Euler Sum

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

$$s(m,n) = \sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^m (k+1)^{-n}$$ (1)

$$= \sum_{k=1}^\infty [\gamma+\psi_0(k+1)]^m (k+1)^{-n}$$ (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

$$\sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^2 k^{-2} = {\textstyle{17\over 4}}\zeta(4),$$ (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

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

where $\sigma_h$ is defined below.

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

$$s_h(m,n) = \sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^m (k+1)^{-n}$$ (6)

$$s_a(m,n) = \sum_{k=1}^\infty \left[{1-{1\over 2}+\ldots+{(-1)^{k+1}\over k}}\right]^m (k+1)^{-n}$$ (7)

$$a_h(m,n) = \sum_{k=1}^\infty \left({1+{1\over 2}+\ldots+{1\over k}}\right)^m (-1)^{k+1}(k+1)^{-n}$$ (8)

$$a_a(m,n) = \sum_{k=1}^\infty \left({1-{1\over 2}+\ldots+{(-1)^{k+1}\over k}}\right)^m (-1)^{k+1}(k+1)^{-n}$$ (9)

$$\sigma_h(m,n) = \sum_{k=1}^\infty \left({1+{1\over 2^m}+\ldots+{1\over k^m}}\right)(k+1)^{-n}$$ (10)

$$\sigma_a(m,n) = \sum_{k=1}^\infty \left({1-{1\over 2^m}+\ldots+{(-1)^{k+1}\over k^m}}\right)(k+1)^{-n}$$ (11)

$$\alpha_h(m,n) = \sum_{k=1}^\infty \left({1+{1\over 2^m}+\ldots+{1\over k^m}}\right)(-1)^{k+1}(k+1)^{-n}$$ (12) (13)

$$\alpha_a(m,n) = \sum_{k=1}^\infty \left({1-{1\over 2^m}+\ldots+{(-1)^{k+1}\over k^m}}\right)(-1)^{k+1}(k+1)^{-n},$$ (14)

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

$$s_h = \sum_{k=1}^\infty [\gamma+\psi_0(n+1)]^m {k+1}^{-n}$$ (15)

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

$$s_h(2,2) = {\textstyle{3\over 2}}\zeta(4)+{\textstyle{1\over 2}}[\zeta(2)]^2$$ (24)

$$= {\textstyle{11\over 360}}\pi^4$$ (25)

$$s_h(2,4) = {\textstyle{2\over 3}}\zeta(6)-{\textstyle{1\over 3}}\zeta(2)\zeta(4)+{\textstyle{1\over 3}}[\zeta(2)]^3-[\zeta(3)]^2$$ (26)

$$= {\textstyle{37\over 22680}}\pi^6-[\zeta(3)]^2$$ (27)

$$s_h(3,2) = {\textstyle{15\over 2}}\zeta(5)+\zeta(2)\zeta(3)$$ (28)

$$s_h(3,3) = -{\textstyle{33\over 16}}\zeta(6)+2[\zeta(3)]^2$$ (29)

$$s_h(3,4) = {\textstyle{119\over 16}}\zeta(7)-{\textstyle{33\over 4}}\zeta(3)\zeta(4)+2\zeta(2)\zeta(5)$$ (30)

$$s_h(3,6) = {\textstyle{197\over 24}}\zeta(9)-{\textstyle{33\over 4}}\zeta(4)\zeta(5)-{\textstyle{37\over 8}}\zeta(3)\zeta(6)+[\zeta(3)]^3+3\zeta(2)\zeta(7)$$ (31)

$$s_h(4,2) = {\textstyle{859\over 24}}\zeta(6)+3[\zeta(3)]^2$$ (32)

$$s_h(4,3) = -{\textstyle{109\over 8}}\zeta(7)+{\textstyle{37\over 2}}\zeta(3)\zeta(4)-5\zeta(2)\zeta(5)$$ (33)

$$s_h(4,5) = -{\textstyle{29\over 2}}\zeta(9)+{\textstyle{37\over 2}}\zeta(4)\... ...3\over 4}}\zeta(3)\zeta(6)-{\textstyle{8\over 3}}[\zeta(3)]^3-7\zeta(2)\zeta(7)$$ (34)

$$s_h(5,2) = {\textstyle{1855\over 16}}\zeta(7)+33\zeta(3)\zeta(4)+{\textstyle{57\over 2}}\zeta(2)\zeta(5)$$ (35)

$$s_h(5,4) = {\textstyle{890\over 9}}\zeta(9)+66\zeta(4)\zeta(5)-{\textstyle{4... ...ver 24}}\zeta(3)\zeta(6)-5[\zeta(3)]^3+{\textstyle{265\over 8}}\zeta(2)\zeta(7)$$ (36)

$$s_h(6,3) = -{\textstyle{3073\over 12}}\zeta(9)-243\zeta(4)\zeta(5)+{\textsty... ...6)+{\textstyle{67\over 3}}[\zeta(3)]^3-{\textstyle{651\over 8}}\zeta(2)\zeta(7)$$ (37)

$$s_h(7,2) = {\textstyle{134701\over 36}}\zeta(9)+{\textstyle{15697\over 8}}\z... ... 24}}\zeta(3)\zeta(6)+56[\zeta(3)]^3+{\textstyle{3287\over 4}}\zeta(2)\zeta(7),$$ (38)

$$s_a(2,2) = 6\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})+{\textstyle{... ...ln 2)^4-{\textstyle{29\over 8}}\zeta(4)+{\textstyle{3\over 2}}\zeta(2)(\ln 2)^2$$ (39)

$$s_a(2,3) = 4\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})-{\textstyle{... ...(\ln 2)^5-{\textstyle{17\over 32}}\zeta(5)-{\textstyle{11\over 8}}\zeta(4)\ln 2$$

$$\phantom{=} +{\textstyle{7\over 4}}\zeta(3)(\ln 2)^2+{\textstyle{1\over 3}}\zeta(2)(\ln 2)^3-{\textstyle{3\over 4}}\zeta(2)\zeta(3),$$ (40)

$$s_a(3,2) = -24\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})+6\ln 2\mat... ...{1\over 2}})+{\textstyle{9\over 20}}(\ln 2)^5+{\textstyle{659\over 32}}\zeta(5)$$

$$\phantom{=} -{\textstyle{285\over 16}}\zeta(4)\ln 2+{\textstyle{5\over 2}}\zeta(2)(\ln 2)^3+{\textstyle{1\over 2}}\zeta(2)\zeta(3),$$ (41)

$$a_h(2,2) = -2\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})-{\textstyle... ...(4)-{\textstyle{7\over 4}}\zeta(3)\ln 2+{\textstyle{1\over 2}}\zeta(2)(\ln 2)^2$$ (42)

$$a_h(2,3) = -4\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})-4(\ln 2)\ma... ...{1\over 2}})-{\textstyle{2\over 15}}(\ln 2)^5+{\textstyle{107\over 32}}\zeta(5)$$

$$\phantom{=} -{\textstyle{7\over 4}}\zeta(3)(\ln 2)^2+{\textstyle{2\over 3}}\zeta(2)(\ln 2)^3+{\textstyle{3\over 8}}\zeta(2)\zeta(3)$$ (43)

$$a_h(3,2) = 6\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})+6(\ln 2)\mat... ...yle{1\over 2}})+{\textstyle{1\over 5}}(\ln 2)^5-{\textstyle{33\over 8}}\zeta(5)$$

$$\phantom{=} +{\textstyle{21\over 8}}\zeta(3)(\ln 2)^2-\zeta(2)(\ln 2)^3-{\textstyle{15\over 16}}\zeta(2)\zeta(3),$$ (44)

and

$$a_a(2,2) = -4\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})-{\textstyle... ...e{37\over 16}}\zeta(4)+{\textstyle{7\over 4}}\zeta(3)(\ln 2)-2\zeta(2)(\ln 2)^2$$ (45)

$$a_a(2,3) = 4(\ln 2)\mathop{\rm Li}\nolimits _4({\textstyle{1\over 2}})+{\tex... ...e{79\over 32}}\zeta(5)+{\textstyle{11\over 8}}\zeta(4)(\ln 2)-\zeta(2)(\ln 2)^3$$

$$\phantom{=} +{\textstyle{3\over 8}}\zeta(2)\zeta(3)$$ (46)

$$a_a(3,2) = 30\mathop{\rm Li}\nolimits _5({\textstyle{1\over 2}})-{\textstyle... ...textstyle{285\over 16}}\zeta(4)(\ln 2)+{\textstyle{21\over 8}}\zeta(3)(\ln 2)^2$$

$$\phantom{=} -{\textstyle{7\over 2}}\zeta(2)(\ln 2)^3+{\textstyle{3\over 4}}\zeta(2)\zeta(3),$$ (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.