2)$ $\textstyle =$ $\displaystyle {\textstyle{15\over 2}}\zeta(5)+\zeta(2)\zeta(3)$ (28)
$\displaystyle s_h(3,3)$ $\textstyle =$ $\displaystyle -{\textstyle{33\over 16}}\zeta(6)+2[\zeta(3)]^2$ (29)
$\displaystyle s_h(3,4)$ $\textstyle =$ $\displaystyle {\textstyle{119\over 16}}\zeta(7)-{\textstyle{33\over 4}}\zeta(3)\zeta(4)+2\zeta(2)\zeta(5)$ (30)
$\displaystyle s_h(3,6)$ $\textstyle =$ $\displaystyle {\textstyle{197\over 24}}\zeta(9)-{\textstyle{33\over 4}}\zeta(4)\zeta(5)-{\textstyle{37\over 8}}\zeta(3)\zeta(6)+[\zeta(3)]^3+3\zeta(2)\zeta(7)$ (31)
$\displaystyle s_h(4,2)$ $\textstyle =$ $\displaystyle {\textstyle{859\over 24}}\zeta(6)+3[\zeta(3)]^2$ (32)
$\displaystyle s_h(4,3)$ $\textstyle =$ $\displaystyle -{\textstyle{109\over 8}}\zeta(7)+{\textstyle{37\over 2}}\zeta(3)\zeta(4)-5\zeta(2)\zeta(5)$ (33)
$\displaystyle s_h(4,5)$ $\textstyle =$ $\displaystyle -{\textstyle{29\over 2}}\zeta(9)+{\textstyle{37\over 2}}\zeta(4)\...
...3\over 4}}\zeta(3)\zeta(6)-{\textstyle{8\over 3}}[\zeta(3)]^3-7\zeta(2)\zeta(7)$ (34)
$\displaystyle s_h(5,2)$ $\textstyle =$ $\displaystyle {\textstyle{1855\over 16}}\zeta(7)+33\zeta(3)\zeta(4)+{\textstyle{57\over 2}}\zeta(2)\zeta(5)$ (35)
$\displaystyle s_h(5,4)$ $\textstyle =$ $\displaystyle {\textstyle{890\over 9}}\zeta(9)+66\zeta(4)\zeta(5)-{\textstyle{4...
...ver 24}}\zeta(3)\zeta(6)-5[\zeta(3)]^3+{\textstyle{265\over 8}}\zeta(2)\zeta(7)$ (36)
$\displaystyle s_h(6,3)$ $\textstyle =$ $\displaystyle -{\textstyle{3073\over 12}}\zeta(9)-243\zeta(4)\zeta(5)+{\textsty...
...6)+{\textstyle{67\over 3}}[\zeta(3)]^3-{\textstyle{651\over 8}}\zeta(2)\zeta(7)$ (37)
$\displaystyle s_h(7,2)$ $\textstyle =$ $\displaystyle {\textstyle{134701\over 36}}\zeta(9)+{\textstyle{15697\over 8}}\z...
... 24}}\zeta(3)\zeta(6)+56[\zeta(3)]^3+{\textstyle{3287\over 4}}\zeta(2)\zeta(7),$ (38)


$\displaystyle s_a(2,2)$ $\textstyle =$ $\displaystyle 6\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})+{\textstyle{...
...ln 2)^4-{\textstyle{29\over 8}}\zeta(4)+{\textstyle{3\over 2}}\zeta(2)(\ln 2)^2$  
      (39)
$\displaystyle s_a(2,3)$ $\textstyle =$ $\displaystyle 4\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})-{\textstyle{...
...(\ln 2)^5-{\textstyle{17\over 32}}\zeta(5)-{\textstyle{11\over 8}}\zeta(4)\ln 2$  
  $\textstyle \phantom{=}$ $\displaystyle +{\textstyle{7\over 4}}\zeta(3)(\ln 2)^2+{\textstyle{1\over 3}}\zeta(2)(\ln 2)^3-{\textstyle{3\over 4}}\zeta(2)\zeta(3),$  
      (40)
$\displaystyle s_a(3,2)$ $\textstyle =$ $\displaystyle -24\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})+6\ln 2\mat...
...{1\over 2}})+{\textstyle{9\over 20}}(\ln 2)^5+{\textstyle{659\over 32}}\zeta(5)$  
  $\textstyle \phantom{=}$ $\displaystyle -{\textstyle{285\over 16}}\zeta(4)\ln 2+{\textstyle{5\over 2}}\zeta(2)(\ln 2)^3+{\textstyle{1\over 2}}\zeta(2)\zeta(3),$ (41)


$\displaystyle a_h(2,2)$ $\textstyle =$ $\displaystyle -2\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})-{\textstyle...
...(4)-{\textstyle{7\over 4}}\zeta(3)\ln 2+{\textstyle{1\over 2}}\zeta(2)(\ln 2)^2$ (42)
$\displaystyle a_h(2,3)$ $\textstyle =$ $\displaystyle -4\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})-4(\ln 2)\ma...
...{1\over 2}})-{\textstyle{2\over 15}}(\ln 2)^5+{\textstyle{107\over 32}}\zeta(5)$  
  $\textstyle \phantom{=}$ $\displaystyle -{\textstyle{7\over 4}}\zeta(3)(\ln 2)^2+{\textstyle{2\over 3}}\zeta(2)(\ln 2)^3+{\textstyle{3\over 8}}\zeta(2)\zeta(3)$  
      (43)
$\displaystyle a_h(3,2)$ $\textstyle =$ $\displaystyle 6\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})+6(\ln 2)\mat...
...yle{1\over 2}})+{\textstyle{1\over 5}}(\ln 2)^5-{\textstyle{33\over 8}}\zeta(5)$  
  $\textstyle \phantom{=}$ $\displaystyle +{\textstyle{21\over 8}}\zeta(3)(\ln 2)^2-\zeta(2)(\ln 2)^3-{\textstyle{15\over 16}}\zeta(2)\zeta(3),$ (44)

and


$\displaystyle a_a(2,2)$ $\textstyle =$ $\displaystyle -4\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})-{\textstyle...
...e{37\over 16}}\zeta(4)+{\textstyle{7\over 4}}\zeta(3)(\ln 2)-2\zeta(2)(\ln 2)^2$ (45)
$\displaystyle a_a(2,3)$ $\textstyle =$ $\displaystyle 4(\ln 2)\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})+{\tex...
...e{79\over 32}}\zeta(5)+{\textstyle{11\over 8}}\zeta(4)(\ln 2)-\zeta(2)(\ln 2)^3$  
  $\textstyle \phantom{=}$ $\displaystyle +{\textstyle{3\over 8}}\zeta(2)\zeta(3)$ (46)
$\displaystyle a_a(3,2)$ $\textstyle =$ $\displaystyle 30\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})-{\textstyle...
...textstyle{285\over 16}}\zeta(4)(\ln 2)+{\textstyle{21\over 8}}\zeta(3)(\ln 2)^2$  
  $\textstyle \phantom{=}$ $\displaystyle -{\textstyle{7\over 2}}\zeta(2)(\ln 2)^3+{\textstyle{3\over 4}}\zeta(2)\zeta(3),$ (47)

where $\mathop{\rm Li}\nolimits _n$ is a Polylogarithm, and $\zeta(z)$ is the Riemann Zeta Function (Bailey and Plouffe). Of these, only $s_h(3,2)$, $s_h(3,3)$ and the identities for $s_a(m,n)$, $a_h(m,n)$ and $a_a(m,n)$ have been rigorously established.


References

Bailey, D. and Plouffe, S. ``Recognizing Numerical Constants.'' http://www.cecm.sfu.ca/organics/papers/bailey/.

Bailey, D. H.; Borwein, J. M.; and Girgensohn, R. ``Experimental Evaluation of Euler Sums.'' Exper. Math. 3, 17-30, 1994.

Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, 1985.

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.

Borwein, D.; Borwein, J. M.; and Girgensohn, R. ``Explicit Evaluation of Euler Sums.'' Proc. Edinburgh Math. Soc. 38, 277-294, 1995.

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.



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© 1996-9 Eric W. Weisstein
1999-05-25