A double sum is a series having terms depending on two indices,
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A finite double series can be written as a product of series
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An infinite double series can be written in terms of a single series
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by reordering as follows,
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Many examples exists of simple double series that cannot be computed analytically, such as the Erdős-Borwein constant
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(OEIS A065442), where is a q-polygamma function.
Another series is
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(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
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where is the Riemann zeta function zeta(2) (B. Cloitre, pers. comm., Dec. 9, 2004).
The double series
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can be evaluated by interchanging and and averaging,
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(Borwein et al. 2004, p. 54).
Identities involving double sums include the following:
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where
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is the floor function, and
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Consider the series
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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
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can also be defined, which gives rise to such impressive formulas as
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(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,
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where is the Dirichlet eta function. Using the analytic form of the lattice sum
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where is the Dirichlet beta function gives the sum
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Borwein and Borwein (1987, p. 291) show that for ,
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where is the Riemann zeta function, and for appropriate ,
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(Borwein and Borwein 1987, p. 305).
Another double series reduction is given by
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where denotes any function (Glasser 1974).