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Euler Sum

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

s_h(m,n)=sum_(k=1)^(infty)(1+1/2+...+1/k)^m(k+1)^(-n)
(1)
=sum_(k=1)^(infty)[gamma+psi_0(k+1)]^m(k+1)^(-n)
(2)

with m>=1 and n>=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>=2, and E. Au-Yeung numerically discovered

sum_(k=1)^infty(1+1/2+...+1/k)^2k^(-2)=(17)/4zeta(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)(1+1/(2^m)+...+1/(k^m))k^(-n)=sum_(k=0)^(infty)[1+1/(2^m)+...+1/((k+1)^m)](k+1)^(-n)
(4)
=sum_(k=1)^(infty)(1+1/(2^m)+...+1/(k^m))(k+1)^(-n)+sum_(k=1)^(infty)k^(-(m+n))
(5)
=sigma_h(m,n)+zeta(m+n)
(6)

and

sum_(k=1)^infty(1+1/2+...+1/k)^2k^(-n) =s_h(2,n)+2s_h(1,n+1)+zeta(n+2),
(7)

where sigma_h is defined below.

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

s_h(m,n)=sum_(k=1)^(infty)(1+1/2+...+1/k)^m(k+1)^(-n)
(8)
s_a(m,n)=sum_(k=1)^(infty)[1-1/2+...+((-1)^(k+1))/k]^m(k+1)^(-n)
(9)
a_h(m,n)=sum_(k=1)^(infty)(1+1/2+...+1/k)^m(-1)^(k+1)(k+1)^(-n)
(10)
a_a(m,n)=sum_(k=1)^(infty)(1-1/2+...+((-1)^(k+1))/k)^m(-1)^(k+1)(k+1)^(-n)
(11)
sigma_h(m,n)=sum_(k=1)^(infty)(1+1/(2^m)+...+1/(k^m))(k+1)^(-n)
(12)
sigma_a(m,n)=sum_(k=1)^(infty)[1-1/(2^m)+...+((-1)^(k+1))/(k^m)](k+1)^(-n)
(13)
alpha_h(m,n)=sum_(k=1)^(infty)(1+1/(2^m)+...+1/(k^m))(-1)^(k+1)(k+1)^(-n)
(14)
alpha_a(m,n)=sum_(k=1)^(infty)(1-1/(2^m)+...+((-1)^(k+1))/(k^m))(-1)^(k+1)(k+1)^(-n),
(15)

where s_h and a_a have the special forms

s_h(m,n)=sum_(k=1)^(infty)[gamma+psi_0(n+1)]^m(k+1)^(-n)
(16)
a_a(m,n)=sum_(k=1)^(infty){ln2+1/2(-1)^n[psi_0(1/2n+1/2)-psi_0(1/2n+1)]}^m(k+1)^(-m)
(17)
alpha_h(m,n)=(1-2^(1-m-n))zeta(m+n)-sum_(k=1)^(infty)((-1)^(k+1)H_(k,m))/(k^n),
(18)

where H_(k,m) is a generalized harmonic number.

A number of these sums can be expressed in terms of the multivariate zeta function, e.g.,

s_a(2,3)=2zeta(3,-1,-1)+zeta(3,2)
(19)

(Bailey et al. 2006a, p. 39, sign corrected; Bailey et al. 2006b).

Special cases include

sigma_h(r,r)=1/2{[zeta(r)]^2-zeta(2r)}
(20)

(P. Simone, pers. comm., Aug. 30, 2004).

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

s_h(1,n)=1/2nzeta(n+1)-1/2sum_(k=1)^(n-2)zeta(n-k)zeta(k+1)
(21)
s_h(2,n)=1/3n(n+1)zeta(n+2)+zeta(2)zeta(n)-1/2nsum_(k=0)^(n-2)zeta(n-k)zeta(k+2)+1/3sum_(k=2)^(n-2)zeta(n-k)sum_(j=1)^(k-1)zeta(j+1)zeta(k+1-j)+sigma_h(2,n)
(22)
s_h(2,2n-1)=1/6(2n^2-7n-3)zeta(2n+1)+zeta(2)zeta(2n-1)-1/2sum_(k=1)^(n-2)(2k-1)zeta(2n-1-2k)zeta(2k+2)+1/3sum_(k=1)^(n-2)zeta(2k+1)sum_(j=1)^(n-2-k)zeta(2j+1)zeta(2n-1-2k-2j)
(23)
sigma_h(1,n)=s_h(1,n)
(24)
sigma_h(2,2n-1)=-1/2(2n^2+n+1)zeta(2n+1)+zeta(2)zeta(2n-1)+sum_(k=1)^(n-1)2kzeta(k+1)zeta(2n-2k)
(25)
sigma_h(m even,n odd)=1/2[(m+n; m)-1]zeta(m+n)+zeta(m)zeta(n)-sum_(j=1)^(m+n)[(2j-2; m-1)+(2j-2; n-1)]zeta(2j-1)zeta(m+n-2j+1)
(26)
sigma_h(m odd,n even)=-1/2[(m+n; m)+1]zeta(m+n)+sum_(k=1)^(m+n)[(2j-2; m-1)+(2j-2; n-1)]zeta(2j-1)zeta(m+n-2j+1),
(27)

where (n; m) is a binomial coefficient. Explicit formulas inferred using the PSLQ algorithm include

s_h(2,2)=3/2zeta(4)+1/2[zeta(2)]^2
(28)
=(11)/(360)pi^4
(29)
s_h(2,4)=2/3zeta(6)-1/3zeta(2)zeta(4)+1/3[zeta(2)]^3-[zeta(3)]^2
(30)
=(37)/(22680)pi^6-[zeta(3)]^2
(31)
s_h(3,2)=(15)/2zeta(5)+zeta(2)zeta(3)
(32)
s_h(3,3)=-(33)/(16)zeta(6)+2[zeta(3)]^2
(33)
s_h(3,4)=(119)/(16)zeta(7)-(33)/4zeta(3)zeta(4)+2zeta(2)zeta(5)
(34)
s_h(3,6)=(197)/(24)zeta(9)-(33)/4zeta(4)zeta(5)-(37)/8zeta(3)zeta(6)+[zeta(3)]^3+3zeta(2)zeta(7)
(35)
s_h(4,2)=(859)/(24)zeta(6)+3[zeta(3)]^2
(36)
s_h(4,3)=-(109)/8zeta(7)+(37)/2zeta(3)zeta(4)-5zeta(2)zeta(5)
(37)
s_h(4,5)=-(29)/2zeta(9)+(37)/2zeta(4)zeta(5)+(33)/4zeta(3)zeta(6)-8/3[zeta(3)]^3-7zeta(2)zeta(7)
(38)
s_h(5,2)=(1855)/(16)zeta(7)+33zeta(3)zeta(4)+(57)/2zeta(2)zeta(5)
(39)
s_h(5,4)=(890)/9zeta(9)+66zeta(4)zeta(5)-(4295)/(24)zeta(3)zeta(6)-5[zeta(3)]^3+(265)/8zeta(2)zeta(7)
(40)
s_h(6,3)=-(3073)/(12)zeta(9)-243zeta(4)zeta(5)+(2097)/4zeta(3)zeta(6)+(67)/3[zeta(3)]^3-(651)/8zeta(2)zeta(7)
(41)
s_h(7,2)=(134701)/(36)zeta(9)+(15697)/8zeta(4)zeta(5)+(29555)/(24)zeta(3)zeta(6)+56[zeta(3)]^3+(3287)/4zeta(2)zeta(7)
(42)

for s_h,

s_a(2,3)=4Li(1/2)-1/(30)ln^52-(17)/(32)zeta(5)-(11)/(720)pi^4ln2+7/4zeta(3)ln^22+1/(18)pi^2ln^32-1/8pi^2zeta(3)
(43)

for s_a, given as a challenge problem by Borwein and Bailey (2003, pp. 24-25) and discussed in Bailey et al. (2006a, p. 39; Bailey et al. 2006b),

a_h(2,2)=-2Li_4(1/2)-1/(12)(ln2)^4+(99)/(48)zeta(4)-7/4zeta(3)ln2+1/2zeta(2)(ln2)^2
(44)
a_h(2,3)=-4Li_5(1/2)-4(ln2)Li_4(1/2)-2/(15)(ln2)^5+(107)/(32)zeta(5)+(107)/(32)zeta(5)-7/4zeta(3)(ln2)^2+2/3zeta(2)(ln2)^3+3/8zeta(2)zeta(3)
(45)
a_h(3,2)=6Li_5(1/2)+6(ln2)Li_4(1/2)+1/5(ln2)^5-(33)/8zeta(5)+(21)/8zeta(3)(ln2)^2-zeta(2)(ln2)^3-(15)/(16)zeta(2)zeta(3)
(46)

for a_h, and

a_a(2,2)=-4Li_4(1/2)-1/6(ln2)^4+(37)/(16)zeta(4)+7/4zeta(3)(ln2)-2zeta(2)(ln2)^2
(47)
a_a(2,3)=4(ln2)Li_4(1/2)+1/6(ln2)^5-(79)/(32)zeta(5)+(11)/8zeta(4)(ln2)-zeta(2)(ln2)^3+3/8zeta(2)zeta(3)
(48)

for a_a, where Li_n is a polylogarithm, and zeta(z) is the Riemann zeta function (Bailey and Plouffe 1997, Bailey et al. 1994). Of these, only s_h(2,4) (P. Simone, pers. comm., Aug. 30, 2004), 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.

See also

Multiple Series, Multivariate Zeta Function

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References

Adamchik, V. "On Stirling Numbers and Euler Sums." J. Comput. Appl. Math. 79, 119-130, 1997. http://www-2.cs.cmu.edu/~adamchik/articles/stirling.htm.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2006. https://web.archive.org/web/20070202172917/http://crd.lbl.gov/~dhbailey/expmath/maa-course/hyper-ema.pdf.Bailey, D. H.; Borwein, J. M.; and Girgensohn, R. "Experimental Evaluation of Euler Sums." Exper. Math. 3, 17-30, 1994.Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." Organic Mathematics. Proceedings of the Workshop Held in Burnaby, BC, December 12-14, 1995 (Ed. J. Borwein, P. Borwein, L. Jörgenson, and R. Corless). Providence, RI: Amer. Math. Soc., pp. 73-88, 1997.Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, 1985.Borwein, D. and Borwein, J. M. "On Some Intriguing Sums Involving zeta(4)." Proc. Amer. Math. Soc. 123, 111-118, 1995.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.Borwein, J. and Bailey, D. "Recognition of Euler Sums." §2.5 in Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 56-58, 2003.Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Lisonek, P. "Special Values of Multidimensional Polylogarithms." Trans. Amer. Math. Soc. 353, 907-941, 2001.Boyadzhiev, K. N. "Evaluation of Euler-Zagier Sums." Int. J. Math. Math. Sci. 27, 407-412, 2001. http://www2.onu.edu/~kboyadzh/e-zagier.pdf.Boyadzhiev, K. N. "Consecutive Evaluation of Euler Sums." Int. J. Math. Math. Sci. 29, 555-561, 2002. http://www2.onu.edu/~kboyadzh/euler-c(1).pdf.Broadhurst, D. J. "On the Enumeration of Irreducible k-Fold Euler Sums and Their Roles in Knot Theory and Field Theory." April 22, 1996. https://arxiv.org/abs/hep-th/9604128.Broadhurst, D. J. "Massive 3-Loop Feynman Diagrams Reducible to SC^* Primitives of Algebras of the Sixth Root of Unity." March 11, 1998. https://arxiv.org/abs/hep-th/9803091.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.Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351-369, 1999.Flajolet, P. and Salvy, B. "Euler Sums and Contour Integral Representation." Experim. Math. 7, 15-35, 1998.

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Euler Sum

Cite this as:

Weisstein, Eric W. "Euler Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulerSum.html

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