## Euler Sum

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

 (1) (2)

with and and where is the Euler-Mascheroni Constant and is the Digamma Function. Euler found explicit formulas in terms of the Riemann Zeta Function for with , and E. Au-Yeung numerically discovered
 (3)

where is the Riemann Zeta Function, which was subsequently rigorously proven true (Borwein and Borwein 1995). Sums involving can be re-expressed in terms of sums the form via
 (4)
and

 (5)

where is defined below.

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

 (6) (7) (8) (9) (10) (11) (12) (13) (14)

where and have the special forms

 (15) (16)

Analytic single or double sums over can be constructed for

 (17) (18) (19) (20) (21) (22) (23)

where is a Binomial Coefficient. Explicit formulas inferred using the PSLQ Algorithm include

 (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38)

 (39) (40) (41)

 (42) (43) (44)

and

 (45) (46) (47)

where is a Polylogarithm, and is the Riemann Zeta Function (Bailey and Plouffe). Of these, only , and the identities for , and 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 .'' 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 and for Certain Values of and .'' J. Comp. Appl. Math. 37, 125-141, 1991.