Double Series
A double sum is a series having terms depending on two indices,
| [画像: sum_(i,j)b_(ij). ] |
(1)
|
A finite double series can be written as a product of series
An infinite double series can be written in terms of a single series
by reordering as follows,
Many examples exists of simple double series that cannot be computed analytically, such as the Erdős-Borwein constant
(OEIS A065442), where psi_q(z) is a q-polygamma function.
Another series is
(OEIS A091349), where H_n is a harmonic number and zeta_k=e^(2piik/3) is a cube root of unity.
A double series that can be done analytically is given by
where zeta(2) is the Riemann zeta function zeta(2) (B. Cloitre, pers. comm., Dec. 9, 2004).
The double series
can be evaluated by interchanging m and n and averaging,
![画像:1/2[sum_(m=1)^(infty)sum_(n=1)^(infty)(m^2n)/(3^m(n·3^m+m·3^n))+sum_(m=1)^(infty)sum_(n=1)^(infty)(n^2m)/(3^n(n·3^m+m·3^n))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/DoubleSeries/Inline48.svg){kind=link}
(Borwein et al. 2004, p. 54).
Identities involving double sums include the following:
where
is the floor function, and
Consider the series
over binary quadratic forms, where the prime indicates that summation occurs over all pairs of m and n but excludes the term (m,n)=(0,0). If S can be decomposed into a linear sum of products of Dirichlet L-series, it is said to be solvable. The related sums
can also be defined, which gives rise to such impressive formulas as
(Glasser and Zucker 1980). 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).
The lattice sum b_2(2s) can be separated into two pieces,
![画像:4[sum_(i,j=1)^(infty)((-1)^(i+j))/((i^2+j^2)^s)+sum_(i=1)^(infty)((-1)^i)/(i^(2s))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/DoubleSeries/Inline81.svg){kind=link}
![画像:4[sum_(i,j=1)^(infty)((-1)^(i+j))/((i^2+j^2)^s)-eta(2s)],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/DoubleSeries/Inline84.svg){kind=link}
where eta(n) is the Dirichlet eta function. Using the analytic form of the lattice sum
where beta(s) is the Dirichlet beta function gives the sum
Borwein and Borwein (1987, p. 291) show that for R[s]>1,
where zeta(s) is the Riemann zeta function, and for appropriate s,
(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 F denotes any function (Glasser 1974).
See also
Euler Sum, Lattice Sum, Madelung Constants, Multiple Series, Multivariate Zeta Function, Series, Triple Series, Weierstrass's Double Series TheoremExplore with Wolfram|Alpha
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 L-Series." J. Phys. A: Math. Gen. 9, 1207-1214, 1976a.Zucker, I. J. and Robertson, M. M. "A Systematic Approach to the Evaluation of sum_((m,n!=0,0))(am^2+bmn+cn^2)^(-s)." J. Phys. A: Math. Gen. 9, 1215-1225, 1976b.Referenced on Wolfram|Alpha
Double SeriesCite this as:
Weisstein, Eric W. "Double Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DoubleSeries.html