“Euler Sum”的概念、定义、习题解题方法及翻译-数学辞典

单词

Euler Sum

释义

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

isthe Digamma Function. Euler found explicit formulas in terms of the Riemann Zeta Functionfor

with

, and E. Au-Yeung numerically discovered

(3)

where

is the Riemann Zeta Function, which was subsequently rigorously proven true (Borwein andBorwein 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 rigorouslyestablished.

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.

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