Double Series

A double sum is a series having terms depending on two indices,

(1)

A finite double series can be written as a product of series

			(2)
			(3)
			(4)
			(5)

An infinite double series can be written in terms of a single series

(6)

by reordering as follows,

			(7)
			(8)
			(9)
			(10)

Many examples exists of simple double series that cannot be computed analytically, such as the Erdős-Borwein constant

			(11)
			(12)
			(13)

(OEIS A065442), where is a q-polygamma function.

Another series is

			(14)
			(15)

(OEIS A091349), where is a harmonic number and is a cube root of unity.

A double series that can be done analytically is given by

(16)

where is the Riemann zeta function zeta(2) (B. Cloitre, pers. comm., Dec. 9, 2004).

The double series

(17)

can be evaluated by interchanging and and averaging,

			(18)
			(19)
			(20)
			(21)

(Borwein et al. 2004, p. 54).

Identities involving double sums include the following:

(22)

where

$|_r/2_|={1/2r r even; 1/2(r-1) r odd$

(23)

is the floor function, and

(24)

Consider the series

(25)

over binary quadratic forms, where the prime indicates that summation occurs over all pairs of and but excludes the term . If can be decomposed into a linear sum of products of Dirichlet L-series, it is said to be solvable. The related sums

			(26)
			(27)
			(28)

can also be defined, which gives rise to such impressive formulas as

(29)

(Glasser and Zucker 1980). A complete table of the principal solutions of all solvable is given in Glasser and Zucker (1980, pp. 126-131).

The lattice sum can be separated into two pieces,

			(30)
			(31)
			(32)
			(33)

where is the Dirichlet eta function. Using the analytic form of the lattice sum

			(34)
			(35)

where is the Dirichlet beta function gives the sum

			(36)
			(37)

Borwein and Borwein (1987, p. 291) show that for ,

			(38)
			(39)

where is the Riemann zeta function, and for appropriate ,

			(40)
			(41)
			(42)
			(43)
			(44)
			(45)

(Borwein and Borwein 1987, p. 305).

Another double series reduction is given by

sum_(m,n=-infty)^infty(F(|2m+2n+1|))/(cosh[(2n+1)u]cosh(2nu))=2sum_(n=0)^infty((2n+1)F(2n+1))/(sinh[(2n+1)u]),

(46)

where denotes any function (Glasser 1974).

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Glasser, M. L. "Reduction Formulas for Multiple Series." Math. Comput. 28, 265-266, 1974.Glasser, M. L. and Zucker, I. J. "Lattice Sums." In Perspectives in Theoretical Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring). New York: Academic Press, pp. 67-139, 1980.Hardy, G. H. "On the Convergence of Certain Multiple Series." Proc. London Math. Soc. 2, 24-28, 1904.Hardy, G. H. "On the Convergence of Certain Multiple Series." Proc. Cambridge Math. Soc. 19, 86-95, 1917.Jeffreys, H. and Jeffreys, B. S. "Double Series." §1.053 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 16-17, 1988.Meyer, B. "On the Convergence of Alternating Double Series." Amer. Math. Monthly 60, 402-404, 1953.Móricz, F. "Some Remarks on the Notion of Regular Convergence of Multiple Series." Acta Math. Hungar. 41, 161-168, 1983.Sloane, N. J. A. Sequences A065442 and A091349 in "The On-Line Encyclopedia of Integer Sequences."Wilansky, A. "On the Convergence of Double Series." Bull. Amer. Math. Soc. 53, 793-799, 1947.Zucker, I. J. and Robertson, M. M. "Some Properties of Dirichlet