In response to a letter from Goldbach, Euler considered Double Sums of the form
(1) | |||
(2) |
(3) |
(4) |
(5) |
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) |
(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) |
(45) | |||
(46) | |||
(47) |
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.
© 1996-9 Eric W. Weisstein