Double Sum

A nested sum over two variables. Identities involving double sums include the following:

$$\sum_{p=0}^\infty \sum_{q=0}^p a_{q,p-q} = \sum_{m=0}^\infty... ... \sum_{r=0}^\infty \sum_{s=0}^{\lfloor r/2\rfloor} a_{s,r-2s},$$ (1)

where

$$\left\lfloor{r/2}\right\rfloor =\cases{ {\textstyle{1\over 2}}r & $r$\ even\cr {\textstyle{1\over 2}}(r-1) & $r$\ odd\cr}$$ (2)

is the Floor Function, and

$$\sum_{i=1}^n \sum_{j=1}^n x_ix_j = n^2\left\langle{x^2}\right\rangle{}.$$ (3)

Consider the sum

$$S(a,b,c;s)=\sum_{(m,n)\not=(0,0)} (am^2+bmn+cn^2)^{-s}$$ (4)

over binary Quadratic Forms. If $S$ can be decomposed into a linear sum of products of Dirichlet L-Series, it is said to be solvable. The related sums

$$S_1(a, b, c; s) = \sum_{(m,n)\not=(0,0)} (-1)^m(am^2+bmn+cn^2)^{-s}$$ (5)

$$S_2(a, b, c; s) = \sum_{(m,n)\not=(0,0)} (-1)^n(am^2+bmn+cn^2)^{-s}$$ (6)

$$S_{1,2}(a, b, c; s) = \sum_{(m,n)\not=(0,0)} (-1)^{m+n}(am^2+bmn+cn^2)^{-s}$$ (7)

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

$$S_1(1,0,58;1)=-{\pi\ln(27+5\sqrt{29}\,)\over\sqrt{58}}.$$ (8)

A complete table of the principal solutions of all solvable $S(a,b,c;s)$ is given in Glasser and Zucker (1980, pp. 126-131).

See also Euler Sum

References

Glasser, M. L. and Zucker, I. J. ``Lattice Sums in Theoretical Chemistry.'' Theoretical Chemistry: Advances and Perspectives, Vol. 5. New York: Academic Press, 1980.

Zucker, I. J. and Robertson, M. M. ``A Systematic Approach to the Evaluation of $\sum_{(m,n\not=0,0)} (am^2+bmn+cn^2)^{-s}$.'' J. Phys. A: Math. Gen. 9, 1215-1225, 1976.