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) |

(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) |

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.

© 1996-9

1999-05-25